Ideal gas law

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Ideal_gas_law 

 

Isotherms of an ideal gas for different temperatures. The curved lines are rectangular hyperbolae of the form y = a/x. They represent the relationship between pressure (on the vertical axis) and volume (on the horizontal axis) for an ideal gas at different temperatures: lines that are farther away from the origin (that is, lines that are nearer to the top right-hand corner of the diagram) correspond to higher temperatures.

The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Benoît Paul Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written in an empirical form:

$${\displaystyle pV=nRT}$$

 

where ${\displaystyle p}$, ${\displaystyle V}$ and ${\displaystyle T}$ are the pressure, volume and temperature respectively; ${\displaystyle n}$ is the amount of substance; and ${\displaystyle R}$ is the ideal gas constant. It can also be derived from the microscopic kinetic theory, as was achieved (apparently independently) by August Krönig in 1856 and Rudolf Clausius in 1857.

Equation

Molecular collisions within a closed container (a propane tank) are shown (right). The arrows represent the random motions and collisions of these molecules. The pressure and temperature of the gas are directly proportional: As temperature increases, the pressure of the propane gas increases by the same factor. A simple consequence of this proportionality is that on a hot summer day, the propane tank pressure will be elevated, and thus propane tanks must be rated to withstand such increases in pressure.

The state of an amount of gas is determined by its pressure, volume, and temperature. The modern form of the equation relates these simply in two main forms. The temperature used in the equation of state is an absolute temperature: the appropriate SI unit is the kelvin.

Common forms

The most frequently introduced forms are:

$${\displaystyle pV=nRT=n{k}_{\text{B}}{N}_{\text{A}}T=N{k}_{\text{B}}T}$$

where:

• ${\displaystyle p}$ is the absolute pressure of the gas,
• ${\displaystyle V}$ is the volume of the gas,
• ${\displaystyle n}$ is the amount of substance of gas (also known as number of moles),
• ${\displaystyle R}$ is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,
• ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant,
• ${\displaystyle N_A}$ is the Avogadro constant,
• ${\displaystyle T}$ is the absolute temperature of the gas,
• ${\displaystyle N}$ is the number of particles (usually atoms or molecules) of the gas.

In SI units, p is measured in pascals, V is measured in cubic metres, n is measured in moles, and T in kelvins (the Kelvin scale is a shifted Celsius scale, where 0.00 K = −273.15 °C, the lowest possible temperature). R has for value 8.314 J/(mol·K) = 1.989 ≈ 2 cal/(mol·K), or 0.0821 L⋅atm/(mol⋅K).

Molar form

How much gas is present could be specified by giving the mass instead of the chemical amount of gas. Therefore, an alternative form of the ideal gas law may be useful. The chemical amount, n (in moles), is equal to total mass of the gas (m) (in kilograms) divided by the molar mass, M (in kilograms per mole):

${\displaystyle n=\frac{m}{M}.}$

By replacing n with m/M and subsequently introducing density ρ = m/V, we get:

${\displaystyle pV=\frac{m}{M}RT}$

${\displaystyle p=\frac{m}{V}\frac{RT}{M}}$

${\displaystyle p=\rho \frac{R}{M}T}$

Defining the specific gas constant R_specific(r) as the ratio R/M,

${\displaystyle p=\rho R_{\text{specific}}T}$

This form of the ideal gas law is very useful because it links pressure, density, and temperature in a unique formula independent of the quantity of the considered gas. Alternatively, the law may be written in terms of the specific volume v, the reciprocal of density, as

${\displaystyle pv=R_{\text{specific}}T.}$

It is common, especially in engineering and meteorological applications, to represent the specific gas constant by the symbol R. In such cases, the universal gas constant is usually given a different symbol such as ${\displaystyle \overline{R}}$ or ${\displaystyle R^{\ast }}$ to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being used.

Statistical mechanics

In statistical mechanics the following molecular equation is derived from first principles

${\displaystyle P=n{k}_{\text{B}}T,}$

where P is the absolute pressure of the gas, n is the number density of the molecules (given by the ratio n = N/V, in contrast to the previous formulation in which n is the number of moles), T is the absolute temperature, and k_B is the Boltzmann constant relating temperature and energy, given by:

${\displaystyle k_{\text{B}}=\frac{R}{N_{\text{A}}}}$

where N_A is the Avogadro constant.

From this we notice that for a gas of mass m, with an average particle mass of μ times the atomic mass constant, m_u, (i.e., the mass is μ u) the number of molecules will be given by

${\displaystyle N=\frac{m}{\mu m_{\text{u}}},}$

and since ρ = m/V = nμm_u, we find that the ideal gas law can be rewritten as

${\displaystyle P=\frac{1}{V}\frac{m}{\mu m_{\text{u}}}{k}_{\text{B}}T=\frac{k_{\text{B}}}{\mu m_{\text{u}}}\rho T.}$

In SI units, P is measured in pascals, V in cubic metres, T in kelvins, and k_B = 1.38×10⁻²³ J⋅K⁻¹ in SI units.

Combined gas law

Combining the laws of Charles, Boyle and Gay-Lussac gives the combined gas law, which takes the same functional form as the ideal gas law says that the number of moles is unspecified, and the ratio of ${\displaystyle PV}$ to ${\displaystyle T}$ is simply taken as a constant:

${\displaystyle \frac{PV}{T}=k,}$

where ${\displaystyle P}$ is the pressure of the gas, ${\displaystyle V}$ is the volume of the gas, ${\displaystyle T}$ is the absolute temperature of the gas, and ${\displaystyle k}$ is a constant. When comparing the same substance under two different sets of conditions, the law can be written as

${\displaystyle \frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}.}$

Energy associated with a gas

According to the assumptions of the kinetic theory of ideal gases, one can consider that there are no intermolecular attractions between the molecules, or atoms, of an ideal gas. In other words, its potential energy is zero. Hence, all the energy possessed by the gas is the kinetic energy of the molecules, or atoms, of the gas.

${\displaystyle E=\frac{3}{2}nRT}$

This corresponds to the kinetic energy of n moles of a monoatomic gas having 3 degrees of freedom; x, y, z. The table here below gives this relationship for different amounts of a monoatomic gas.

Energy of a monoatomic gas	Mathematical expression
Energy associated with one mole	${\displaystyle E=\frac{3}{2}RT}$
Energy associated with one gram	${\displaystyle E=\frac{3}{2}rT}$
Energy associated with one atom	${\displaystyle E=\frac{3}{2}{k}_{B}T}$

Applications to thermodynamic processes

The table below essentially simplifies the ideal gas equation for a particular processes, thus making this equation easier to solve using numerical methods.

A thermodynamic process is defined as a system that moves from state 1 to state 2, where the state number is denoted by subscript. As shown in the first column of the table, basic thermodynamic processes are defined such that one of the gas properties (P, V, T, S, or H) is constant throughout the process.

For a given thermodynamics process, in order to specify the extent of a particular process, one of the properties ratios (which are listed under the column labeled "known ratio") must be specified (either directly or indirectly). Also, the property for which the ratio is known must be distinct from the property held constant in the previous column (otherwise the ratio would be unity, and not enough information would be available to simplify the gas law equation).

In the final three columns, the properties (p, V, or T) at state 2 can be calculated from the properties at state 1 using the equations listed.

