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 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 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 (i.e. the ideal gas law in its theoretical form) 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. The form can be further simplified by defining the kinetic energy corresponding to the temperature:

${\displaystyle T:=k_{\text{B}}T,}$

so the ideal gas law is more simply expressed as:

${\displaystyle p=nT,}$

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 Da) 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	${\displaystyle E=\frac{3}{2}{k}_{\mathrm{B}}T}$

Applications to thermodynamic processes

The table below essentially simplifies the ideal gas equation for a particular process, making the 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 a 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 thermodynamic 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(p1/p2)	T2 = T1
		V2/V1	p2 = p1(V1/V2)	V2 = V1(V2/V1)	T2 = T1
Isentropic process (Reversible adiabatic process)	Entropy[a]	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[b]	p2 − p1	p2 = p1 + (p2 − p1)		T2 = T1 + μJT(p2 − p1)
		T2 − T1	p2 = p1 + (T2 − T1)/μJT		T2 = T1 + (T2 − T1)

^{^} a. In an isentropic process, system entropy (S) is constant. Under these conditions, p₁V₁^γ = p₂V₂^γ, where γ is defined as the heat capacity ratio, which is constant for a calorifically perfect gas. The value used for γ is typically 1.4 for diatomic gases like nitrogen (N₂) and oxygen (O₂), (and air, which is 99% diatomic). Also γ is typically 1.6 for mono atomic gases like the noble gases helium (He), and argon (Ar). In internal combustion engines γ varies between 1.35 and 1.15, depending on constitution gases and temperature.

^{^} b. In an isenthalpic process, system enthalpy (H) is constant. In the case of free expansion for an ideal gas, there are no molecular interactions, and the temperature remains constant. For real gasses, the molecules do interact via attraction or repulsion depending on temperature and pressure, and heating or cooling does occur. This is known as the Joule–Thomson effect. For reference, the Joule–Thomson coefficient μ_JT for air at room temperature and sea level is 0.22 °C/bar.

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 = ⁠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

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

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. If all molecules move with the same velocity ${\displaystyle v}$, denote the corresponding pressure by ${\displaystyle P_0}$. 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 ${\displaystyle NvtS}$ 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 ${\displaystyle NvtS/6}$.

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 ${\displaystyle |\mathrm{\Delta }\mathbf{p}|=2mv{N}^{\prime }/V=NtSm{v}^2/(3V)=ntSm{v}^2/3}$. From ${\displaystyle F=|\mathrm{\Delta }\mathbf{p}|/t}$ and ${\displaystyle P_0=F/S}$ we get

${\displaystyle P_0=\frac{1}{3}nm{v}^2.}$

We considered a situation where all molecules move with the same velocity ${\displaystyle v}$. Now we consider a situation where they can move with different velocities, so we apply an "averaging transformation" to the above equation, effectively replacing ${\displaystyle P_0}$ by a new pressure ${\displaystyle P}$ and ${\displaystyle v^2}$ by the arithmetic mean of all squares of all velocities of the molecules, i.e. by ${\displaystyle v_{\text{rms}}^2.}$ Therefore

${\displaystyle P=\frac{1}{3}nm{v}_{\text{rms}}^2}$

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

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

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

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

Using the integration formula

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

it follows that

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

from which we get the ideal gas law:

${\displaystyle P=\frac{1}{3}nm\left(\frac{3k_{\mathrm{B}}T}{m}\right)=n{k}_{\mathrm{B}}T.}$

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 (two times) 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

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

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:

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

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:

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

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

Asteroseismology

From Wikipedia, the free encyclopedia

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

Different oscillation modes have different sensitivities to the structure of a star. By observing multiple modes, one can therefore partially infer a star's internal structure.

Asteroseismology is the study of oscillations in stars. Stars have many resonant modes and frequencies, and the path of sound waves passing through a star depends on the local speed of sound, which in turn depends on local temperature and chemical composition. Because the resulting oscillation modes are sensitive to different parts of the star, they inform astronomers about the internal structure of the star, which is otherwise not directly possible from overall properties like brightness and surface temperature.

Asteroseismology is closely related to helioseismology, the study of stellar pulsation specifically in the Sun. Though both are based on the same underlying physics, more and qualitatively different information is available for the Sun because its surface can be resolved.

Theoretical background

A propagation diagram for a standard solar model showing where oscillations have a g-mode character (blue) or where dipole modes have a p-mode character (orange). Between about 100 and 400 μHz, modes would potentially have two oscillating regions: these are known as mixed modes. The dashed line shows the acoustic cut-off frequency, computed from more precise modelling, and above which modes are not trapped in the star, and roughly speaking do not resonate.

