Kinetic theory of gases

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Kinetic_theory_of_gases

The temperature of the ideal gas is proportional to the average kinetic energy of its particles. The size of helium atoms relative to their spacing is shown to scale under 1,950 atmospheres of pressure. The atoms have an average speed relative to their size slowed down here two trillion fold from that at room temperature.

The basic version of the model describes an ideal gas. It treats the collisions as perfectly elastic and as the only interaction between the particles, which are additionally assumed to be much smaller than their average distance apart.

The theory's introduction allowed many principal concepts of thermodynamics to be established. It explains the macroscopic properties of gases, such as volume, pressure, and temperature, as well as transport properties such as viscosity, thermal conductivity and mass diffusivity. Due to the time reversibility of microscopic dynamics (microscopic reversibility), the kinetic theory is also connected to the principle of detailed balance, in terms of the fluctuation-dissipation theorem (for Brownian motion) and the Onsager reciprocal relations.

The theory was historically significant as the first explicit exercise of the ideas of statistical mechanics.

History

In about 50 BCE, the Roman philosopher Lucretius proposed that apparently static macroscopic bodies were composed on a small scale of rapidly moving atoms all bouncing off each other. This Epicurean atomistic point of view was rarely considered in the subsequent centuries, when Aristotlean ideas were dominant.

Hydrodynamica front cover

In 1738 Daniel Bernoulli published Hydrodynamica, which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the pressure of the gas, and that their average kinetic energy determines the temperature of the gas. The theory was not immediately accepted, in part because conservation of energy had not yet been established, and it was not obvious to physicists how the collisions between molecules could be perfectly elastic.

Other pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were Mikhail Lomonosov (1747), Georges-Louis Le Sage (ca. 1780, published 1818), John Herapath (1816) and John James Waterston (1843), which connected their research with the development of mechanical explanations of gravitation.

In 1856 August Krönig created a simple gas-kinetic model, which only considered the translational motion of the particles. In 1857 Rudolf Clausius developed a similar, but more sophisticated version of the theory, which included translational and, contrary to Krönig, also rotational and vibrational molecular motions. In this same work he introduced the concept of mean free path of a particle. In 1859, after reading a paper about the diffusion of molecules by Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. In his 1873 thirteen page article 'Molecules', Maxwell states: "we are told that an 'atom' is a material point, invested and surrounded by 'potential forces' and that when 'flying molecules' strike against a solid body in constant succession it causes what is called pressure of air and other gases." In 1871, Ludwig Boltzmann generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution. The logarithmic connection between entropy and probability was also first stated by Boltzmann.

At the beginning of the 20th century, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was Albert Einstein's (1905) and Marian Smoluchowski's (1906) papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.

Following the development of the Boltzmann equation, a framework for its use in developing transport equations was developed independently by David Enskog and Sydney Chapman in 1917 and 1916. The framework provided a route to prediction of the transport properties of dilute gases, and became known as Chapman–Enskog theory. The framework was gradually expanded throughout the following century, eventually becoming a route to prediction of transport properties in real, dense gases.

Assumptions

The application of kinetic theory to ideal gases makes the following assumptions:

• The gas consists of very small particles. This smallness of their size is such that the sum of the volume of the individual gas molecules is negligible compared to the volume of the container of the gas. This is equivalent to stating that the average distance separating the gas particles is large compared to their size, and that the elapsed time during a collision between particles and the container's wall is negligible when compared to the time between successive collisions.
• The number of particles is so large that a statistical treatment of the problem is well justified. This assumption is sometimes referred to as the thermodynamic limit.
• The rapidly moving particles constantly collide among themselves and with the walls of the container, and all these collisions are perfectly elastic.
• Interactions (i.e. collisions) between particles are strictly binary and uncorrelated, meaning that there are no three-body (or higher) interactions, and the particles have no memory.
• Except during collisions, the interactions among molecules are negligible. They exert no other forces on one another.

Thus, the dynamics of particle motion can be treated classically, and the equations of motion are time-reversible.

As a simplifying assumption, the particles are usually assumed to have the same mass as one another; however, the theory can be generalized to a mass distribution, with each mass type contributing to the gas properties independently of one another in agreement with Dalton's Law of partial pressures. Many of the model's predictions are the same whether or not collisions between particles are included, so they are often neglected as a simplifying assumption in derivations (see below).

More modern developments, such as Revised Enskog Theory and the Extended BGK model, relax one or more of the above assumptions. These can accurately describe the properties of dense gases, and gases with internal degrees of freedom, because they include the volume of the particles as well as contributions from intermolecular and intramolecular forces as well as quantized molecular rotations, quantum rotational-vibrational symmetry effects, and electronic excitation. While theories relaxing the assumptions that the gas particles occupy negligible volume and that collisions are strictly elastic have been successful, it has been shown that relaxing the requirement of interactions being binary and uncorrelated will eventually lead to divergent results.

Equilibrium properties

Pressure and kinetic energy

In the kinetic theory of gases, the pressure is assumed to be equal to the force (per unit area) exerted by the individual gas atoms or molecules hitting and rebounding from the gas container's surface.

Consider a gas particle traveling at velocity, $v_i$, along the ${\displaystyle \hat{i}}$-direction in an enclosed volume with characteristic length, ${\displaystyle L_i}$, cross-sectional area, ${\displaystyle A_i}$, and volume, ${\displaystyle V=A_i{L}_i}$. The gas particle encounters a boundary after characteristic time

$${\displaystyle t=L_i/v_i.}$$

The momentum of the gas particle can then be described as

$${\displaystyle p_i=m{v}_i=m{L}_i/t.}$$

We combine the above with Newton's second law, which states that the force experienced by a particle is related to the time rate of change of its momentum, such that

$${\displaystyle F_i=\frac{\mathrm{d}{p}_i}{\mathrm{d}t}=\frac{m{L}_i}{t^2}=\frac{m{v}_i^2}{L_i}.}$$

Now consider a large number, N, of gas particles with random orientation in a three-dimensional volume. Because the orientation is random, the average particle speed, $v$, in every direction is identical

$${\displaystyle v^2={\overrightarrow{v}}_x^2={\overrightarrow{v}}_y^2={\overrightarrow{v}}_z^2.}$$

Further, assume that the volume is symmetrical about its three dimensions, ${\displaystyle \hat{i},\hat{j},\hat{k}}$, such that

$${\displaystyle v=v_i=v_j=v_k,}$$ $${\displaystyle F=F_i=F_j=F_k,}$$ $${\displaystyle A_i=A_j=A_k.}$$ The total surface area on which the gas particles act is therefore $${\displaystyle A=3\ast A_i.}$$

The pressure exerted by the collisions of the N gas particles with the surface can then be found by adding the force contribution of every particle and dividing by the interior surface area of the volume,

$${\displaystyle P=\frac{N\overline{F}}{A}=\frac{NLF}{V}}$$ $${\displaystyle \Rightarrow PV=NLF=\frac{N}{3}m{v}^2.}$$

The total translational kinetic energy ${\displaystyle K_t}$ of the gas is defined as $${\displaystyle K_t=\frac{N}{2}m{v}^2,}$$ providing the result $${\displaystyle PV=\frac{2}{3}{K}_t.}$$

This is an important, non-trivial result of the kinetic theory because it relates pressure, a macroscopic property, to the translational kinetic energy of the molecules, which is a microscopic property.