Process	Constant	Known ratio or delta	p2	V2	T2
Isobaric process	Pressure	V2/V1	p2 = p1	V2 = V1(V2/V1)	T2 = T1(V2/V1)
		T2/T1	p2 = p1	V2 = V1(T2/T1)	T2 = T1(T2/T1)
Isochoric process (Isovolumetric process) (Isometric process)	Volume	p2/p1	p2 = p1(p2/p1)	V2 = V1	T2 = T1(p2/p1)
		T2/T1	p2 = p1(T2/T1)	V2 = V1	T2 = T1(T2/T1)
Isothermal process	Temperature	p2/p1	p2 = p1(p2/p1)	V2 = V1/(p2/p1)	T2 = T1
		V2/V1	p2 = p1/(V2/V1)	V2 = V1(V2/V1)	T2 = T1
Isentropic process (Reversible adiabatic process)	Entropy	p2/p1	p2 = p1(p2/p1)	V2 = V1(p2/p1)(−1/γ)	T2 = T1(p2/p1)(γ − 1)/γ
		V2/V1	p2 = p1(V2/V1)−γ	V2 = V1(V2/V1)	T2 = T1(V2/V1)(1 − γ)
		T2/T1	p2 = p1(T2/T1)γ/(γ − 1)	V2 = V1(T2/T1)1/(1 − γ)	T2 = T1(T2/T1)
Polytropic process	P Vn	p2/p1	p2 = p1(p2/p1)	V2 = V1(p2/p1)(−1/n)	T2 = T1(p2/p1)(n − 1)/n
		V2/V1	p2 = p1(V2/V1)−n	V2 = V1(V2/V1)	T2 = T1(V2/V1)(1 − n)
		T2/T1	p2 = p1(T2/T1)n/(n − 1)	V2 = V1(T2/T1)1/(1 − n)	T2 = T1(T2/T1)
Isenthalpic process (Irreversible adiabatic process)	Enthalpy	p2 − p1	p2 = p1 + (p2 − p1)		T2 = T1 + μJT(p2 − p1)
		T2 − T1	p2 = p1 + (T2 − T1)/μJT		T2 = T1 + (T2 − T1)

Deviations from ideal behavior of real gases

The equation of state given here (PV = nRT) applies only to an ideal gas, or as an approximation to a real gas that behaves sufficiently like an ideal gas. There are in fact many different forms of the equation of state. Since the ideal gas law neglects both molecular size and intermolecular attractions, it is most accurate for monatomic gases at high temperatures and low pressures. The neglect of molecular size becomes less important for lower densities, i.e. for larger volumes at lower pressures, because the average distance between adjacent molecules becomes much larger than the molecular size. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy, i.e., with increasing temperatures. More detailed equations of state, such as the van der Waals equation, account for deviations from ideality caused by molecular size and intermolecular forces.

Derivations

Empirical

The empirical laws that led to the derivation of the ideal gas law were discovered with experiments that changed only 2 state variables of the gas and kept every other one constant.

All the possible gas laws that could have been discovered with this kind of setup are:

• Boyle's law (Equation 1)
  $${\displaystyle PV=C_1\text{or}{P}_1{V}_1=P_2{V}_2}$$

  
• Charles's law (Equation 2)
  $${\displaystyle \frac{V}{T}=C_2\text{or}\frac{V_1}{T_1}=\frac{V_2}{T_2}}$$

  
• Avogadro's law (Equation 3)
  $${\displaystyle \frac{V}{N}=C_3\text{or}\frac{V_1}{N_1}=\frac{V_2}{N_2}}$$

  
• Gay-Lussac's law (Equation 4)
  $${\displaystyle \frac{P}{T}=C_4\text{or}\frac{P_1}{T_1}=\frac{P_2}{T_2}}$$

  
• Equation 5
  $${\displaystyle NT=C_5\text{or}{N}_1{T}_1=N_2{T}_2}$$

  
• Equation 6
  $${\displaystyle \frac{P}{N}=C_6\text{or}\frac{P_1}{N_1}=\frac{P_2}{N_2}}$$

  

Relationships between Boyle's, Charles's, Gay-Lussac's, Avogadro's, combined and ideal gas laws, with the Boltzmann constant k_B = R/N_A = n R/N  (in each law, properties circled are variable and properties not circled are held constant)

where P stands for pressure, V for volume, N for number of particles in the gas and T for temperature; where ${\displaystyle C_1,C_2,C_3,C_4,C_5,C_6}$ are constants in this context because of each equation requiring only the parameters explicitly noted in them changing.

To derive the ideal gas law one does not need to know all 6 formulas, one can just know 3 and with those derive the rest or just one more to be able to get the ideal gas law, which needs 4.

Since each formula only holds when only the state variables involved in said formula change while the others (which are a property of the gas but are not explicitly noted in said formula) remain constant, we cannot simply use algebra and directly combine them all. This is why: Boyle did his experiments while keeping N and T constant and this must be taken into account (in this same way, every experiment kept some parameter as constant and this must be taken into account for the derivation).

Keeping this in mind, to carry the derivation on correctly, one must imagine the gas being altered by one process at a time (as it was done in the experiments). The derivation using 4 formulas can look like this:

at first the gas has parameters ${\displaystyle P_1,V_1,N_1,T_1}$

Say, starting to change only pressure and volume, according to Boyle's law (Equation 1), then:

$${\displaystyle P_1{V}_1=P_2{V}_2}$$

(7)

After this process, the gas has parameters ${\displaystyle P_2,V_2,N_1,T_1}$

Using then equation (5) to change the number of particles in the gas and the temperature,

$${\displaystyle N_1{T}_1=N_2{T}_2}$$

(8)

After this process, the gas has parameters ${\displaystyle P_2,V_2,N_2,T_2}$

Using then equation (6) to change the pressure and the number of particles,

$${\displaystyle \frac{P_2}{N_2}=\frac{P_3}{N_3}}$$

(9)

After this process, the gas has parameters ${\displaystyle P_3,V_2,N_3,T_2}$

Using then Charles's law (equation 2) to change the volume and temperature of the gas,

$${\displaystyle \frac{V_2}{T_2}=\frac{V_3}{T_3}}$$

(10)

After this process, the gas has parameters ${\displaystyle P_3,V_3,N_3,T_3}$

Using simple algebra on equations (7), (8), (9) and (10) yields the result:

$${\displaystyle \frac{P_1{V}_1}{N_1{T}_1}=\frac{P_3{V}_3}{N_3{T}_3}}$$

or

$${\displaystyle \frac{PV}{NT}=k_{\text{B}},}$$

where ${\displaystyle k_{\text{B}}}$ stands for the Boltzmann constant.

Another equivalent result, using the fact that ${\displaystyle nR=N{k}_{\text{B}}}$, where n is the number of moles in the gas and R is the universal gas constant, is:

$${\displaystyle PV=nRT,}$$

which is known as the ideal gas law.

If three of the six equations are known, it may be possible to derive the remaining three using the same method. However, because each formula has two variables, this is possible only for certain groups of three. For example, if you were to have equations (1), (2) and (4) you would not be able to get any more because combining any two of them will only give you the third. However, if you had equations (1), (2) and (3) you would be able to get all six equations because combining (1) and (2) will yield (4), then (1) and (3) will yield (6), then (4) and (6) will yield (5), as well as would the combination of (2) and (3) as is explained in the following visual relation:

Relationship between the six gas laws

where the numbers represent the gas laws numbered above.

