Question

The degree of freedom of an ideal gas is n. The internal energy of 1 mole of the gas is U_n and the speed of sound n the gas is V_n. At a fixed temperature and pressure, which of the following is correct?

• A. V₅>V₇ and U₅>U₇
• B. V₅>V₃ and U₃>U₅
• C. V₃<V₆ and U₃>U₆
• D. V₆>V₇ and U₆<U₇

Answer 1

The Correct Option is D

Sound Velocity and Internal Energy: Explanation

The correct option is (D): \( V_6 > V_7 \) and \( U_6 < U_7 \). 

The formulas provided for the sound velocity and internal energy are:

\(V = \sqrt{\frac{\gamma RT}{M_0}}\)
where \( \gamma = 1 + \frac{2}{n} \),

and

\(U = n' \frac{n}{2RT}\)

Explanation:

Let's break down the formulas and the effect of the number of degrees of freedom (\(n\)):

1. Sound Velocity Equation:

The formula for sound velocity \(V\) is:

\(V = \sqrt{\frac{\gamma RT}{M_0}}\)

Here, \(V\) is the sound velocity, \(R\) is the gas constant, \(T\) is the temperature, and \(M_0\) is the molar mass.

As \(n\) (the number of degrees of freedom) increases, \(\gamma\) decreases because:

\(\gamma = 1 + \frac{2}{n}\)

Since \(\gamma\) appears in the numerator of the sound velocity equation, a decrease in \(\gamma\) leads to a decrease in the sound velocity \(V\).

2. Internal Energy Equation:

The formula for internal energy \(U\) is:

\(U = n' \frac{n}{2RT}\)

Here, \(n'\) is a constant and \(n\) represents the number of degrees of freedom. As \(n\) increases, the internal energy \(U\) increases because \(U\) is directly proportional to \(n\).

Conclusion:

Thus, when \(n\) increases:

• \(\gamma\) decreases, which results in a decrease in the sound velocity (\(V\)).
• The internal energy \(U\) increases because it is directly proportional to \(n\).

Therefore, the correct option (D) is confirmed: \( V_6 > V_7 \) and \( U_6 < U_7 \).