Temperature and kinetic energy

Rewriting the above result for the pressure as $PV=\frac{Nm{v}^2}{3}$, we may combine it with the ideal gas law

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

(1)

where ${\displaystyle k_{\mathrm{B}}}$ is the Boltzmann constant and ${\displaystyle T}$ the absolute temperature defined by the ideal gas law, to obtain

$${\displaystyle k_{\mathrm{B}}T=\frac{m{v}^2}{3},}$$ which leads to a simplified expression of the average translational kinetic energy per molecule, $${\displaystyle \frac{1}{2}m{v}^2=\frac{3}{2}{k}_{\mathrm{B}}T.}$$ The translational kinetic energy of the system is ${\displaystyle N}$ times that of a molecule, namely $K_t=\frac{1}{2}Nm{v}^2$. The temperature, ${\displaystyle T}$ is related to the translational kinetic energy by the description above, resulting in

${\displaystyle T=\frac{1}{3}\frac{m{v}^2}{k_{\mathrm{B}}}}$

(2)

which becomes

${\displaystyle T=\frac{2}{3}\frac{K_t}{N{k}_{\mathrm{B}}}.}$

(3)

Equation (3) is one important result of the kinetic theory: The average molecular kinetic energy is proportional to the ideal gas law's absolute temperature. From equations (1) and (3), we have

${\displaystyle PV=\frac{2}{3}{K}_t.}$

(4)

Thus, the product of pressure and volume per mole is proportional to the average translational molecular kinetic energy.

Equations (1) and (4) are called the "classical results", which could also be derived from statistical mechanics; for more details, see:

The equipartition theorem requires that kinetic energy is partitioned equally between all kinetic degrees of freedom, D. A monatomic gas is axially symmetric about each spatial axis, so that D = 3 comprising translational motion along each axis. A diatomic gas is axially symmetric about only one axis, so that D = 5, comprising translational motion along three axes and rotational motion along two axes. A polyatomic gas, like water, is not radially symmetric about any axis, resulting in D = 6, comprising 3 translational and 3 rotational degrees of freedom.

Because the equipartition theorem requires that kinetic energy is partitioned equally, the total kinetic energy is

$${\displaystyle K=D{K}_t=\frac{D}{2}Nm{v}^2.}$$

Thus, the energy added to the system per gas particle kinetic degree of freedom is

$${\displaystyle \frac{K}{ND}=\frac{1}{2}{k}_{B}T.}$$

Therefore, the kinetic energy per kelvin of one mole of monatomic ideal gas (D = 3) is

$${\displaystyle K=\frac{D}{2}{k}_B{N}_{A}T=\frac{3}{2}R,}$$

where ${\displaystyle N_A}$ is the Avogadro constant, and R is the ideal gas constant.

Thus, the kinetic energy per unit kelvin of an ideal monatomic gas can be calculated easily:

• per mole: 12.47 J / K
• per molecule: 20.7 yJ / K = 129 μeV / K

At standard temperature (273.15 K), the kinetic energy can also be obtained:

• per mole: 3406 J
• per molecule: 5.65 zJ = 35.2 meV.

At higher temperatures (typically thousands of kelvins), vibrational modes become active to provide additional degrees of freedom, creating a temperature-dependence on D and the total molecular energy. Quantum statistical mechanics is needed to accurately compute these contributions.

Collisions with container wall

For an ideal gas in equilibrium, the rate of collisions with the container wall and velocity distribution of particles hitting the container wall can be calculated based on naive kinetic theory, and the results can be used for analyzing effusive flow rates, which is useful in applications such as the gaseous diffusion method for isotope separation.

Assume that in the container, the number density (number per unit volume) is ${\displaystyle n=N/V}$ and that the particles obey Maxwell's velocity distribution: $${\displaystyle f_{\text{Maxwell}}(v_x,v_y,v_z)d{v}_{x}d{v}_{y}d{v}_z={\left(\frac{m}{2\pi k_{B}T}\right)}^{3/2}{e}^{-\frac{m{v}^2}{2k_{B}T}}d{v}_{x}d{v}_{y}d{v}_z}$$

Then for a small area ${\displaystyle dA}$ on the container wall, a particle with speed ${\displaystyle v}$ at angle ${\displaystyle \theta }$ from the normal of the area ${\displaystyle dA}$, will collide with the area within time interval ${\displaystyle dt}$, if it is within the distance ${\displaystyle vdt}$ from the area ${\displaystyle dA}$. Therefore, all the particles with speed ${\displaystyle v}$ at angle ${\displaystyle \theta }$ from the normal that can reach area ${\displaystyle dA}$ within time interval ${\displaystyle dt}$ are contained in the tilted pipe with a height of ${\displaystyle v\cos (\theta )dt}$ and a volume of ${\displaystyle v\cos (\theta )dAdt}$.

The total number of particles that reach area ${\displaystyle dA}$ within time interval ${\displaystyle dt}$ also depends on the velocity distribution; All in all, it calculates to be:$${\displaystyle nv\cos (\theta )dAdt\times {\left(\frac{m}{2\pi k_{B}T}\right)}^{3/2}{e}^{-\frac{m{v}^2}{2k_{B}T}}\left(v^2\sin (\theta )dvd\theta d\varphi \right).}$$

Integrating this over all appropriate velocities within the constraint ${\displaystyle v>0,0<\theta <\pi /2,0<\varphi <2\pi }$ yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time: $${\displaystyle J_{\text{collision}}=\frac{{\int }_0^{\pi /2}\cos \theta \sin \theta d\theta }{{\int }_0^{\pi }\sin \theta d\theta }\times n\overline{v}=\frac{1}{4}n\overline{v}=\frac{n}{4}\sqrt{\frac{8k_{\mathrm{B}}T}{\pi m}}.}$$

This quantity is also known as the "impingement rate" in vacuum physics. Note that to calculate the average speed ${\displaystyle \overline{v}}$ of the Maxwell's velocity distribution, one has to integrate over${\displaystyle v>0,0<\theta <\pi ,0<\varphi <2\pi }$.

