Sci-fi physics: gamma equal to 2

Problem

Imagine a gas with $\gamma = 2$. What would its characteristics be? How many quadratic degrees of freedom would it have, and what physical situation would it correspond to?

Solution

From the specific heats. Mayer’s relation requires $c_p - c_V = R$. With $\gamma = c_p/c_V = 2$ we have $c_p = 2\,c_V$, hence:

$2\,c_V - c_V = R\quad\Rightarrow\quad c_V = R,\qquad c_p = 2R$

Degrees of freedom. By the equipartition theorem $c_V = \frac{f}{2}R$, so:

$\frac{f}{2}R = R\quad\Rightarrow\quad f = 2$

only two quadratic degrees of freedom.

Physical situation. This corresponds to a gas whose motion is confined to a plane (2D), for example molecules constrained to slide over a wall or a surface: only two translational directions contribute to the energy.

Adiabatic. The law $pV^{\gamma} = pV^2 = \text{const}$ would be steeper than that of a monatomic gas ($\gamma = 5/3$): for the same compression, the pressure would rise more rapidly.

$\ev{c_V = R,\ c_p = 2R,\ f = 2\ \text{(2D motion)}}$

Links

Topics: Thermodynamics Concepts: Specific heat and heat capacity Skills: Analysis of limiting cases and science-fiction physics Objects: Ideal gas