Question

A gas has n degrees of freedom. The ratio of specific heat of gas at constant volume to the specific heat of gas at constant pressure will be

• A. \(\frac{n}{n+2}\)
• B. \(\frac{n+2}{n}\)
• C. \(\frac{n}{2n+2}\)
• D. \(\frac{n}{n-2}\)

Answer 1

The Correct Option is A

To determine the ratio of the specific heat of a gas at constant volume, \(C_v\), to the specific heat of the gas at constant pressure, \(C_p\), we start by revisiting the relationships given by the kinetic theory of gases.

The degrees of freedom of a gas, represented by \(n\), is directly related to the capacities \(C_v\) and \(C_p\). For a general polyatomic gas with \(n\) degrees of freedom:

• The specific heat at constant volume is given by: C_v = \frac{n}{2} R
• The specific heat at constant pressure is: C_p = C_v + R

Using the formula for the degrees of freedom, we have:

1. Calculate \(C_p\):
   • \(C_p = C_v + R = \frac{n}{2} R + R = \left(\frac{n}{2} + 1\right) R\)
2. Find the ratio \(\frac{C_v}{C_p}\):
   • Substituting the values from steps 1 and 2: \(\frac{C_v}{C_p} = \frac{\frac{n}{2} R}{\left(\frac{n}{2} + 1\right) R}\)
   • This simplifies to: \(\frac{n}{n + 2}\)

Thus, the correct answer is \(\frac{n}{n+2}\).

Let's rule out the incorrect options:

• \(\frac{n+2}{n}\): This incorrectly inverts the relationship.
• \(\frac{n}{2n+2}\): This calculation does not align with the standard formula for the degrees of freedom in relation to specific heat capacities.
• \(\frac{n}{n-2}\): This does not fit with the calculation where \(C_v\) is reduced instead of increased by \(R\).

Therefore, based on the standard formulas used in the kinetic theory of gases, the ratio \(\frac{C_v}{C_p}\) is \(\frac{n}{n+2}\).