The momentum transfer to the container wall from particles hitting the area ${\displaystyle dA}$ with speed ${\displaystyle v}$ at angle ${\displaystyle \theta }$ from the normal, in time interval ${\displaystyle dt}$ is:$${\displaystyle [2mv\cos (\theta )]\times nv\cos (\theta )dAdt\times {\left(\frac{m}{2\pi k_{B}T}\right)}^{3/2}{e}^{-\frac{m{v}^2}{2k_{B}T}}\left(v^2\sin (\theta )dvd\theta d\varphi \right).}$$Integrating this over all appropriate velocities within the constraint ${\displaystyle v>0,0<\theta <\pi /2,0<\varphi <2\pi }$ yields the pressure (consistent with Ideal gas law):$${\displaystyle P=\frac{2{\int }_0^{\pi /2}{\cos }^2\theta \sin \theta d\theta }{{\int }_0^{\pi }\sin \theta d\theta }\times nm{v}_{\text{rms}}^2=\frac{1}{3}nm{v}_{\text{rms}}^2=\frac{2}{3}n⟨E_{kin}⟩=n{k}_{\mathrm{B}}T}$$If this small area ${\displaystyle A}$ is punched to become a small hole, the effusive flow rate will be: $${\displaystyle {\mathrm{\Phi }}_{\text{effusion}}=J_{\text{collision}}A=nA\sqrt{\frac{k_{\mathrm{B}}T}{2\pi m}}.}$$

Combined with the ideal gas law, this yields $${\displaystyle {\mathrm{\Phi }}_{\text{effusion}}=\frac{PA}{\sqrt{2\pi m{k}_{\mathrm{B}}T}}.}$$

The above expression is consistent with Graham's law.

To calculate the velocity distribution of particles hitting this small area, we must take into account that all the particles with ${\displaystyle (v,\theta ,\varphi )}$ that hit the area ${\displaystyle dA}$ within the time interval ${\displaystyle dt}$ are contained in the tilted pipe with a height of ${\displaystyle v\cos (\theta )dt}$ and a volume of ${\displaystyle v\cos (\theta )dAdt}$; Therefore, compared to the Maxwell distribution, the velocity distribution will have an extra factor of ${\displaystyle v\cos \theta }$: $${\displaystyle \begin{array}{rl}f(v,\theta ,\varphi )dvd\theta d\varphi & =\lambda v\cos \theta \times {\left(\frac{m}{2\pi kT}\right)}^{3/2}{e}^{-\frac{m{v}^2}{2k_{\mathrm{B}}T}}(v^2\sin \theta dvd\theta d\varphi )\end{array}}$$ with the constraint $v>0,0<\theta <\frac{\pi }{2},0<\varphi <2\pi$. The constant ${\displaystyle \lambda }$ can be determined by the normalization condition ${\displaystyle \int f(v,\theta ,\varphi )dvd\theta d\varphi =1}$ to be $4/\overline{v}$, and overall:$${\displaystyle \begin{array}{rl}f(v,\theta ,\varphi )dvd\theta d\varphi & =\frac{1}{2\pi }{\left(\frac{m}{k_{\mathrm{B}}T}\right)}^2{e}^{-\frac{m{v}^2}{2k_{\mathrm{B}}T}}(v^3\sin \theta \cos \theta dvd\theta d\varphi )\end{array};v>0,0<\theta <\frac{\pi }{2},0<\varphi <2\pi }$$

Speed of molecules

From the kinetic energy formula it can be shown that $${\displaystyle v_{\text{p}}=\sqrt{2\cdot \frac{k_{\mathrm{B}}T}{m}},}$$ $${\displaystyle \overline{v}=\frac{2}{\sqrt{\pi }}{v}_p=\sqrt{\frac{8}{\pi }\cdot \frac{k_{\mathrm{B}}T}{m}},}$$ $${\displaystyle v_{\text{rms}}=\sqrt{\frac{3}{2}}{v}_p=\sqrt{3\cdot \frac{k_{\mathrm{B}}T}{m}},}$$ where v is in m/s, T is in kelvin, and m is the mass of one molecule of gas in kg. The most probable (or mode) speed ${\displaystyle v_{\text{p}}}$ is 81.6% of the root-mean-square speed ${\displaystyle v_{\text{rms}}}$, and the mean (arithmetic mean, or average) speed ${\displaystyle \overline{v}}$ is 92.1% of the rms speed (isotropic distribution of speeds).

See:

• Average,
• Root-mean-square speed
• Arithmetic mean
• Mean
• Mode (statistics)

Mean free path

Main article: Mean free path

In kinetic theory of gases, the mean free path is the average distance traveled by a molecule, or a number of molecules per volume, before they make their first collision. Let ${\displaystyle \sigma }$ be the collision cross section of one molecule colliding with another. As in the previous section, the number density ${\displaystyle n}$ is defined as the number of molecules per (extensive) volume, or ${\displaystyle n=N/V}$. The collision cross section per volume or collision cross section density is ${\displaystyle n\sigma }$, and it is related to the mean free path ${\displaystyle l}$ by$${\displaystyle l=\frac{1}{n\sigma \sqrt{2}}}$$

Notice that the unit of the collision cross section per volume ${\displaystyle n\sigma }$ is reciprocal of length.

Transport properties

See also: Transport phenomena

The kinetic theory of gases deals not only with gases in thermodynamic equilibrium, but also very importantly with gases not in thermodynamic equilibrium. This means using Kinetic Theory to consider what are known as "transport properties", such as viscosity, thermal conductivity, mass diffusivity and thermal diffusion.

In its most basic form, Kinetic gas theory is only applicable to dilute gases. The extension of Kinetic gas theory to dense gas mixtures, Revised Enskog Theory, was developed in 1983-1987 by E. G. D. Cohen, J. M. Kincaid and M. Lòpez de Haro, building on work by H. van Beijeren and M. H. Ernst.

