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An ideal gas is in thermodynamic equilibrium at fixed temperature and pressure. A molecule of the gas has n degrees of freedom. Let Uₙ denote the molar internal energy and vₙ denote the speed of sound in the gas when the degrees of freedom are n. Which of the following statements is correct?
- v₃ < v₆ and U₃ > U₆
- v₅ > v₃ and U₃ > U₅
- v₅ > v₇ and U₅ < U₇
- v₆ < v₇ and U₆ < U₇
Correct answer: v₅ > v₇ and U₅ < U₇
Solution
Since gammaₙ = 1 + 2/n decreases as n increases, the speed of sound vₙ = sqrt(gammaₙ*RT/M) also decreases with n. Meanwhile Uₙ = (n/2)RT increases with n. So v₅ > v₇ (since 5 < 7) and U₅ < U₇ (since 5 < 7), making this the only self-consistent option.
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