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 𝑈𝑛 and the speed of sound in the gas is v𝑛. At a fixed temperature and pressure, which of the following is the correct option?

• A. v3 < v6 and 𝑈3 > 𝑈6
• B. v5 > v3 and 𝑈3 > 𝑈5
• C. v5 > v7 and 𝑈5 < 𝑈7
• D. v6 < v7 and 𝑈6 < 𝑈7

Answer 1

Answer: (C)

v5 > v7 and 𝑈5 < 𝑈7

Answer 2

For an ideal gas, the number of degrees of freedom (n) determines the value of γ. For a monatomic gas (such as helium or argon), the number of degrees of freedom is 3, and γ is equal to $\frac{5}{3}$.
For a diatomic gas (such as oxygen or nitrogen), the number of degrees of freedom is 5, and γ is equal to$\frac{7}{5}$.
Now, let's analyze the options:(3) $v_5 > v_7$ and $𝑈_5 < 𝑈_7$:
This option states that the speed of sound in gas 5 is greater than in gas 7, which is true because a gas with a higher number of degrees of freedom will have a higher speed of sound. It also states that the internal energy of gas 5 is less than gas 7, which is true as well because higher degrees of freedom correspond to higher internal energy. Therefore, this option is correct.
Therefore, the correct option is (3) $v_5 > v_7$ and $𝑈_5 < 𝑈_7.$