Viscosity and kinetic momentum

See also: Viscosity § Momentum transport

In books on elementary kinetic theory one can find results for dilute gas modeling that are used in many fields. Derivation of the kinetic model for shear viscosity usually starts by considering a Couette flow where two parallel plates are separated by a gas layer. The upper plate is moving at a constant velocity to the right due to a force F. The lower plate is stationary, and an equal and opposite force must therefore be acting on it to keep it at rest. The molecules in the gas layer have a forward velocity component ${\displaystyle u}$ which increase uniformly with distance ${\displaystyle y}$ above the lower plate. The non-equilibrium flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Inside a dilute gas in a Couette flow setup, let ${\displaystyle u_0}$ be the forward velocity of the gas at a horizontal flat layer (labeled as ${\displaystyle y=0}$); ${\displaystyle u_0}$ is along the horizontal direction. The number of molecules arriving at the area ${\displaystyle dA}$ on one side of the gas layer, with speed ${\displaystyle v}$ at angle ${\displaystyle \theta }$ from the normal, in time interval ${\displaystyle dt}$ is $${\displaystyle nv\cos (\theta )dAdt\times {\left(\frac{m}{2\pi k_{\mathrm{B}}T}\right)}^{3/2}{e}^{-\frac{m{v}^2}{2k_{\mathrm{B}}T}}(v^2\sin \theta dvd\theta d\varphi )}$$

These molecules made their last collision at ${\displaystyle y=\pm l\cos \theta }$, where ${\displaystyle l}$ is the mean free path. Each molecule will contribute a forward momentum of $${\displaystyle p_x^{\pm }=m\left(u_0\pm l\cos \theta \frac{du}{dy}\right),}$$ where plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient ${\displaystyle du/dy}$ can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint $${\displaystyle \{\begin{array}{l}v>0\\ 0<\theta <\pi /2\\ 0<\varphi <2\pi \end{array}}$$ yields the forward momentum transfer per unit time per unit area (also known as shear stress): $${\displaystyle {\tau }^{\pm }=\frac{1}{4}\overline{v}n\cdot m\left(u_0\pm \frac{2}{3}l\frac{du}{dy}\right)}$$

The net rate of momentum per unit area that is transported across the imaginary surface is thus $${\displaystyle \tau ={\tau }^{+}-{\tau }^{-}=\frac{1}{3}\overline{v}nm\cdot l\frac{du}{dy}}$$

Combining the above kinetic equation with Newton's law of viscosity $${\displaystyle \tau =\eta \frac{du}{dy}}$$ gives the equation for shear viscosity, which is usually denoted ${\displaystyle {\eta }_0}$ when it is a dilute gas: $${\displaystyle {\eta }_0=\frac{1}{3}\overline{v}nml}$$

Combining this equation with the equation for mean free path gives $${\displaystyle {\eta }_0=\frac{1}{3\sqrt{2}}\frac{m\cdot \overline{v}}{\sigma }}$$

Maxwell-Boltzmann distribution gives the average (equilibrium) molecular speed as $${\displaystyle \overline{v}=\frac{2}{\sqrt{\pi }}{v}_p=2\sqrt{\frac{2}{\pi }\cdot \frac{k_{\mathrm{B}}T}{m}}}$$ where ${\displaystyle v_p}$ is the most probable speed. We note that $${\displaystyle k_B\cdot N_A=R\text{and}M=m\cdot N_A}$$

and insert the velocity in the viscosity equation above. This gives the well known equation (with ${\displaystyle \sigma }$ subsequently estimated below) for shear viscosity for dilute gases:

$${\displaystyle {\eta }_0=\frac{2}{3\sqrt{\pi }}\cdot \frac{\sqrt{m{k}_{\mathrm{B}}T}}{\sigma }=\frac{2}{3\sqrt{\pi }}\cdot \frac{\sqrt{MRT}}{\sigma \cdot N_A}}$$

and ${\displaystyle M}$ is the molar mass. The equation above presupposes that the gas density is low (i.e. the pressure is low). This implies that the transport of momentum through the gas due to the translational motion of molecules is much larger than the transport due to momentum being transferred between molecules during collisions. The transfer of momentum between molecules is explicitly accounted for in Revised Enskog theory, which relaxes the requirement of a gas being dilute. The viscosity equation further presupposes that there is only one type of gas molecules, and that the gas molecules are perfect elastic and hard core particles of spherical shape. This assumption of elastic, hard core spherical molecules, like billiard balls, implies that the collision cross section of one molecule can be estimated by

$${\displaystyle \sigma =\pi {\left(2r\right)}^2=\pi d^2}$$

The radius ${\displaystyle r}$ is called collision cross section radius or kinetic radius, and the diameter ${\displaystyle d}$ is called collision cross section diameter or kinetic diameter of a molecule in a monomolecular gas. There are no simple general relation between the collision cross section and the hard core size of the (fairly spherical) molecule. The relation depends on shape of the potential energy of the molecule. For a real spherical molecule (i.e. a noble gas atom or a reasonably spherical molecule) the interaction potential is more like the Lennard-Jones potential or Morse potential which have a negative part that attracts the other molecule from distances longer than the hard core radius. The radius for zero Lennard-Jones potential may then be used as a rough estimate for the kinetic radius. However, using this estimate will typically lead to an erroneous temperature dependency of the viscosity. For such interaction potentials, significantly more accurate results are obtained by numerical evaluation of the required collision integrals.

The expression for viscosity obtained from Revised Enskog Theory reduces to the above expression in the limit of infinite dilution, and can be written as

$${\displaystyle \eta =(1+{\alpha }_{\eta }){\eta }_0+{\eta }_c}$$

where ${\displaystyle {\alpha }_{\eta }}$ is a term that tends to zero in the limit of infinite dilution that accounts for excluded volume, and ${\displaystyle {\eta }_c}$ is a term accounting for the transfer of momentum over a non-zero distance between particles during a collision.

Thermal conductivity and heat flux

See also: Thermal conductivity

Following a similar logic as above, one can derive the kinetic model for thermal conductivity of a dilute gas:

Consider two parallel plates separated by a gas layer. Both plates have uniform temperatures, and are so massive compared to the gas layer that they can be treated as thermal reservoirs. The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy ${\displaystyle \epsilon }$ which increases uniformly with distance ${\displaystyle y}$ above the lower plate. The non-equilibrium energy flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Let ${\displaystyle {\epsilon }_0}$ be the molecular kinetic energy of the gas at an imaginary horizontal surface inside the gas layer. The number of molecules arriving at an area ${\displaystyle dA}$ on one side of the gas layer, with speed ${\displaystyle v}$ at angle ${\displaystyle \theta }$ from the normal, in time interval ${\displaystyle dt}$ is $${\displaystyle nv\cos (\theta )dAdt\times {\left(\frac{m}{2\pi k_{\mathrm{B}}T}\right)}^{3/2}{e}^{-\frac{m{v}^2}{2k_{B}T}}(v^2\sin (\theta )dvd\theta d\varphi )}$$