If you were to use the same method used above on 2 of the 3 laws on the vertices of one triangle that has a "O" inside it, you would get the third.

For example:

Change only pressure and volume first:

$${\displaystyle P_1{V}_1=P_2{V}_2}$$

(1')

then only volume and temperature:

$${\displaystyle \frac{V_2}{T_1}=\frac{V_3}{T_2}}$$

(2')

then as we can choose any value for ${\displaystyle V_3}$, if we set ${\displaystyle V_1=V_3}$, equation (2') becomes:

$${\displaystyle \frac{V_2}{T_1}=\frac{V_1}{T_2}}$$

(3')

combining equations (1') and (3') yields ${\displaystyle \frac{P_1}{T_1}=\frac{P_2}{T_2}}$, which is equation (4), of which we had no prior knowledge until this derivation.

Theoretical

Kinetic theory

Main article: Kinetic theory of gases

The ideal gas law can also be derived from first principles using the kinetic theory of gases, in which several simplifying assumptions are made, chief among which are that the molecules, or atoms, of the gas are point masses, possessing mass but no significant volume, and undergo only elastic collisions with each other and the sides of the container in which both linear momentum and kinetic energy are conserved.

First we show that the fundamental assumptions of the kinetic theory of gases imply that

${\displaystyle PV=\frac{1}{3}Nm{v}_{\text{rms}}^2.}$

Consider a container in the ${\displaystyle xyz}$ Cartesian coordinate system. For simplicity, we assume that a third of the molecules moves parallel to the ${\displaystyle x}$-axis, a third moves parallel to the ${\displaystyle y}$-axis and a third moves parallel to the ${\displaystyle z}$-axis. Next we temporarily assume that all molecules move with the same velocity ${\displaystyle v}$. We choose an area ${\displaystyle S}$ on a wall of the container, perpendicular to the ${\displaystyle x}$-axis. When time ${\displaystyle t}$ elapses, all molecules in the volume ${\displaystyle vtS}$ moving in the positive direction of the ${\displaystyle x}$-axis will hit the area. There are ${\displaystyle NvtS/V}$ molecules in a part of volume ${\displaystyle vtS}$ of the container, but only one sixth (i.e. a half of a third) of them moves in the positive direction of the ${\displaystyle x}$-axis. Therefore the number of molecules ${\displaystyle N^{\prime }}$ that will hit the area ${\displaystyle S}$ when the time ${\displaystyle t}$ elapses is ${\displaystyle NvtS/(6V)}$.

When a molecule bounces off the wall of the container, it changes its momentum ${\displaystyle {\mathbf{p}}_1}$ to ${\displaystyle {\mathbf{p}}_2=-{\mathbf{p}}_1}$. Hence the magnitude of change of the momentum of one molecule is ${\displaystyle |{\mathbf{p}}_2-{\mathbf{p}}_1|=2mv}$. The magnitude of the change of momentum of all molecules that bounce off the area ${\displaystyle S}$ when time ${\displaystyle t}$ elapses is then ${\displaystyle |\mathrm{\Delta }\mathbf{p}|=2mv{N}^{\prime }=NtSm{v}^2/(3V)}$. From ${\displaystyle F=|\mathrm{\Delta }\mathbf{p}|/t}$ and ${\displaystyle P=F/S}$ we get

${\displaystyle P=\frac{1}{3}Nm\frac{v^2}{V}.}$

We supposed that all molecules move with the same velocity ${\displaystyle v}$, but in fact they move with different velocities, so we replace ${\displaystyle v^2}$ in the equation by the arithmetic mean of all squares of all velocities of the molecules, i.e. by ${\displaystyle v_{\text{rms}}^2.}$ Therefore

${\displaystyle P=\frac{1}{3}Nm\frac{v_{\text{rms}}^2}{V}}$

which gives the desired formula.

Using the Maxwell–Boltzmann distribution, the fraction of molecules that have a speed in the range ${\displaystyle v}$ to ${\displaystyle v+dv}$ is ${\displaystyle f(v)dv}$, where

${\displaystyle f(v)=4\pi {\left(\frac{m}{2\pi kT}\right)}^{\frac{3}{2}}{v}^2{e}^{-\frac{m{v}^2}{2kT}}}$

and ${\displaystyle k}$ denotes the Boltzmann constant. The root-mean-square speed can be calculated by

${\displaystyle v_{\text{rms}}^2={\int }_0^{\mathrm{\infty }}{v}^{2}f(v)dv=4\pi {\left(\frac{m}{2\pi kT}\right)}^{\frac{3}{2}}{\int }_0^{\mathrm{\infty }}{v}^4{e}^{-\frac{m{v}^2}{2kT}}dv.}$

Using the integration formula

${\displaystyle {\int }_0^{\mathrm{\infty }}{x}^{2n}{e}^{-\frac{x^2}{a^2}}dx=\sqrt{\pi }\frac{(2n)!}{n!}{\left(\frac{a}{2}\right)}^{2n+1},}$

it follows that

${\displaystyle v_{\text{rms}}^2=4\pi {\left(\frac{m}{2\pi kT}\right)}^{\frac{3}{2}}\sqrt{\pi }\frac{4!}{2!}{\left(\frac{\sqrt{\frac{2kT}{m}}}{2}\right)}^5=\frac{3kT}{m},}$

from which we get the ideal gas law:

${\displaystyle PV=\frac{1}{3}Nm\left(\frac{3kT}{m}\right)=NkT.}$

Statistical mechanics

Main article: Statistical mechanics

Let q = (q_x, q_y, q_z) and p = (p_x, p_y, p_z) denote the position vector and momentum vector of a particle of an ideal gas, respectively. Let F denote the net force on that particle. Then the time-averaged kinetic energy of the particle is:

${\displaystyle \begin{array}{rl}⟨\mathbf{q}\cdot \mathbf{F}⟩& =⟨q_x\frac{d{p}_x}{dt}⟩+⟨q_y\frac{d{p}_y}{dt}⟩+⟨q_z\frac{d{p}_z}{dt}⟩\\ & =-⟨q_x\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_x}⟩-⟨q_y\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_y}⟩-⟨q_z\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_z}⟩=-3k_{\text{B}}T,\end{array}}$

where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition theorem. Summing over a system of N particles yields

${\displaystyle 3N{k}_{B}T=-⟨\sum _{k=1}^N{\mathbf{q}}_k\cdot {\mathbf{F}}_k⟩.}$

By Newton's third law and the ideal gas assumption, the net force of the system is the force applied by the walls of the container, and this force is given by the pressure P of the gas. Hence

${\displaystyle -⟨\sum _{k=1}^N{\mathbf{q}}_k\cdot {\mathbf{F}}_k⟩=P{\oint }_{\text{surface}}\mathbf{q}\cdot d\mathbf{S},}$

where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is

${\displaystyle \mathrm{\nabla }\cdot \mathbf{q}=\frac{\mathrm{\partial }{q}_x}{\mathrm{\partial }{q}_x}+\frac{\mathrm{\partial }{q}_y}{\mathrm{\partial }{q}_y}+\frac{\mathrm{\partial }{q}_z}{\mathrm{\partial }{q}_z}=3,}$

the divergence theorem implies that

${\displaystyle P{\oint }_{\text{surface}}\mathbf{q}\cdot d\mathbf{S}=P{\int }_{\text{volume}}\left(\mathrm{\nabla }\cdot \mathbf{q}\right)dV=3PV,}$

where dV is an infinitesimal volume within the container and V is the total volume of the container.