By linearly perturbing the equations defining the mechanical equilibrium of a star (i.e. mass conservation and hydrostatic equilibrium) and assuming that the perturbations are adiabatic, one can derive a system of four differential equations whose solutions give the frequency and structure of a star's modes of oscillation. The stellar structure is usually assumed to be spherically symmetric, so the horizontal (i.e. non-radial) component of the oscillations is described by spherical harmonics, indexed by an angular degree ${\displaystyle \ell }$ and azimuthal order ${\displaystyle m}$. In non-rotating stars, modes with the same angular degree must all have the same frequency because there is no preferred axis. The angular degree indicates the number of nodal lines on the stellar surface, so for large values of ${\displaystyle \ell }$, the opposing sectors roughly cancel out, making it difficult to detect light variations. As a consequence, modes can only be detected up to an angular degree of about 3 in intensity and about 4 if observed in radial velocity.

By additionally assuming that the perturbation to the gravitational potential is negligible (the Cowling approximation) and that the star's structure varies more slowly with radius than the oscillation mode, the equations can be reduced approximately to one second-order equation for the radial component of the displacement eigenfunction ${\displaystyle {\xi }_r}$, $${\displaystyle \frac{d^2{\xi }_r}{d{r}^2}=\frac{{\omega }^2}{c_s^2}\left(1-\frac{N^2}{{\omega }^2}\right)\left(\frac{S_{\ell }^2}{{\omega }^2}-1\right){\xi }_r}$$ where ${\displaystyle r}$ is the radial co-ordinate in the star, ${\displaystyle \omega }$ is the angular frequency of the oscillation mode, ${\displaystyle c_s}$ is the sound speed inside the star, ${\displaystyle N}$ is the Brunt–Väisälä or buoyancy frequency and ${\displaystyle S_{\ell }}$ is the Lamb frequency. The last two are defined by $${\displaystyle N^2=g\left(\frac{1}{{\mathrm{\Gamma }}_{1}P}\frac{dp}{dr}-\frac{1}{\rho }\frac{d\rho }{dr}\right)}$$ and $${\displaystyle S_{\ell }^2=\frac{\ell (\ell +1)c_s^2}{r^2}}$$ respectively. By analogy with the behaviour of simple harmonic oscillators, this implies that oscillating solutions exist when the frequency is either greater or less than both ${\displaystyle S_{\ell }}$ and ${\displaystyle N}$. We identify the former case as high-frequency pressure modes (p-modes) and the latter as low-frequency gravity modes (g-modes).

This basic separation allows us to determine (to reasonable accuracy) where we expect what kind of mode to resonate in a star. By plotting the curves ${\displaystyle \omega =N}$ and ${\displaystyle \omega =S_{\ell }}$ (for given ${\displaystyle \ell }$), we expect p-modes to resonate at frequencies below both curves or frequencies above both curves.

Excitation mechanisms

Kappa-mechanism

Main article: Kappa-mechanism

Under fairly specific conditions, some stars have regions where heat is transported by radiation and the opacity is a sharply decreasing function of temperature. This opacity bump can drive oscillations through the ${\displaystyle \kappa }$-mechanism (or Eddington valve). Suppose that, at the beginning of an oscillation cycle, the stellar envelope has contracted. By expanding and cooling slightly, the layer in the opacity bump becomes more opaque, absorbs more radiation, and heats up. This heating causes expansion, further cooling and the layer becomes even more opaque. This continues until the material opacity stops increasing so rapidly, at which point the radiation trapped in the layer can escape. The star contracts and the cycle prepares to commence again. In this sense, the opacity acts like a valve that traps heat in the star's envelope.

Pulsations driven by the ${\displaystyle \kappa }$-mechanism are coherent and have relatively large amplitudes. It drives the pulsations in many of the longest-known variable stars, including the Cepheid and RR Lyrae variables.

Surface convection

In stars with surface convection zones, turbulent fluids motions near the surface simultaneously excite and damp oscillations across a broad range of frequency. Because the modes are intrinsically stable, they have low amplitudes and are relatively short-lived. This is the driving mechanism in all solar-like oscillators.

Convective blocking

If the base of a surface convection zone is sharp and the convective timescales slower than the pulsation timescales, the convective flows react too slowly to perturbations that can build up into large, coherent pulsations. This mechanism is known as convective blocking and is believed to drive pulsations in the ${\displaystyle \gamma }$ Doradus variables.

Tidal excitation

Observations from the Kepler satellite revealed eccentric binary systems in which oscillations are excited during the closest approach. These systems are known as heartbeat stars because of the characteristic shape of the lightcurves.

Types of oscillators

A Hertzsprung–Russell diagram highlighting regions where many classes of pulsating star are found.

Solar-like oscillators

Main article: Solar-like oscillations

Because solar oscillations are driven by near-surface convection, any stellar oscillations caused similarly are known as solar-like oscillations and the stars themselves as solar-like oscillators. However, solar-like oscillations also occur in evolved stars (subgiants and red giants), which have convective envelopes, even though the stars are not Sun-like.

Cepheid variables

Main article: Cepheid variable

Cepheid variables are one of the most important classes of pulsating star. They are core-helium burning stars with masses above about 5 solar masses. They principally oscillate at their fundamental modes, with typical periods ranging from days to months. Their pulsation periods are closely related to their luminosities, so it is possible to determine the distance to a Cepheid by measuring its oscillation period, computing its luminosity, and comparing this to its observed brightness.