These molecules made their last collision at a distance ${\displaystyle l\cos \theta }$ above and below the gas layer, and each will contribute a molecular kinetic energy of $${\displaystyle {\epsilon }^{\pm }=\left({\epsilon }_0\pm m{c}_{v}l\cos \theta \frac{dT}{dy}\right),}$$ where ${\displaystyle c_v}$ is the specific heat capacity. Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient ${\displaystyle dT/dy}$ can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint $${\displaystyle \{\begin{array}{l}v>0\\ 0<\theta <\pi /2\\ 0<\varphi <2\pi \end{array}}$$

yields the energy transfer per unit time per unit area (also known as heat flux): $${\displaystyle q_y^{\pm }=-\frac{1}{4}\overline{v}n\cdot \left({\epsilon }_0\pm \frac{2}{3}m{c}_{v}l\frac{dT}{dy}\right)}$$

Note that the energy transfer from above is in the ${\displaystyle -y}$ direction, and therefore the overall minus sign in the equation. The net heat flux across the imaginary surface is thus $${\displaystyle q=q_y^{+}-q_y^{-}=-\frac{1}{3}\overline{v}nm{c}_{v}l\frac{dT}{dy}}$$

Combining the above kinetic equation with Fourier's law $${\displaystyle q=-\kappa \frac{dT}{dy}}$$ gives the equation for thermal conductivity, which is usually denoted ${\displaystyle {\kappa }_0}$ when it is a dilute gas: $${\displaystyle {\kappa }_0=\frac{1}{3}\overline{v}nm{c}_{v}l}$$

Similarly to viscosity, Revised Enskog Theory yields an expression for thermal conductivity that reduces to the above expression in the limit of infinite dilution, and which can be written as $${\displaystyle \kappa ={\alpha }_{\kappa }{\kappa }_0+{\kappa }_c}$$

where ${\displaystyle {\alpha }_{\kappa }}$ is a term that tends to unity in the limit of infinite dilution, accounting for excluded volume, and ${\displaystyle {\kappa }_c}$ is a term accounting for the transfer of energy across a non-zero distance between particles during a collision.

Diffusion Coefficient and diffusion flux

See also: Fick's laws of diffusion

Following a similar logic as above, one can derive the kinetic model for mass diffusivity of a dilute gas:

Consider a steady diffusion between two regions of the same gas with perfectly flat and parallel boundaries separated by a layer of the same gas. Both regions have uniform number densities, but the upper region has a higher number density than the lower region. In the steady state, the number density at any point is constant (that is, independent of time). However, the number density ${\displaystyle n}$ in the layer increases uniformly with distance ${\displaystyle y}$ above the lower plate. The non-equilibrium molecular flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Let ${\displaystyle n_0}$ be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area ${\displaystyle dA}$ on one side of the gas layer, with speed ${\displaystyle v}$ at angle ${\displaystyle \theta }$ from the normal, in time interval ${\displaystyle dt}$ is $${\displaystyle nv\cos (\theta )dAdt\times {\left(\frac{m}{2\pi k_{\mathrm{B}}T}\right)}^{3/2}{e}^{-\frac{m{v}^2}{2k_{B}T}}(v^2\sin (\theta )dvd\theta d\varphi )}$$

These molecules made their last collision at a distance ${\displaystyle l\cos \theta }$ above and below the gas layer, where the local number density is $${\displaystyle n^{\pm }=\left(n_0\pm l\cos \theta \frac{dn}{dy}\right)}$$

Again, plus sign applies to molecules from above, and minus sign below. Note that the number density gradient ${\displaystyle dn/dy}$ can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint

$${\displaystyle \{\begin{array}{l}v>0\\ 0<\theta <\pi /2\\ 0<\varphi <2\pi \end{array}}$$

yields the molecular transfer per unit time per unit area (also known as diffusion flux): $${\displaystyle J_y^{\pm }=-\frac{1}{4}\overline{v}\cdot \left(n_0\pm \frac{2}{3}l\frac{dn}{dy}\right)}$$

Note that the molecular transfer from above is in the ${\displaystyle -y}$ direction, and therefore the overall minus sign in the equation. The net diffusion flux across the imaginary surface is thus $${\displaystyle J=J_y^{+}-J_y^{-}=-\frac{1}{3}\overline{v}l\frac{dn}{dy}}$$

Combining the above kinetic equation with Fick's first law of diffusion $${\displaystyle J=-D\frac{dn}{dy}}$$ gives the equation for mass diffusivity, which is usually denoted ${\displaystyle D_0}$ when it is a dilute gas: $${\displaystyle D_0=\frac{1}{3}\overline{v}l}$$

The corresponding expression obtained from Revised Enskog Theory may be written as $${\displaystyle D={\alpha }_D{D}_0}$$ where ${\displaystyle {\alpha }_D}$ is a factor that tends to unity in the limit of infinite dilution, which accounts for excluded volume and the variation chemical potentials with density.

Detailed balance

Fluctuation and dissipation

Main article: Fluctuation-dissipation theorem

The kinetic theory of gases entails that due to the microscopic reversibility of the gas particles' detailed dynamics, the system must obey the principle of detailed balance. Specifically, the fluctuation-dissipation theorem applies to the Brownian motion (or diffusion) and the drag force, which leads to the Einstein–Smoluchowski equation:$${\displaystyle D=\mu k_{\text{B}}T, }$$where

• D is the mass diffusivity;
• μ is the "mobility", or the ratio of the particle's terminal drift velocity to an applied force, μ = v_d/F;
• k_B is the Boltzmann constant;
• T is the absolute temperature.

Note that the mobility μ = v_d/F can be calculated based on the viscosity of the gas; Therefore, the Einstein–Smoluchowski equation also provides a relation between the mass diffusivity and the viscosity of the gas.

Onsager reciprocal relations

Main article: Onsager reciprocal relations

The mathematical similarities between the expressions for shear viscocity, thermal conductivity and diffusion coefficient of the ideal (dilute) gas is not a coincidence; It is a direct result of the Onsager reciprocal relations (i.e. the detailed balance of the reversible dynamics of the particles), when applied to the convection (matter flow due to temperature gradient, and heat flow due to pressure gradient) and advection (matter flow due to the velocity of particles, and momentum transfer due to pressure gradient) of the ideal (dilute) gas.

History of molecular theory

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/History_of_molecular_theory

Space-filling model of the H₂O molecule.

In chemistry, the history of molecular theory traces the origins of the concept or idea of the existence of strong chemical bonds between two or more atoms.