Putting these equalities together yields

${\displaystyle 3N{k}_{\text{B}}T=-⟨\sum _{k=1}^N{\mathbf{q}}_k\cdot {\mathbf{F}}_k⟩=3PV,}$

which immediately implies the ideal gas law for N particles:

${\displaystyle PV=N{k}_{B}T=nRT,}$

where n = N/N_A is the number of moles of gas and R = N_Ak_B is the gas constant.

Other dimensions

For a d-dimensional system, the ideal gas pressure is:

${\displaystyle P^{(d)}=\frac{N{k}_{B}T}{L^d},}$

where ${\displaystyle L^d}$ is the volume of the d-dimensional domain in which the gas exists. Note that the dimensions of the pressure changes with dimensionality.

Intermolecular force

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Intermolecular_force

An intermolecular force (IMF) (or secondary force) is the force that mediates interaction between molecules, including the electromagnetic forces of attraction or repulsion which act between atoms and other types of neighbouring particles, e.g. atoms or ions. Intermolecular forces are weak relative to intramolecular forces – the forces which hold a molecule together. For example, the covalent bond, involving sharing electron pairs between atoms, is much stronger than the forces present between neighboring molecules. Both sets of forces are essential parts of force fields frequently used in molecular mechanics.

The first reference to the nature of microscopic forces is found in Alexis Clairaut's work Théorie de la figure de la Terre, published in Paris in 1743. Other scientists who have contributed to the investigation of microscopic forces include: Laplace, Gauss, Maxwell and Boltzmann.

Attractive intermolecular forces are categorized into the following types:

• Hydrogen bonding
• Ion–dipole forces and ion–induced dipole forces
• Van der Waals forces – Keesom force, Debye force, and London dispersion force

Information on intermolecular forces is obtained by macroscopic measurements of properties like viscosity, pressure, volume, temperature (PVT) data. The link to microscopic aspects is given by virial coefficients and Lennard-Jones potentials.

Hydrogen bonding

Main article: Hydrogen bond

A hydrogen bond is an extreme form of dipole-dipole bonding, referring to the attraction between a hydrogen atom that is bonded to an element with high electronegativity, usually nitrogen, oxygen, or fluorine. The hydrogen bond is often described as a strong electrostatic dipole–dipole interaction. However, it also has some features of covalent bonding: it is directional, stronger than a van der Waals force interaction, produces interatomic distances shorter than the sum of their van der Waals radii, and usually involves a limited number of interaction partners, which can be interpreted as a kind of valence. The number of Hydrogen bonds formed between molecules is equal to the number of active pairs. The molecule which donates its hydrogen is termed the donor molecule, while the molecule containing lone pair participating in H bonding is termed the acceptor molecule. The number of active pairs is equal to the common number between number of hydrogens the donor has and the number of lone pairs the acceptor has.

Hydrogen bonding in water

Though both not depicted in the diagram, water molecules have four active bonds. The oxygen atom’s two lone pairs interact with a hydrogen each, forming two additional hydrogen bonds, and the second hydrogen atom also interacts with a neighbouring oxygen. Intermolecular hydrogen bonding is responsible for the high boiling point of water (100 °C) compared to the other group 16 hydrides, which have little capability to hydrogen bond. Intramolecular hydrogen bonding is partly responsible for the secondary, tertiary, and quaternary structures of proteins and nucleic acids. It also plays an important role in the structure of polymers, both synthetic and natural.

Beta bonding

Main article: Ionic bonding

The attraction between cationic and anionic sites is a noncovalent, or intermolecular interaction which is usually referred to as ion pairing or salt bridge. It is essentially due to electrostatic forces, although in aqueous medium the association is driven by entropy and often even endothermic. Most salts form crystals with characteristic distances between the ions; in contrast to many other noncovalent interactions, salt bridges are not directional and show in the solid state usually contact determined only by the van der Waals radii of the ions. Inorganic as well as organic ions display in water at moderate ionic strength I similar salt bridge as association ΔG values around 5 to 6 kJ/mol for a 1:1 combination of anion and cation, almost independent of the nature (size, polarizability, etc.) of the ions. The ΔG values are additive and approximately a linear function of the charges, the interaction of e.g. a doubly charged phosphate anion with a single charged ammonium cation accounts for about 2x5 = 10 kJ/mol. The ΔG values depend on the ionic strength I of the solution, as described by the Debye-Hückel equation, at zero ionic strength one observes ΔG = 8 kJ/mol.

Dipole–dipole and similar interactions

Dipole–dipole interactions (or Keesom interactions) are electrostatic interactions between molecules which have permanent dipoles. This interaction is stronger than the London forces but is weaker than ion-ion interaction because only partial charges are involved. These interactions tend to align the molecules to increase attraction (reducing potential energy). An example of a dipole–dipole interaction can be seen in hydrogen chloride (HCl): the positive end of a polar molecule will attract the negative end of the other molecule and influence its position. Polar molecules have a net attraction between them. Examples of polar molecules include hydrogen chloride (HCl) and chloroform (CHCl₃).

${\displaystyle \stackrel{\delta +}{\text{H}}-\stackrel{\delta -}{\text{Cl}}\cdots \stackrel{\delta +}{\text{H}}-\stackrel{\delta -}{\text{Cl}}}$

Often molecules contain dipolar groups of atoms, but have no overall dipole moment on the molecule as a whole. This occurs if there is symmetry within the molecule that causes the dipoles to cancel each other out. This occurs in molecules such as tetrachloromethane and carbon dioxide. The dipole–dipole interaction between two individual atoms is usually zero, since atoms rarely carry a permanent dipole.

The Keesom interaction is a van der Waals force. It is discussed further in the section "Van der Waals forces".

Ion–dipole and ion–induced dipole forces

Ion–dipole and ion–induced dipole forces are similar to dipole–dipole and dipole–induced dipole interactions but involve ions, instead of only polar and non-polar molecules. Ion–dipole and ion–induced dipole forces are stronger than dipole–dipole interactions because the charge of any ion is much greater than the charge of a dipole moment. Ion–dipole bonding is stronger than hydrogen bonding.

An ion–dipole force consists of an ion and a polar molecule interacting. They align so that the positive and negative groups are next to one another, allowing maximum attraction. An important example of this interaction is hydration of ions in water which give rise to hydration enthalpy. The polar water molecules surround themselves around ions in water and the energy released during the process is known as hydration enthalpy. The interaction has its immense importance in justifying the stability of various ions (like Cu²⁺) in water.

An ion–induced dipole force consists of an ion and a non-polar molecule interacting. Like a dipole–induced dipole force, the charge of the ion causes distortion of the electron cloud on the non-polar molecule.

Van der Waals forces

Main article: van der Waals force

The van der Waals forces arise from interaction between uncharged atoms or molecules, leading not only to such phenomena as the cohesion of condensed phases and physical absorption of gases, but also to a universal force of attraction between macroscopic bodies.

