Thermodynamics Terms

The correct option is v₅ > v₃ and 𝑈₃ > 𝑈₅. To understand why, we need to analyze how speed of sound and internal energy relate to the degrees of freedom, n, of the gas molecules.

1. Internal Energy (U): For an ideal gas, the internal energy per mole in terms of degrees of freedom, n, is given by:

U = (n/2)RT

where R is the ideal gas constant and T is the temperature. This implies that internal energy increases with degrees of freedom.

2. Speed of Sound (v): The speed of sound in an ideal gas is given by:

v = √(γRT/M)

where γ is the adiabatic index (Cp/Cv), R is the ideal gas constant, T is the temperature, and M is the molar mass. Here, the adiabatic index γ is related to the degrees of freedom by:

γ = (n + 2)/n

Because the adiabatic index decreases with an increase in degrees of freedom, the speed of sound decreases as the degrees of freedom increase.

In conclusion, as the number of degrees of freedom increases, internal energy increases but the speed of sound decreases. Therefore, for a gas with degrees of freedom 5 (n=5) and another with 3 (n=3), we have:

v₅ > v₃ (speed of sound decreases with more degrees of freedom)

U₃ > U₅ (internal energy increases with more degrees of freedom)