Cepheid pulsations are excited by the kappa mechanism acting on the second ionization zone of helium.

RR Lyrae variables

Main article: RR Lyrae variable

RR Lyraes are similar to Cepheid variables but of lower metallicity (i.e. Population II) and much lower masses (about 0.6 to 0.8 time solar). They are core helium-burning giants that oscillate in one or both of their fundamental mode or first overtone. The oscillation are also driven by the kappa mechanism acting through the second ionization of helium. Many RR Lyraes, including RR Lyrae itself, show long period amplitude modulations, known as the Blazhko effect.

Delta Scuti and Gamma Doradus stars

Main articles: Delta Scuti variable and Gamma Doradus variable

Delta Scuti variables are found roughly where the classical instability strip intersects the main sequence. They are typically A- to early F-type dwarfs and subgiants and the oscillation modes are low-order radial and non-radial pressure modes, with periods ranging from 0.25 to 8 hours and magnitude variations anywhere between. Like Cepheid variables, the oscillations are driven by the kappa mechanism acting on the second ionization of helium.

SX Phoenicis variables are regarded as metal-poor relatives of Delta Scuti variables.

Gamma Doradus variables occur in similar stars to the red end of the Delta Scuti variables, usually of early F-type. The stars show multiple oscillation frequencies between about 0.5 and 3 days, which is much slower than the low-order pressure modes. Gamma Doradus oscillations are generally thought to be high-order gravity modes, excited by convective blocking.

Following results from Kepler, it appears that many Delta Scuti stars also show Gamma Doradus oscillations and are therefore hybrids.

Rapidly oscillating Ap (roAp) stars

Main article: Rapidly oscillating Ap star

Rapidly oscillating Ap stars have similar parameters to Delta Scuti variables, mostly being A- and F-type, but they are also strongly magnetic and chemically peculiar (hence the p spectral subtype). Their dense mode spectra are understood in terms of the oblique pulsator model: the mode's frequencies are modulated by the magnetic field, which is not necessarily aligned with the star's rotation (as is the case in the Earth). The oscillation modes have frequencies around 1500 μHz and amplitudes of a few mmag.

Slowly pulsating B stars and Beta Cephei variables

Main articles: Slowly pulsating B-type star and Beta Cephei variable

Slowly pulsating B (SPB) stars are B-type stars with oscillation periods of a few days, understood to be high-order gravity modes excited by the kappa mechanism. Beta Cephei variables are slightly hotter (and thus more massive), also have modes excited by the kappa mechanism and additionally oscillate in low-order gravity modes with periods of several hours. Both classes of oscillators contain only slowly rotating stars.

Variable subdwarf B stars

Main article: Subdwarf B star

Subdwarf B (sdB) stars are in essence the cores of core-helium burning giants who have somehow lost most of their hydrogen envelopes, to the extent that there is no hydrogen-burning shell. They have multiple oscillation periods that range between about 1 and 10 minutes and amplitudes anywhere between 0.001 and 0.3 mag in visible light. The oscillations are low-order pressure modes, excited by the kappa mechanism acting on the iron opacity bump.

White dwarfs

Main article: Pulsating white dwarf

White dwarfs are characterized by spectral type, much like ordinary stars, except that the relationship between spectral type and effective temperature does not correspond in the same way. Thus, white dwarfs are known by types DO, DA and DB. Cooler types are physically possible but the Universe is too young for them to have cooled enough. White dwarfs of all three types are found to pulsate. The pulsators are known as GW Virginis stars (DO variables, sometimes also known as PG 1159 stars), V777 Herculis stars (DB variables) and ZZ Ceti stars (DA variables). All pulsate in low-degree, high-order g-modes. The oscillation periods broadly decrease with effective temperature, ranging from about 30 min down to about 1 minute. GW Virginis and ZZ Ceti stars are thought to be excited by the kappa mechanism; V777 Herculis stars by convective blocking.

Space missions

A number of past, present and future spacecraft have asteroseismology studies as a significant part of their missions (order chronological).

• WIRE – A NASA satellite launched in 1999. A failed large infrared telescope, the two-inch aperture star tracker was used for more than a decade as a bright-star asteroseismology instrument. Re-entered Earth's atmosphere 2011.
• MOST – A Canadian satellite launched in 2003. The first spacecraft dedicated to asteroseismology.
• CoRoT – A French led ESA planet-finder and asteroseismology satellite launched in 2006.
• Kepler space telescope – A NASA planet-finder spacecraft launched in 2009, repurposed as K2 since the failure of a second reaction wheel prevented the telescope from continuing to monitor the same field.
• BRITE – A constellation of nanosatellites used to study the brightest oscillating stars. First two satellites launched Feb 25, 2013.
• TESS – A NASA planet-finder that will survey bright stars across most of the sky launched in 2018.
• PLATO – A planned ESA mission that will specifically exploit asteroseismology to obtain accurate masses and radii of transiting planets.