A modern conceptualization of molecules began to develop in the 19th century along with experimental evidence for pure chemical elements and how individual atoms of different chemical elements such as hydrogen and oxygen can combine to form chemically stable molecules such as water molecules.

Ancient world

The modern concept of molecules can be traced back towards pre-scientific and Greek philosophers such as Leucippus and Democritus who argued that all the universe is composed of atoms and voids.

Circa 450 BC Empedocles imagined fundamental elements (fire (), earth (), air (), and water ()) and "forces" of attraction and repulsion allowing the elements to interact. Prior to this, Heraclitus had claimed that fire or change was fundamental to our existence, created through the combination of opposite properties.

In the Timaeus, Plato, following Pythagoras, considered mathematical entities such as number, point, line and triangle as the fundamental building blocks or elements of this ephemeral world, and considered the four elements of fire, air, water and earth as states of substances through which the true mathematical principles or elements would pass. A fifth element, the incorruptible quintessence aether, was considered to be the fundamental building block of the heavenly bodies.

The viewpoint of Leucippus and Empedocles, along with the aether, was accepted by Aristotle and passed to medieval and renaissance Europe.

Greek atomism

The earliest views on the shapes and connectivity of atoms was that proposed by Leucippus, Democritus, and Epicurus who reasoned that the solidness of the material corresponded to the shape of the atoms involved. Thus, iron atoms are solid and strong with hooks that lock them into a solid; water atoms are smooth and slippery; salt atoms, because of their taste, are sharp and pointed; and air atoms are light and whirling, pervading all other materials.

It was Democritus that was the main proponent of this view. Using analogies based on the experiences of the senses, he gave a picture or an image of an atom in which atoms were distinguished from each other by their shape, their size, and the arrangement of their parts. Moreover, connections were explained by material links in which single atoms were supplied with attachments: some with hooks and eyes others with balls and sockets (see diagram).

A water molecule as hook-and-eye model might have represented it. Leucippus, Democritus, Epicurus, Lucretius and Gassendi adhered to such conception. Note that the composition of water was not known before Avogadro (c. 1811).

17th century

With the rise of scholasticism and the decline of the Roman Empire, the atomic theory was abandoned for many ages in favor of the various four element theories and later alchemical theories. The 17th century, however, saw a resurgence in the atomic theory primarily through the works of Gassendi, and Newton.

Among other scientists of that time Gassendi deeply studied ancient history, wrote major works about Epicurus natural philosophy and was a persuasive propagandist of it. He reasoned that to account for the size and shape of atoms moving in a void could account for the properties of matter. Heat was due to small, round atoms; cold, to pyramidal atoms with sharp points, which accounted for the pricking sensation of severe cold; and solids were held together by interlacing hooks.

Newton, though he acknowledged the various atom attachment theories in vogue at the time, i.e. "hooked atoms", "glued atoms" (bodies at rest), and the "stick together by conspiring motions" theory, rather believed, as famously stated in "Query 31" of his 1704 Opticks, that particles attract one another by some force, which "in immediate contact is extremely strong, at small distances performs the chemical operations, and reaches not far from particles with any sensible effect."

In a more concrete manner, however, the concept of aggregates or units of bonded atoms, i.e. "molecules", traces its origins to Robert Boyle's 1661 hypothesis, in his famous treatise The Sceptical Chymist, that matter is composed of clusters of particles and that chemical change results from the rearrangement of the clusters. Boyle argued that matter's basic elements consisted of various sorts and sizes of particles, called "corpuscles", which were capable of arranging themselves into groups.

In 1680, using the corpuscular theory as a basis, French chemist Nicolas Lemery stipulated that the acidity of any substance consisted in its pointed particles, while alkalis were endowed with pores of various sizes. A molecule, according to this view, consisted of corpuscles united through a geometric locking of points and pores.

18th century

Étienne François Geoffroy’s 1718 Affinity Table: at the head of the column is a substance with which all the substances below can combine.

An early precursor to the idea of bonded "combinations of atoms", was the theory of "combination via chemical affinity". For example, in 1718, building on Boyle's conception of combinations of clusters, the French chemist Étienne François Geoffroy developed theories of chemical affinity to explain combinations of particles, reasoning that a certain alchemical "force" draws certain alchemical components together. Geoffroy's name is best known in connection with his tables of "affinities" (tables des rapports), which he presented to the French Academy in 1718 and 1720.

These were lists, prepared by collating observations on the actions of substances one upon another, showing the varying degrees of affinity exhibited by analogous bodies for different reagents. These tables retained their vogue for the rest of the century, until displaced by the profounder conceptions introduced by CL Berthollet.

In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica, which laid the basis for the kinetic theory of gases. In this work, Bernoulli positioned the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion. The theory was not immediately accepted, in part because conservation of energy had not yet been established, and it was not obvious to physicists how the collisions between molecules could be perfectly elastic.

In 1789, William Higgins published views on what he called combinations of "ultimate" particles, which foreshadowed the concept of valency bonds. If, for example, according to Higgins, the force between the ultimate particle of oxygen and the ultimate particle of nitrogen were 6, then the strength of the force would be divided accordingly, and similarly for the other combinations of ultimate particles:

William Higgins' combinations of ultimate particles (1789)

19th century

John Dalton's union of atoms combined in ratios (1808)

Similar to these views, in 1803 John Dalton took the atomic weight of hydrogen, the lightest element, as unity, and determined, for example, that the ratio for nitrous anhydride was 2 to 3 which gives the formula N₂O₃. Dalton incorrectly imagined that atoms "hooked" together to form molecules. Later, in 1808, Dalton published his famous diagram of combined "atoms":

Amedeo Avogadro created the word "molecule". His 1811 paper "Essay on Determining the Relative Masses of the Elementary Molecules of Bodies", he essentially states, i.e. according to Partington's A Short History of Chemistry, that:

The smallest particles of gases are not necessarily simple atoms, but are made up of a certain number of these atoms united by attraction to form a single molecule.

Note that this quote is not a literal translation. Avogadro uses the name "molecule" for both atoms and molecules. Specifically, he uses the name "elementary molecule" when referring to atoms and to complicate the matter also speaks of "compound molecules" and "composite molecules".

During his stay in Vercelli, Avogadro wrote a concise note (memoria) in which he declared the hypothesis of what we now call Avogadro's law: equal volumes of gases, at the same temperature and pressure, contain the same number of molecules. This law implies that the relationship occurring between the weights of same volumes of different gases, at the same temperature and pressure, corresponds to the relationship between respective molecular weights. Hence, relative molecular masses could now be calculated from the masses of gas samples.

