Kinetic theory

The temperature of an ideal monatomic gas is a measure related to the average kinetic energy of its atoms as they move. In this animation, the size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. These room-temperature atoms have a certain, average speed (slowed down here two trillion fold).

Kinetic theory (or kinetic theory of gases) attempts to explain macroscopic properties of gases, such as pressure, temperature, or volume, by considering their molecular composition and motion. Essentially, the theory posits that pressure is due not to static repulsion between molecules, as was Isaac Newton's conjecture, but due to collisions between molecules moving at different velocities. Kinetic theory is also known as the kinetic-molecular theory or the collision theory.

Contents

Postulates

The theory for ideal gases makes the following assumptions:

In addition, if the gas is in a container, the collisions with the walls are assumed to be instantaneous and elastic.

More modern developments relax these assumptions and are based on the Boltzmann equation. These can accurately describe the properties of dense gases, because they include the volume of the molecules. The necessary assumptions are the absence of quantum effects, molecular chaos and small gradients in bulk properties. Expansions to higher orders in the density are known as virial expansions. The definitive work is the book by Chapman and Enskog but there have been many modern developments and there is an alternative approach developed by Grad based on moment expansions. In the other limit, for extremely rarefied gases, the gradients in bulk properties are not small compared to the mean free paths. This is known as the Knudsen regime and expansions can be performed in the Knudsen number.

The kinetic theory has also been extended to include inelastic collisions in granular matter by Jenkins, Jwang and others.

Pressure

Pressure is explained by kinetic theory as arising from the force exerted by gas molecules impacting on the walls of the container. Consider a gas of N molecules, each of mass m, enclosed in a cuboidal container of volume V. When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision), then the momentum lost by the particle and gained by the wall is:

\Delta p_x = p_i - p_f = 2 m v_x\,

where vx is the x-component of the initial velocity of the particle.

The particle impacts the wall once every 2l/vx time units (where l is the length of the container). Although the particle impacts a side wall once every 1l/vx time units, only the momentum change on one wall is considered so that the particle produces a momentum change on a particular wall once every 2l/vx time units.

\Delta t = \frac{2l}{v_x}

The force due to this particle is:

F = \frac{\Delta p}{\Delta t} = \frac{2 m v_x}{\frac{2l}{v_x}} = \frac{m v_x^2}{l}

The total force acting on the wall is:

F = \frac{m\sum_j v_{jx}^2}{l}

where the summation is over all the gas molecules in the container.

The magnitude of the velocity for each particle will follow:

v^2 = v_x^2 + v_y^2 + v_z^2

Now considering the total force acting on all six walls, adding the contributions from each direction we have:

\mbox{Total Force} = 2 \cdot \frac{m}{l}(\sum_j v_{jx}^2 + \sum_j v_{jy}^2 + \sum_j v_{jz}^2) = 2 \cdot \frac{m}{l} \sum_j (v_{jx}^2 + v_{jy}^2 + v_{jz}^2) = 2 \cdot \frac{m \sum_j v_{j}^2}{l}

where the factor of two arises from now considering both walls in a given direction.

Assuming there are a large number of particles moving sufficiently randomly, the force on each of the walls will be approximately the same and now considering the force on only one wall we have:

F = \frac{1}{6} \left(2 \cdot \frac{m \sum_j v_{j}^2}{l}\right) = \frac{m \sum_j v_{j}^2}{3l}

The quantity \sum_j v_{j}^2 can be written as {N} \overline{v^2}, where the bar denotes an average, in this case an average over all particles. This quantity is also denoted by v_{rms}^2 where v_{rms} is the root-mean-square velocity of the collection of particles.

Thus the force can be written as:

F = \frac{Nmv_{rms}^2}{3l}

Pressure, which is force per unit area, of the gas can then be written as:

P = \frac{F}{A} = \frac{Nmv_{rms}^2}{3Al}

where A is the area of the wall of which the force exerted on is considered.

Thus, as cross-sectional area multiplied by length is equal to volume, we have the following expression for the pressure

P = {Nmv_{rms}^2 \over 3V}

where V is the volume. Also, as Nm is the total mass of the gas, and mass divided by volume is density

P = {1 \over 3} \rho\ v_{rms}^2

where ρ is the density of the gas.