Keesom force (permanent dipole – permanent dipole)

The first contribution to van der Waals forces is due to electrostatic interactions between rotating permanent dipoles, quadrupoles (all molecules with symmetry lower than cubic), and multipoles. It is termed the Keesom interaction, named after Willem Hendrik Keesom. These forces originate from the attraction between permanent dipoles (dipolar molecules) and are temperature dependent.

They consist of attractive interactions between dipoles that are ensemble averaged over different rotational orientations of the dipoles. It is assumed that the molecules are constantly rotating and never get locked into place. This is a good assumption, but at some point molecules do get locked into place. The energy of a Keesom interaction depends on the inverse sixth power of the distance, unlike the interaction energy of two spatially fixed dipoles, which depends on the inverse third power of the distance. The Keesom interaction can only occur among molecules that possess permanent dipole moments, i.e., two polar molecules. Also Keesom interactions are very weak van der Waals interactions and do not occur in aqueous solutions that contain electrolytes. The angle averaged interaction is given by the following equation:

${\displaystyle \frac{-d_1^2{d}_2^2}{24{\pi }^2{\epsilon }_0^2{\epsilon }_r^2{k}_{\text{B}}T{r}^6}=V,}$

where d = electric dipole moment, ${\displaystyle {\epsilon }_0}$ = permitivity of free space, ${\displaystyle {\epsilon }_r}$ = dielectric constant of surrounding material, T = temperature, ${\displaystyle k_{\text{B}}}$ = Boltzmann constant, and r = distance between molecules.

Debye force (permanent dipoles–induced dipoles)

The second contribution is the induction (also termed polarization) or Debye force, arising from interactions between rotating permanent dipoles and from the polarizability of atoms and molecules (induced dipoles). These induced dipoles occur when one molecule with a permanent dipole repels another molecule's electrons. A molecule with permanent dipole can induce a dipole in a similar neighboring molecule and cause mutual attraction. Debye forces cannot occur between atoms. The forces between induced and permanent dipoles are not as temperature dependent as Keesom interactions because the induced dipole is free to shift and rotate around the polar molecule. The Debye induction effects and Keesom orientation effects are termed polar interactions.

The induced dipole forces appear from the induction (also termed polarization), which is the attractive interaction between a permanent multipole on one molecule with an induced (by the former di/multi-pole) 31 on another. This interaction is called the Debye force, named after Peter J. W. Debye.

One example of an induction interaction between permanent dipole and induced dipole is the interaction between HCl and Ar. In this system, Ar experiences a dipole as its electrons are attracted (to the H side of HCl) or repelled (from the Cl side) by HCl. The angle averaged interaction is given by the following equation:

${\displaystyle \frac{-d_1^2{\alpha }_2}{16{\pi }^2{\epsilon }_0^2{\epsilon }_r^2{r}^6}=V,}$

where ${\displaystyle {\alpha }_2}$ = polarizability.

This kind of interaction can be expected between any polar molecule and non-polar/symmetrical molecule. The induction-interaction force is far weaker than dipole–dipole interaction, but stronger than the London dispersion force.

London dispersion force (fluctuating dipole–induced dipole interaction)

Main article: London dispersion force

The third and dominant contribution is the dispersion or London force (fluctuating dipole–induced dipole), which arises due to the non-zero instantaneous dipole moments of all atoms and molecules. Such polarization can be induced either by a polar molecule or by the repulsion of negatively charged electron clouds in non-polar molecules. Thus, London interactions are caused by random fluctuations of electron density in an electron cloud. An atom with a large number of electrons will have a greater associated London force than an atom with fewer electrons. The dispersion (London) force is the most important component because all materials are polarizable, whereas Keesom and Debye forces require permanent dipoles. The London interaction is universal and is present in atom-atom interactions as well. For various reasons, London interactions (dispersion) have been considered relevant for interactions between macroscopic bodies in condensed systems. Hamaker developed the theory of van der Waals between macroscopic bodies in 1937 and showed that the additivity of these interactions renders them considerably more long-range.

Relative strength of forces

Bond type	Dissociation energy (kcal/mol)	Dissociation energy (kJ/mol)	Note
Ionic lattice	250–4000	1100–20000
Covalent bond	30–260	130–1100
Hydrogen bond	1–12	4–50	About 5 kcal/mol (21 kJ/mol) in water
Dipole–dipole	0.5–2	2–8
London dispersion forces	<1 to 15	<4 to 63	Estimated from the enthalpies of vaporization of hydrocarbons

This comparison is approximate. The actual relative strengths will vary depending on the molecules involved. For instance, the presence of water creates competing interactions that greatly weaken the strength of both ionic and hydrogen bonds. We may consider that for static systems, Ionic bonding and covalent bonding will always be stronger than intermolecular forces in any given substance. But it is not so for big moving systems like enzyme molecules interacting with substrate molecules. Here the numerous intramolecular (most often - hydrogen bonds) bonds form an active intermediate state where the intermolecular bonds cause some of the covalent bond to be broken, while the others are formed, in this way procceding the thousands of enzymatic reactions, so important for living organisms.

Effect on the behavior of gases

Intermolecular forces are repulsive at short distances and attractive at long distances (see the Lennard-Jones potential). In a gas, the repulsive force chiefly has the effect of keeping two molecules from occupying the same volume. This gives a real gas a tendency to occupy a larger volume than an ideal gas at the same temperature and pressure. The attractive force draws molecules closer together and gives a real gas a tendency to occupy a smaller volume than an ideal gas. Which interaction is more important depends on temperature and pressure (see compressibility factor).

In a gas, the distances between molecules are generally large, so intermolecular forces have only a small effect. The attractive force is not overcome by the repulsive force, but by the thermal energy of the molecules. Temperature is the measure of thermal energy, so increasing temperature reduces the influence of the attractive force. In contrast, the influence of the repulsive force is essentially unaffected by temperature.

When a gas is compressed to increase its density, the influence of the attractive force increases. If the gas is made sufficiently dense, the attractions can become large enough to overcome the tendency of thermal motion to cause the molecules to disperse. Then the gas can condense to form a solid or liquid, i.e., a condensed phase. Lower temperature favors the formation of a condensed phase. In a condensed phase, there is very nearly a balance between the attractive and repulsive forces.

Quantum mechanical theories

Main article: Covalent bond § Quantum mechanical description

Intermolecular forces observed between atoms and molecules can be described phenomenologically as occurring between permanent and instantaneous dipoles, as outlined above. Alternatively, one may seek a fundamental, unifying theory that is able to explain the various types of interactions such as hydrogen bonding, van der Waals force and dipole–dipole interactions. Typically, this is done by applying the ideas of quantum mechanics to molecules, and Rayleigh–Schrödinger perturbation theory has been especially effective in this regard. When applied to existing quantum chemistry methods, such a quantum mechanical explanation of intermolecular interactions provides an array of approximate methods that can be used to analyze intermolecular interactions. One of the most helpful methods to visualize this kind of intermolecular interactions, that we can find in quantum chemistry, is the non-covalent interaction index, which is based on the electron density of the system. London dispersion forces play a big role with this.

Concerning electron density topology, recent methods based on electron density gradient methods have emerged recently, notably with the development of IBSI (Intrinsic Bond Strength Index), relying on the IGM (Independent Gradient Model) methodology.

