An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is n . The internal energy of one mole of the gas is U n and the speed of sound in the gas is v

Question

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is n . The internal energy of one mole of the gas is U n and the speed of sound in the gas is v n . At a fixed temperature and pressure, which of the following is the correct option?

TestHub · Accepted Answer

As we know, internal energy of one mole of the gas $U_{n} = \frac{n R T}{2}$ .
Clearly, $U_{n} \propto n$ , where $n$ is degree of freedom. Therefore, as $n$ increases, $U_{n}$ also increases.
Now, speed of sound in gas
$v_{n} = \sqrt{\frac{γ R T}{M}} = \sqrt{\frac{(1 + \frac{2}{n}) R T}{M}}$
Clearly, as $n$ increases and $v_{n}$ decreases.
Hence,
$U_{7} > U_{5}$ and $U_{7} > U_{6}$
and $v_{5} > v_{7}$ .

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