Avogadro developed this hypothesis to reconcile Joseph Louis Gay-Lussac's 1808 law on volumes and combining gases with Dalton's 1803 atomic theory. The greatest difficulty Avogadro had to resolve was the huge confusion at that time regarding atoms and molecules—one of the most important contributions of Avogadro's work was clearly distinguishing one from the other, admitting that simple particles too could be composed of molecules and that these are composed of atoms. Dalton, by contrast, did not consider this possibility. Curiously, Avogadro considers only molecules containing even numbers of atoms; he does not say why odd numbers are left out.

In 1826, building on the work of Avogadro, the French chemist Jean-Baptiste Dumas states:

Gases in similar circumstances are composed of molecules or atoms placed at the same distance, which is the same as saying that they contain the same number in the same volume.

In coordination with these concepts, in 1833 the French chemist Marc Antoine Auguste Gaudin presented a clear account of Avogadro's hypothesis, regarding atomic weights, by making use of "volume diagrams", which clearly show both semi-correct molecular geometries, such as a linear water molecule, and correct molecular formulas, such as H₂O:

Marc Antoine Auguste Gaudin's volume diagrams of molecules in the gas phase (1833)

In two papers outlining his "theory of atomicity of the elements" (1857–58), Friedrich August Kekulé was the first to offer a theory of how every atom in an organic molecule was bonded to every other atom. He proposed that carbon atoms were tetravalent, and could bond to themselves to form the carbon skeletons of organic molecules.

In 1856, Scottish chemist Archibald Couper began research on the bromination of benzene at the laboratory of Charles Wurtz in Paris. One month after Kekulé's second paper appeared, Couper's independent and largely identical theory of molecular structure was published. He offered a very concrete idea of molecular structure, proposing that atoms joined to each other like modern-day Tinkertoys in specific three-dimensional structures. Couper was the first to use lines between atoms, in conjunction with the older method of using brackets, to represent bonds, and also postulated straight chains of atoms as the structures of some molecules, ring-shaped molecules of others, such as in tartaric acid and cyanuric acid. In later publications, Couper's bonds were represented using straight dotted lines (although it is not known if this is the typesetter's preference) such as with alcohol and oxalic acid below:

Archibald Couper's molecular structures, for alcohol and oxalic acid, using elemental symbols for atoms and lines for bonds (1858)

In 1861, an unknown Vienna high-school teacher named Joseph Loschmidt published, at his own expense, a booklet entitled Chemische Studien I, containing pioneering molecular images which showed both "ringed" structures as well as double-bonded structures, such as:

Joseph Loschmidt's molecule drawings of ethylene H₂C=CH₂ and acetylene HC≡CH (1861)

Loschmidt also suggested a possible formula for benzene, but left the issue open. The first proposal of the modern structure for benzene was due to Kekulé, in 1865. The cyclic nature of benzene was finally confirmed by the crystallographer Kathleen Lonsdale. Benzene presents a special problem in that, to account for all the bonds, there must be alternating double carbon bonds:

Benzene molecule with alternating double bonds

In 1865, German chemist August Wilhelm von Hofmann was the first to make stick-and-ball molecular models, which he used in lecture at the Royal Institution of Great Britain, such as methane shown below:

Hofmann's 1865 stick-and-ball model of methane CH₄.

The basis of this model followed the earlier 1855 suggestion by his colleague William Odling that carbon is tetravalent. Hofmann's color scheme, to note, is still used to this day: carbon = black, nitrogen = blue, oxygen = red, chlorine = green, sulfur = yellow, hydrogen = white. The deficiencies in Hofmann's model were essentially geometric: carbon bonding was shown as planar, rather than tetrahedral, and the atoms were out of proportion, e.g. carbon was smaller in size than the hydrogen.

In 1864, Scottish organic chemist Alexander Crum Brown began to draw pictures of molecules, in which he enclosed the symbols for atoms in circles, and used broken lines to connect the atoms together in a way that satisfied each atom's valence.

The year 1873, by many accounts, was a seminal point in the history of the development of the concept of the "molecule". In this year, the renowned Scottish physicist James Clerk Maxwell published his famous thirteen page article 'Molecules' in the September issue of Nature. In the opening section to this article, Maxwell clearly states:

An atom is a body which cannot be cut in two; a molecule is the smallest possible portion of a particular substance.

After speaking about the atomic theory of Democritus, Maxwell goes on to tell us that the word 'molecule' is a modern word. He states, "it does not occur in Johnson's Dictionary. The ideas it embodies are those belonging to modern chemistry." We are told that an 'atom' is a material point, invested and surrounded by 'potential forces' and that when 'flying molecules' strike against a solid body in constant succession it causes what is called pressure of air and other gases. At this point, however, Maxwell notes that no one has ever seen or handled a molecule.

In 1874, Jacobus Henricus van 't Hoff and Joseph Achille Le Bel independently proposed that the phenomenon of optical activity could be explained by assuming that the chemical bonds between carbon atoms and their neighbors were directed towards the corners of a regular tetrahedron. This led to a better understanding of the three-dimensional nature of molecules.

Emil Fischer developed the Fischer projection technique for viewing 3-D molecules on a 2-D sheet of paper:

In 1898, Ludwig Boltzmann, in his Lectures on Gas Theory, used the theory of valence to explain the phenomenon of gas phase molecular dissociation, and in doing so drew one of the first rudimentary yet detailed atomic orbital overlap drawings. Noting first the known fact that molecular iodine vapor dissociates into atoms at higher temperatures, Boltzmann states that we must explain the existence of molecules composed of two atoms, the "double atom" as Boltzmann calls it, by an attractive force acting between the two atoms. Boltzmann states that this chemical attraction, owing to certain facts of chemical valence, must be associated with a relatively small region on the surface of the atom called the sensitive region.

Boltzmann states that this "sensitive region" will lie on the surface of the atom, or may partially lie inside the atom, and will firmly be connected to it. Specifically, he states "only when two atoms are situated so that their sensitive regions are in contact, or partly overlap, will there be a chemical attraction between them. We then say that they are chemically bound to each other." This picture is detailed below, showing the α-sensitive region of atom-A overlapping with the β-sensitive region of atom-B:

Boltzmann’s 1898 I₂ molecule diagram showing atomic "sensitive region" (α, β) overlap.

