An ideal gas of N monatomic molecules is in thermal equilibrium with an ideal gas of the same number of diatomic molecules and equilibrium is maintained as temperature is increased.The ratio of the changes in the internal energies ΔE_{dia /}ΔE_mon is: A)1/2 B)3/5 C)1 D)5/3 E)2

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Quizplus · Accepted Answer

The internal energy change for an ideal gas depends on its degrees of freedom. A monatomic gas has 3 translational degrees of freedom, while a diatomic gas, at sufficiently high temperatures where vibrational modes are excited, can have up to 7 degrees of freedom (3 translational, 2 rotational, and 2 vibrational). The change in internal energy ($\Delta U$) for an ideal gas is given by $\Delta U = nC_v\Delta T$, where $C_v$ is the molar specific heat at constant volume, and $n$ is the amount of substance. Since $C_v$ is proportional to the degrees of freedom ($f$), we can say $\Delta U \propto f\Delta T$. For monatomic gas $f = 3$, and for diatomic gas at high temperatures $f = 7$. Therefore, the ratio of the changes in internal energies is $\frac{7}{3} = \frac{5}{3}$.

An Ideal Gas of N Monatomic Molecules Is in Thermal