Azeotrope

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Azeotrope 

Vapour–liquid equilibrium of 2-propanol/water showing positive azeotropic behaviour.

An azeotrope (/əˈziːəˌtroʊp/) or a constant heating point mixture is a mixture of two or more components in fluidic states whose proportions cannot be altered or changed by simple distillation. This happens because when an azeotrope is boiled, the vapour has the same proportions of constituents as the unboiled mixture. Azeotropic mixture behavior is important for fluid separation processes.

Some azeotropic mixtures of pairs of compounds are known, and many azeotropes of three or more compounds are also known. There are two types of azeotropes: minimum boiling azeotrope and maximum boiling azeotrope. For technical applications, the pressure-temperature-composition behavior of a mixture is the most important. Yet, also other important thermophysical properties are strongly influenced by azeotropy, e.g. the surface tension and transport properties.

Etymology

The term azeotrope is derived from the Greek words ζέειν (boil) and τρόπος (turning) with the prefix α- (no) to give the overall meaning, "no change on boiling". The term was coined in 1911 by English chemist John Wade and Richard William Merriman. Because their composition is unchanged by distillation, azeotropes are also called (especially in older texts) constant boiling point mixtures.

Types

Positive and negative azeotropes

Positive azeotrope – mixture of chloroform and methanol

 

Negative azeotrope – mixture of formic acid and water

Each azeotrope has a characteristic boiling point. The boiling point of an azeotrope is either less than the boiling point temperatures of any of its constituents (a positive azeotrope), or greater than the boiling point of any of its constituents (a negative azeotrope). For both positive and negative azeotropes, it is not possible to separate the components by fractional distillation and azeotropic distillation is usually used instead.

A solution that shows greater positive deviation from Raoult's law forms a minimum boiling azeotrope at a specific composition. For example, an ethanol–water mixture (obtained by fermentation of sugars) on fractional distillation yields a solution containing at most 95% (by volume) of ethanol. Once this composition has been achieved, the liquid and vapour have the same composition, and no further separation occurs. A solution that shows large negative deviation from Raoult's law forms a maximum boiling azeotrope at a specific composition. Nitric acid and water is an example of this class of azeotrope. This azeotrope has an approximate composition of 68% nitric acid and 32% water by mass, with a boiling point of 393.5 K (120.4 °C).

Construction of the p-v-x diagram which shows azeotropic behavior is shown in the accompanying diagram 

Construction of the p-v-x diagram appropriate for an azeotrope

A well-known example of a positive azeotrope is 95.63% ethanol and 4.37% water (by mass), which boils at 78.2 °C. Ethanol boils at 78.4 °C, water boils at 100 °C, but the azeotrope boils at 78.2 °C, which is lower than either of its constituents. Indeed, 78.2 °C is the minimum temperature at which any ethanol/water solution can boil at atmospheric pressure. In general, a positive azeotrope boils at a lower temperature than any other ratio of its constituents. Positive azeotropes are also called minimum boiling mixtures or pressure maximum azeotropes.

In general, a negative azeotrope boils at a higher temperature than any other ratio of its constituents. Negative azeotropes are also called maximum boiling mixtures or pressure minimum azeotropes. An example of a negative azeotrope is hydrochloric acid at a concentration of 20.2% and 79.8% water (by mass). Hydrogen chloride boils at −84 °C and water at 100 °C, but the azeotrope boils at 110 °C, which is higher than either of its constituents. The maximum temperature at which any hydrochloric acid solution can boil is 110 °C. Other examples:

• hydrofluoric acid (35.6%) / water, boils at 111.35 °C
• nitric acid (68%) / water, boils at 120.2 °C at 1 atm
• perchloric acid (71.6%) / water, boils at 203 °C
• sulfuric acid (98.3%) / water, boils at 338 °C

Azeotropes consisting of two constituents are called binary azeotropes such as diethyl ether (33%) / halothane (66%) a mixture once commonly used in anesthesia. Azeotropes consisting of three constituents are called ternary azeotropes, e.g. acetone / methanol / chloroform. Azeotropes of more than three constituents are also known.

Heterogeneous azeotropes

Phase diagram of a heteroazeotrope. Vertical axis is temperature, horizontal axis is composition. The dotted vertical line indicates the composition of the combined layers of the distillate whenever both layers are present in the original mixture.

If the components of a mixture are not completely miscible, an azeotrope can be found inside the miscibility gap. This type of azeotrope is called heterogeneous azeotrope or heteroazeotrope. A heteroazeotropic distillation will have two liquid phases. Heterogeneous azeotropes are only known in combination with temperature-minimum azeotropic behavior.

If the constituents of a mixture are completely miscible in all proportions with each other, the type of azeotrope is called a homogeneous azeotrope. Homogeneous azeotropes can be of the low-boiling or high-boiling azeotropic type (see above). For example, any amount of ethanol can be mixed with any amount of water to form a homogeneous solution.

For example, if equal volumes of chloroform (water solubility 0.8 g/100 ml at 20 °C) and water are shaken together and then left to stand, the liquid will separate into two layers. Analysis of the layers shows that the top layer is mostly water with a small amount of chloroform dissolved in it, and the bottom layer is mostly chloroform with a small amount of water dissolved in it. If the two layers are heated together, the system of layers will boil at 53.3 °C, which is lower than either the boiling point of chloroform (61.2 °C) or the boiling point of water (100 °C). The vapor will consist of 97.0% chloroform and 3.0% water regardless of how much of each liquid layer is present provided both layers are indeed present. If the vapor is re-condensed, the layers will reform in the condensate, and will do so in a fixed ratio, which in this case is 4.4% of the volume in the top layer and 95.6% in the bottom layer.

The diagram illustrates how the various phases of a heteroazeotrope are related.

Double azeotrope of benzene and hexafluorobenzene. Proportions are by weight.

Double azeotrope

Also more complex azeotropes exist, which comprise both a minimum-boiling and a maximum-boiling point (see figure). Such a system is called a double azeotrope, and will have two azeotropic compositions and boiling points. An example is water and N-methylethylenediamine as well as benzene and hexafluorobenzene.

Zeotropy

Combinations of solvents that do not form an azeotrope when mixed in any proportion are said to be zeotropic. Azeotropes are useful in separating zeotropic mixtures. An example is acetic acid and water, which do not form an azeotrope. Despite this, it is very difficult to separate pure acetic acid (boiling point: 118.1 °C) from a solution of acetic acid and water by distillation alone. As progressive distillations produce solutions with less and less water, each further distillation becomes less effective at removing the remaining water. Distilling the solution to dry acetic acid is therefore economically impractical. But ethyl acetate forms an azeotrope with water that boils at 70.4 °C. By adding ethyl acetate as an entrainer, it is possible to distill away the azeotrope and leave nearly pure acetic acid as the residue.

Mechanism

Condition of existence

The condition relates activity coefficients in liquid phase to total pressure and the vapour pressures of pure components.

Total vapor pressure of mixtures as a function of composition at a chosen constant temperature

Azeotropes can form only when a mixture deviates from Raoult's law, the equality of compositions in liquid phase and vapor phases, in vapour-liquid equilibrium and Dalton's law the equality of pressures for total pressure being equal to the sum of the partial pressures in real mixtures.

