Problem — Sci-fi physics: gas in five dimensions

Problem

In a space with $d = 5$ spatial dimensions, how many translational degrees of freedom does a monatomic molecule have? Calculate $\bar E$ per molecule and the molar $c_V$.

Solution

In a $d$-dimensional space a molecule can move along $d$ independent directions, so it has $d$ translational degrees of freedom: $f = d = 5$ By the equipartition theorem each degree of freedom carries a mean energy $\frac{1}{2}k_B T$, so the mean energy per molecule is $\bar E = \frac{f}{2}\,k_B T = \frac{5}{2}\,k_B T$ The molar heat capacity at constant volume is $c_V = \frac{f}{2}R$: $c_V = \frac{5}{2}R$ $\ev{f = 5, \quad \bar E = \tfrac{5}{2}k_B T, \quad c_V = \tfrac{5}{2}R}$ Curiously, in $d = 5$ a monatomic gas has the same $c_V$ as a diatomic gas in $d = 3$ (where $f = 5$ between translations and rotations): what matters is only the total number of quadratic degrees of freedom.

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Topics: Kinetic theory of gases Concepts: Internal energy Skills: Analysis of limiting cases and sci-fi physics