Question

A polyatomic ideal gas has 24 vibrational modes. What is the value of \[\gamma \] ?
A. 1.03
B. 1.30
C. 10.3
D. 1.37

Answer 1

Hint:Before we start addressing the problem, we need to know about the polyatomic gas and degrees of freedom. A polyatomic gas is one that has more than 2 atoms in one molecule and it must have more than or equal to 6 degrees of freedom. The degree of freedom is defined as the number of coordinates required to specify the position of all the atoms in a molecule.

Formula Used:
To find the formula for \[\gamma \]we have,
\[\gamma = 1 + \dfrac{2}{f}\]
Where, f is the number of degrees of freedom.

Complete step by step solution:
The equation for \[\gamma \] is,
\[\gamma = 1 + \dfrac{2}{f}\] ………… (1)
Since, the number of degrees of freedom is due to translation, rotation, vibration, then
\[f = T + R + V\]
That is since each vibrational mode has 2 degrees of freedom hence total vibrational degrees of freedom is 48.
\[f = 3 + 3 + 48\]
\[\Rightarrow f = 54\]
Now, substitute the value of f in the equation (1) we get,
\[\gamma = 1 + \dfrac{2}{{54}}\]
\[\therefore \gamma = 1.037\]
Therefore, the value of \[\gamma \] is 1.03.

Hence, Option A is the correct answer

Note:The ratio of the specific heats \[\gamma = \dfrac{{{C_P}}}{{{C_V}}}\] is a factor that determines the speed of sound in a gas. A polyatomic gas has basically three translations that are, translational, rotational, and vibrational mode as used in this question. Hence, the degree of freedom for polyatomic gas is greater than or equal to 6. The value of \[\gamma \] for the polyatomic gas is greater than one. There are some examples of polyatomic gases, they are helium, radon, neon, xenon, argon, etc.