In other words: Raoult's law predicts the vapor pressures of ideal mixtures as a function of composition ratio. More simply: per Raoult's law molecules of the constituents stick to each other to the same degree as they do to themselves. For example, if the constituents are X and Y, then X sticks to Y with roughly equal energy as X does with X and Y does with Y. A so-called positive deviation from Raoult's law results when the constituents have a disaffinity for each other – that is X sticks to X and Y to Y better than X sticks to Y. Because this results in the mixture having less total affinity of the molecules than the pure constituents, they more readily escape from the stuck-together phase, which is to say the liquid phase, and into the vapor phase. When X sticks to Y more aggressively than X does to X and Y does to Y, the result is a negative deviation from Raoult's law. In this case because the molecules in the mixture are sticking together more than in the pure constituents, they are more reluctant to escape the stuck-together liquid phase.

When the deviation is great enough to cause a maximum or minimum in the vapor pressure versus composition function, it is a mathematical consequence that at that point, the vapor will have the same composition as the liquid, resulting in an azeotrope.

The adjacent diagram illustrates total vapor pressure of three hypothetical mixtures of constituents, X, and Y. The temperature throughout the plot is assumed to be constant.

The center trace is a straight line, which is what Raoult's law predicts for an ideal mixture. In general solely mixtures of chemically similar solvents, such as n-hexane with n-heptane, form nearly ideal mixtures that come close to obeying Raoult's law. The top trace illustrates a nonideal mixture that has a positive deviation from Raoult's law, where the total combined vapor pressure of constituents, X and Y, is greater than what is predicted by Raoult's law. The top trace deviates sufficiently that there is a point on the curve where its tangent is horizontal. Whenever a mixture has a positive deviation and has a point at which the tangent is horizontal, the composition at that point is a positive azeotrope. At that point the total vapor pressure is at a maximum. Likewise the bottom trace illustrates a nonideal mixture that has a negative deviation from Raoult's law, and at the composition where tangent to the trace is horizontal there is a negative azeotrope. This is also the point where total vapor pressure is minimum.

Minimum-boiling or Positive azeotrope

Phase diagram of a positive azeotrope. Vertical axis is temperature, horizontal axis is composition.

The boiling and recondensation of a mixture of two solvents are changes of chemical state; as such, they are best illustrated with a phase diagram. If the pressure is held constant, the two variable parameters are the temperature and the composition.

The phase diagram on the right shows a positive azeotrope of hypothetical constituents, X and Y. The bottom trace illustrates the boiling temperature of various compositions. Below the bottom trace, only the liquid phase is in equilibrium. The top trace illustrates the vapor composition above the liquid at a given temperature. Above the top trace, only the vapor is in equilibrium. Between the two traces, liquid and vapor phases exist simultaneously in equilibrium: for example, heating a 25% X : 75% Y mixture to temperature AB would generate vapor of composition B over liquid of composition A. The azeotrope is the point on the diagram where the two curves touch. The horizontal and vertical steps show the path of repeated distillations. Point A is the boiling point of a nonazeotropic mixture. The vapor that separates at that temperature has composition B. The shape of the curves requires that the vapor at B be richer in constituent X than the liquid at point A. The vapor is physically separated from the VLE (vapor-liquid equilibrium) system and is cooled to point C, where it condenses. The resulting liquid (point C) is now richer in X than it was at point A. If the collected liquid is boiled again, it progresses to point D, and so on. The stepwise progression shows how repeated distillation can never produce a distillate that is richer in constituent X than the azeotrope. Note that starting to the right of the azeotrope point results in the same stepwise process closing in on the azeotrope point from the other direction.

Maximum-boiling or Negative azeotrope

Phase diagram of a negative azeotrope. Vertical axis is temperature, horizontal axis is composition.

The phase diagram on the right shows a negative azeotrope of ideal constituents, X and Y. Again the bottom trace illustrates the boiling temperature at various compositions, and again, below the bottom trace the mixture must be entirely liquid phase. The top trace again illustrates the condensation temperature of various compositions, and again, above the top trace the mixture must be entirely vapor phase. The point, A, shown here is a boiling point with a composition chosen very near to the azeotrope. The vapor is collected at the same temperature at point B. That vapor is cooled, condensed, and collected at point C. Because this example is a negative azeotrope rather than a positive one, the distillate is farther from the azeotrope than the original liquid mixture at point A was. So the distillate is poorer in constituent X and richer in constituent Y than the original mixture. Because this process has removed a greater fraction of Y from the liquid than it had originally, the residue must be poorer in Y and richer in X after distillation than before.

If the point, A had been chosen to the right of the azeotrope rather than to the left, the distillate at point C would be farther to the right than A, which is to say that the distillate would be richer in X and poorer in Y than the original mixture. So in this case too, the distillate moves away from the azeotrope and the residue moves toward it. This is characteristic of negative azeotropes. No amount of distillation, however, can make either the distillate or the residue arrive on the opposite side of the azeotrope from the original mixture. This is characteristic of all azeotropes.

Traces

The traces in the phase diagrams separate whenever the composition of the vapor differs from the composition of the liquid at the same temperature. Suppose the total composition were 50/50%. You could make this composition using 50% of 50/50% vapor and 50% of 50/50% liquid, but you could also make it from 83.33% of 45/55% vapor and 16.67% of 75%/25% liquid, as well as from many other combinations. The separation of the two traces represents the range of combinations of liquid and vapor that can make each total composition.

Temperature-pressure dependence

For both the top and bottom traces, the temperature point of the azeotrope is the constant temperature chosen for the graph. If the ambient pressure is controlled to be equal to the total vapor pressure at the azeotropic mixture, then the mixture will boil at this fixed temperature.

Vapor pressure of both pure liquids as well as mixtures is a sensitive function of temperature. As a rule, vapor pressure of a liquid increases nearly exponentially as a function of temperature. If the graph were replotted for a different fixed temperature, then the total vapor pressure at the azeotropic composition will certainly change, but it is also possible that the composition at which the azeotrope occurs will change. This implies that the composition of an azeotrope is affected by the pressure chosen at which to boil the mixture. Ordinarily distillation is done at atmospheric pressure, but with proper equipment it is possible to carry out distillation at a wide variety of pressures, both above and below atmospheric pressure.

Azeotropy in fluid separation processes

If the two solvents can form a negative azeotrope, then distillation of any mixture of those constituents will result in the residue being closer to the composition at the azeotrope than the original mixture. For example, if a hydrochloric acid solution contains less than 20.2% hydrogen chloride, boiling the mixture will leave behind a solution that is richer in hydrogen chloride than the original. If the solution initially contains more than 20.2% hydrogen chloride, then boiling will leave behind a solution that is poorer in hydrogen chloride than the original. Boiling of any hydrochloric acid solution long enough will cause the solution left behind to approach the azeotropic ratio. On the other hand, if two solvents can form a positive azeotrope, then distillation of any mixture of those constituents will result in the residue away from the composition at the azeotrope than the original mixture. For example, if a 50/50 mixture of ethanol and water is distilled once, the distillate will be 80% ethanol and 20% water, which is closer to the azeotropic mixture than the original, which means the solution left behind will be poorer in ethanol. Distilling the 80/20% mixture produces a distillate that is 87% ethanol and 13% water. Further repeated distillations will produce mixtures that are progressively closer to the azeotropic ratio of 95.5/4.5%. No numbers of distillations will ever result in a distillate that exceeds the azeotropic ratio. Likewise, when distilling a mixture of ethanol and water that is richer in ethanol than the azeotrope, the distillate (contrary to intuition) will be poorer in ethanol than the original but still richer than the azeotrope.