This result is interesting and significant, because it relates pressure, a macroscopic property, to the average (translational) kinetic energy per molecule (1/2mvrms2), which is a microscopic property. Note that the product of pressure and volume is simply two thirds of the total kinetic energy.

Temperature and kinetic energy

From the ideal gas law,

\displaystyle PV = N k_B T

(1)

where \displaystyle k_B is the Boltzmann constant, and \displaystyle T the absolute temperature, it follows from the above result that the temperature \displaystyle T takes the form

\displaystyle PV = N k_B T = \frac {N m v_{rms}^2} {3} \Longrightarrow T = \frac {m v_{rms}^2} {3 k_B}

(2)

and the kinetic energy \displaystyle K of the system can now be written as

\displaystyle K = \frac {1} {2} N m v_{rms}^2 = \frac {3} {2} N k_B T \ {\rm and} \ T = \frac {2} {3} \frac {K} {N k_B}

(3)

Eq.(3)1 is one important result of the kinetic theory: The average molecular kinetic energy is proportional to the absolute temperature.

From Eq.(1) and Eq.(3)1, we have

\displaystyle PV = \frac {2} {3} K

(4)

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

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

Since there are \displaystyle 3N degrees of freedom (dofs) in a monoatomic-gas system with \displaystyle N particles, the kinetic energy per dof is

\displaystyle \frac {K} {3 N} = \frac {k_B T} {2}

(5)

In the kinetic energy per dof, the constant of proportionality of temperature is 1/2 times Boltzmann constant. This result is related to the equipartition theorem.

As noted in the article on heat capacity, diatomic gases should have 7 degrees of freedom, but the lighter gases act as if they have only 5.

Thus the kinetic energy per kelvin (monatomic ideal gas) is:

At standard temperature (273.15 K), we get:

Number of collisions with wall

One can calculate the number of atomic or molecular collisions with a wall of a container per unit area per unit time.

Assuming an ideal gas, a derivation[2] results in an equation for total number of collisions per unit time per area:

A = \frac{1}{4}\frac{N}{V} v_{avg} = \frac{\rho}{4} \sqrt{\frac{8 k T}{\pi m}} \frac{1}{m} \,

RMS speeds of molecules

From the kinetic energy formula it can be shown that

v_{rms}^2 = \frac{3RT}{\mbox{molar mass}}

with v in m/s, T in kelvins, and R is the gas constant. The molar mass is given as kg/mol. The most probable speed is 81.6% of the rms speed, and the mean speeds 92.1% (distribution of speeds).

History

In 1740 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.

Other pioneers of the kinetic theory (which were neglected by their contemporaries) were Mikhail Lomonosov (1747),[3] Georges-Louis Le Sage (ca. 1780, published 1818),[4] John Herapath (1816)[5] and John James Waterston (1843),[6] which connected their research with the development of mechanical explanations of gravitation. In 1856 August Krönig (probably after reading a paper of Waterston) created a simple gas-kinetic model, which only considered the translational motion of the particles. [7]

In 1857 Rudolf Clausius, according to his own words independently of Krönig, developed a similar, but much 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. [8] In 1859, after reading a paper by Clausius, 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.[9] 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.”[10] In 1871, Ludwig Boltzmann generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution. Also the logarithmic connection between entropy and probability was first stated by him.

In the beginning of twentieth century, however, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was Albert Einstein's (1905)[11] and Marian Smoluchowski's (1906)[12] papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.

See also

References

Endnotes
  1. ↑ Configuration integral (statistical mechanics)
  2. ↑ Collisions With a Surface
  3. ↑ Lomovosov 1758
  4. ↑ Le Sage 1780/1818
  5. ↑ Herapath 1816, 1821
  6. ↑ Waterston 1843
  7. ↑ Krönig 1856
  8. ↑ Clausius 1857
  9. ↑ Mahon 2003
  10. ↑ Maxwell 1875
  11. ↑ Einstein 1905
  12. ↑ Smoluchowski 1906

The Mathematical Theory of Non-uniform Gases : An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases Sydney Chapman, T. G. Cowling

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