20th century

In the early 20th century, the American chemist Gilbert N. Lewis began to use dots in lecture, while teaching undergraduates at Harvard, to represent the electrons around atoms. His students favored these drawings, which stimulated him in this direction. From these lectures, Lewis noted that elements with a certain number of electrons seemed to have a special stability. This phenomenon was pointed out by the German chemist Richard Abegg in 1904, to which Lewis referred to as "Abegg's law of valence" (now generally known as Abegg's rule). To Lewis it appeared that once a core of eight electrons has formed around a nucleus, the layer is filled, and a new layer is started. Lewis also noted that various ions with eight electrons also seemed to have a special stability. On these views, he proposed the rule of eight or octet rule: Ions or atoms with a filled layer of eight electrons have a special stability.

Moreover, noting that a cube has eight corners Lewis envisioned an atom as having eight sides available for electrons, like the corner of a cube. Subsequently, in 1902 he devised a conception in which cubic atoms can bond on their sides to form cubic-structured molecules.

In other words, electron-pair bonds are formed when two atoms share an edge, as in structure C below. This results in the sharing of two electrons. Similarly, charged ionic-bonds are formed by the transfer of an electron from one cube to another, without sharing an edge A. An intermediate state B where only one corner is shared was also postulated by Lewis.

Lewis cubic-atoms bonding to form cubic-molecules

Hence, double bonds are formed by sharing a face between two cubic atoms. This results in the sharing of four electrons.

In 1913, while working as the chair of the department of chemistry at the University of California, Berkeley, Lewis read a preliminary outline of paper by an English graduate student, Alfred Lauck Parson, who was visiting Berkeley for a year. In this paper, Parson suggested that the electron is not merely an electric charge but is also a small magnet (or "magneton" as he called it) and furthermore that a chemical bond results from two electrons being shared between two atoms. This, according to Lewis, meant that bonding occurred when two electrons formed a shared edge between two complete cubes.

On these views, in his famous 1916 article The Atom and the Molecule, Lewis introduced the "Lewis structure" to represent atoms and molecules, where dots represent electrons and lines represent covalent bonds. In this article, he developed the concept of the electron-pair bond, in which two atoms may share one to six electrons, thus forming the single electron bond, a single bond, a double bond, or a triple bond.

Lewis-type Chemical bond

In Lewis' own words:

An electron may form a part of the shell of two different atoms and cannot be said to belong to either one exclusively.

Moreover, he proposed that an atom tended to form an ion by gaining or losing the number of electrons needed to complete a cube. Thus, Lewis structures show each atom in the structure of the molecule using its chemical symbol. Lines are drawn between atoms that are bonded to one another; occasionally, pairs of dots are used instead of lines. Excess electrons that form lone pairs are represented as pair of dots, and are placed next to the atoms on which they reside:

Lewis dot structures of the Nitrite-ion

To summarize his views on his new bonding model, Lewis states:

Two atoms may conform to the rule of eight, or the octet rule, not only by the transfer of electrons from one atom to another, but also by sharing one or more pairs of electrons...Two electrons thus coupled together, when lying between two atomic centers, and held jointly in the shells of the two atoms, I have considered to be the chemical bond. We thus have a concrete picture of that physical entity, that "hook and eye" which is part of the creed of the organic chemist.

The following year, in 1917, an unknown American undergraduate chemical engineer named Linus Pauling was learning the Dalton hook-and-eye bonding method at the Oregon Agricultural College, which was the vogue description of bonds between atoms at the time. Each atom had a certain number of hooks that allowed it to attach to other atoms, and a certain number of eyes that allowed other atoms to attach to it. A chemical bond resulted when a hook and eye connected. Pauling, however, wasn't satisfied with this archaic method and looked to the newly emerging field of quantum physics for a new method.

In 1927, the physicists Fritz London and Walter Heitler applied the new quantum mechanics to the deal with the saturable, nondynamic forces of attraction and repulsion, i.e., exchange forces, of the hydrogen molecule. Their valence bond treatment of this problem, in their joint paper, was a landmark in that it brought chemistry under quantum mechanics. Their work was an influence on Pauling, who had just received his doctorate and visited Heitler and London in Zürich on a Guggenheim Fellowship.

Subsequently, in 1931, building on the work of Heitler and London and on theories found in Lewis' famous article, Pauling published his ground-breaking article "The Nature of the Chemical Bond" (see: manuscript) in which he used quantum mechanics to calculate properties and structures of molecules, such as angles between bonds and rotation about bonds. On these concepts, Pauling developed hybridization theory to account for bonds in molecules such as CH₄, in which four sp³ hybridised orbitals are overlapped by hydrogen's 1s orbital, yielding four sigma (σ) bonds. The four bonds are of the same length and strength, which yields a molecular structure as shown below:

A schematic presentation of hybrid orbitals overlapping hydrogens' s orbitals

Owing to these exceptional theories, Pauling won the 1954 Nobel Prize in Chemistry. Notably he has been the only person to ever win two unshared Nobel prizes, winning the Nobel Peace Prize in 1963.

In 1926, French physicist Jean Perrin received the Nobel Prize in physics for proving, conclusively, the existence of molecules. He did this by calculating the Avogadro number using three different methods, all involving liquid phase systems. First, he used a gamboge soap-like emulsion, second by doing experimental work on Brownian motion, and third by confirming Einstein's theory of particle rotation in the liquid phase.

In 1937, chemist K.L. Wolf introduced the concept of supermolecules (Übermoleküle) to describe hydrogen bonding in acetic acid dimers. This would eventually lead to the area of supermolecular chemistry, which is the study of non-covalent bonding.

In 1951, physicist Erwin Wilhelm Müller invents the field ion microscope and is the first to see atoms, e.g. bonded atomic arrangements at the tip of a metal point.

In 1968-1970 Leroy Cooper, PhD of the University of California at Davis completed his thesis which showed what molecules looked like. He used x-ray deflection off crystals and a complex computer program written by Bill Pentz of the UC Davis Computer Center. This program took the mapped deflections and used them to calculate the basic shapes of crystal molecules. His work showed that actual molecular shapes in quartz crystals and other tested crystals looked similar to the long envisioned merged various sized soap bubbles theorized, except instead of being merged spheres of different sizes, actual shapes were rigid mergers of more tear dropped shapes that stayed fixed in orientation. This work verified for the first time that crystal molecules are actually linked or stacked merged tear drop constructions. 

In 1999, researchers from the University of Vienna reported results from experiments on wave-particle duality for C₆₀ molecules. The data published by Anton Zeilinger et al. were consistent with Louis de Broglie's matter waves. This experiment was noted for extending the applicability of wave–particle duality by about one order of magnitude in the macroscopic direction.

In 2009, researchers from IBM managed to take the first picture of a real molecule. Using an atomic force microscope every single atom and bond of a pentacene molecule could be imaged.