Inversion temperature

The inversion temperature in liquefaction of gases.

Theory

The Joule-Thomson effect cannot be described in the theory of Van der Waals force between interacting particles that becomes much stronger as a gas becomes a liquid.

For a statistical mechanics as

$H = \frac{5}{2} N k_B T + \frac{N^2}{V} (b k_B T - 2a)$ ,

where $N$ is the number of molecules, $V$ is volume, $T$ is temperature (in the Kelvin scale), $k B$ is Boltzmann's constant, and $a$ and $b$ are constants depending on intermolecular forces and molecular volume, respectively.

From this equation, we note that if we keep enthalpy constant and increase volume, temperature must change depending on the sign of $b k B T - 2 a$ . Therefore, our inversion temperature is given where the sign flips at zero, or

$k_B T_\textrm{inv} = 2a / b = \frac{27}{4} k_B T_c$ ,

where $T c$ is the critical temperature of the substance. So for $T > T inv$ , an expansion at constant enthalpy increases temperature as the work done by the repulsive interactions of the gas is dominant, and so the change in energy is negative. But for $T < T inv$ , expansion causes temperature to decrease because the work of attractive intermolecular forces dominates, giving a positive change in energy. ^[1]

References

^ Charles Kittel and Herbert Kroemer (1980). Thermal Physics, 2nd Edition, W.H. Freeman. ISBN 0-7167-1088-9.

Categories: Chemical engineering

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Inversion temperature

Theory

See also

References