Distillation is one of the primary tools that chemists and chemical engineers use to separate mixtures into their constituents. Because distillation cannot separate the constituents of an azeotrope, the separation of azeotropic mixtures (also called azeotrope breaking) is a topic of considerable interest. Indeed, this difficulty led some early investigators to believe that azeotropes were actually compounds of their constituents. But there are two reasons for believing that this is not the case. One is that the molar ratio of the constituents of an azeotrope is not generally the ratio of small integers. For example, the azeotrope formed by water and acetonitrile contains 2.253 moles (or 9/4 with a relative error of just 2%) of acetonitrile for each mole of water. A more compelling reason for believing that azeotropes are not compounds is, as discussed in the last section, that the composition of an azeotrope can be affected by pressure. Contrast that with a true compound, carbon dioxide for example, which is two moles of oxygen for each mole of carbon no matter what pressure the gas is observed at. That azeotropic composition can be affected by pressure suggests a means by which such a mixture can be separated.

Pressure swing distillation

Azeotrope composition shift due to pressure swing.

A hypothetical azeotrope of constituents X and Y is shown in the adjacent diagram. Two sets of curves on a phase diagram one at an arbitrarily chosen low pressure and another at an arbitrarily chosen, but higher, pressure. The composition of the azeotrope is substantially different between the high- and low-pressure plots – higher in X for the high-pressure system. The goal is to separate X in as high a concentration as possible starting from point A. At the low pressure, it is possible by progressive distillation to reach a distillate at the point, B, which is on the same side of the azeotrope as A. Note that successive distillation steps near the azeotropic composition exhibit very little difference in boiling temperature. If this distillate is now exposed to the high pressure, it boils at point C. From C, by progressive distillation it is possible to reach a distillate at the point D, which is on the same side of the high-pressure azeotrope as C. If that distillate is then exposed again to the low pressure, it boils at point E, which is on the opposite side of the low-pressure azeotrope to A. So, by means of the pressure swing, it is possible to cross over the low-pressure azeotrope.

When the solution is boiled at point E, the distillate is poorer in X than the residue at point E. This means that the residue is richer in X than the distillate at point E. Indeed, progressive distillation can produce a residue as rich in X as is required.

In summary:

1. Low-pressure rectification (A to B) 2. High-pressure rectification (C to D) 3. Low-pressure stripping (E to target purity)

Rectification: the distillate, or "tops", is retained and exhibits an increasingly lower boiling point. Stripping: the residue, or "bottoms", is retained and exhibits an increasingly higher boiling point.

Note that both azeotropes above are of the positive, or minimum boiling type; care must be taken to ensure that the correct component of the separation step is retained, i.e. the binary phase-envelope diagram (boiling-point curve) must be correctly read.

A mixture of 5% water with 95% tetrahydrofuran is an example of an azeotrope that can be economically separated using a pressure swing – a swing in this case between 1 atm and 8 atm. By contrast the composition of the water to ethanol azeotrope discussed earlier is not affected enough by pressure to be easily separated using pressure swings and instead, an entrainer may be added that either modifies the azeotropic composition and exhibits immiscibility with one of the components, or extractive distillation may be used.

Azeotropic distillation

Main article: Azeotropic distillation

Other methods of separation involve introducing an additional agent, called an entrainer, that will affect the volatility of one of the azeotrope constituents more than another. When an entrainer is added to a binary azeotrope to form a ternary azeotrope, and the resulting mixture distilled, the method is called azeotropic distillation. The best known example is adding benzene or cyclohexane to the water/ethanol azeotrope. With cyclohexane as the entrainer, the ternary azeotrope is 7% water, 17% ethanol, and 76% cyclohexane, and boils at 62.1 °C. Just enough cyclohexane is added to the water/ethanol azeotrope to engage all of the water into the ternary azeotrope. When the mixture is then boiled, the azeotrope vaporizes leaving a residue composed almost entirely of the excess ethanol.

Chemical action separation

Another type of entrainer is one that has a strong chemical affinity for one of the constituents. Using again the example of the water/ethanol azeotrope, the liquid can be shaken with calcium oxide, which reacts strongly with water to form the nonvolatile compound, calcium hydroxide. Nearly all of the calcium hydroxide can be separated by filtration and the filtrate redistilled to obtain 100% pure ethanol.

A more extreme example is the azeotrope of 1.2% water with 98.8% diethyl ether. Ether holds the last bit of water so tenaciously that only a very powerful desiccant such as sodium metal added to the liquid phase can result in completely dry ether.

Anhydrous calcium chloride is used as a desiccant for drying a wide variety of solvents since it is inexpensive and does not react with most nonaqueous solvents. Chloroform is an example of a solvent that can be effectively dried using calcium chloride.

Distillation using a dissolved salt

Main article: Salt-effect distillation

When a salt is dissolved in a solvent, it always has the effect of raising the boiling point of that solvent – that is it decreases the volatility of the solvent. When the salt is readily soluble in one constituent of a mixture but not in another, the volatility of the constituent in which it is soluble is decreased and the other constituent is unaffected. In this way, for example, it is possible to break the water/ethanol azeotrope by dissolving potassium acetate in it and distilling the result.

Extractive distillation

Extractive distillation is similar to azeotropic distillation, except in this case the entrainer is less volatile than any of the azeotrope's constituents. For example, the azeotrope of 20% acetone with 80% chloroform can be broken by adding water and distilling the result. The water forms a separate layer in which the acetone preferentially dissolves. The result is that the distillate is richer in chloroform than the original azeotrope.

Pervaporation and other membrane methods

The pervaporation method uses a membrane that is more permeable to the one constituent than to another to separate the constituents of an azeotrope as it passes from liquid to vapor phase. The membrane is rigged to lie between the liquid and vapor phases. Another membrane method is vapor permeation, where the constituents pass through the membrane entirely in the vapor phase. In all membrane methods, the membrane separates the fluid passing through it into a permeate (that which passes through) and a retentate (that which is left behind). When the membrane is chosen so that is it more permeable to one constituent than another, then the permeate will be richer in that first constituent than the retentate.

Complex systems

Saddle azeotropic system Methanol/Acetone/Chloroform calculated with mod. UNIFAC

The rules for positive and negative azeotropes apply to all the examples discussed so far, but there are some examples that don't fit into the categories of positive or negative azeotropes. The best known of these is the ternary azeotrope formed by 30% acetone, 47% chloroform, and 23% methanol, which boils at 57.5 °C. Each pair of these constituents forms a binary azeotrope, but chloroform/methanol and acetone/methanol both form positive azeotropes while chloroform/acetone forms a negative azeotrope. The resulting ternary azeotrope is neither positive nor negative. Its boiling point falls between the boiling points of acetone and chloroform, so it is neither a maximum nor a minimum boiling point. This type of system is called a saddle azeotrope. Only systems of three or more constituents can form saddle azeotropes.