Kvant Physics Problem 4

A fixed mass $m = 1,\mathrm{kg}$ of an unknown gas is considered under two thermodynamic processes: heating at constant pressure and heating at constant volume.

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Problem

In a cylinder there is a purified gas, but its identity is unknown. To raise the temperature of 1 kg of this gas by one degree at constant pressure requires 958.4 J, and at constant volume — 704.6 J. What gas is it?

III All-Union Physics Olympiad

Setup and Assumptions

A fixed mass $m = 1,\mathrm{kg}$ of an unknown gas is considered under two thermodynamic processes: heating at constant pressure and heating at constant volume. The measured molar properties are given in mass-specific form as heat capacities per unit mass: $C_p = 958.4,\mathrm{J,kg^{-1},K^{-1}}$ and $C_v = 704.6,\mathrm{J,kg^{-1},K^{-1}}$. The task is to determine the identity of the gas.

The gas is treated as ideal, so intermolecular interactions are neglected except insofar as they determine macroscopic thermodynamic constants. The heat capacities are assumed constant over the temperature interval of interest. The gas is assumed chemically pure and thermodynamically stable in the relevant range.

Physical Principles

For an ideal gas, the difference between specific heat capacities at constant pressure and constant volume equals the specific gas constant,

$C_p - C_v = R_{\mathrm{spec}}.$

The specific gas constant is related to the universal gas constant $R$ and the molar mass $M$ by

$R_{\mathrm{spec}} = \frac{R}{M}.$

The universal gas constant is

$R = 8.314,\mathrm{J,mol^{-1},K^{-1}}.$

Derivation

The specific gas constant is obtained directly from the measured heat capacities,

$R_{\mathrm{spec}} = C_p - C_v = 958.4,\mathrm{J,kg^{-1},K^{-1}} - 704.6,\mathrm{J,kg^{-1},K^{-1}} = 253.8,\mathrm{J,kg^{-1},K^{-1}}.$

The molar mass follows from the relation between $R$ and $R_{\mathrm{spec}}$,

$M = \frac{R}{R_{\mathrm{spec}}}.$

Substituting the numerical values,

$M = \frac{8.314,\mathrm{J,mol^{-1},K^{-1}}}{253.8,\mathrm{J,kg^{-1},K^{-1}}} = 0.03276,\mathrm{kg,mol^{-1}}.$

Converting to grams per mole,

$M = 32.76,\mathrm{g,mol^{-1}}.$

Result

The molar mass of the gas is

$M = \frac{R}{C_p - C_v} = \frac{8.314,\mathrm{J,mol^{-1},K^{-1}}}{253.8,\mathrm{J,kg^{-1},K^{-1}}} = 0.03276,\mathrm{kg,mol^{-1}} = 32.76,\mathrm{g,mol^{-1}}.$

The molar mass closest to this value corresponds to oxygen gas,

$\boxed{\mathrm{O_2}}.$

Sanity Checks

The difference $C_p - C_v$ has units $\mathrm{J,kg^{-1},K^{-1}}$, matching the required units of a specific gas constant, and the ratio $R/R_{\mathrm{spec}}$ yields units of $\mathrm{kg,mol^{-1}}$, confirming dimensional consistency.

The computed molar mass $32.76,\mathrm{g,mol^{-1}}$ is very close to the tabulated value for molecular oxygen, $32,\mathrm{g,mol^{-1}}$, and significantly closer to this value than to nearby common gases such as nitrogen at $28,\mathrm{g,mol^{-1}}$ or carbon monoxide at $28,\mathrm{g,mol^{-1}}$. The small discrepancy is consistent with experimental rounding in the given heat capacities.