The question in the image asks about the dependence of entropy on temperature for an ideal gas at constant volume and offers four possible answers:

a. Logarithmic function
b. Quadratic function
c. Exponential function
d. Linear function

Correct Answer:

a. Logarithmic function

Explanation:

For an ideal gas, the entropy $S$ at constant volume depends on the logarithm of the temperature $T$. The general relation for entropy at constant volume is derived from thermodynamic principles:

$S = S_0 + nC_v \ln T$

Where:

• $S_0$ is the reference entropy,
• $n$ is the number of moles,
• $C_v$ is the molar heat capacity at constant volume,
• $T$ is the temperature.

This relationship clearly shows that the entropy depends logarithmically on temperature when volume is held constant.

Would you like me to expand on this with additional derivations or examples?

Related Questions:

1. What is the physical meaning of entropy in thermodynamics?
2. How does entropy change at constant pressure for an ideal gas?
3. What is the role of $C_v$ in determining entropy?
4. How does the logarithmic relationship affect real-world systems?
5. What other factors can influence entropy besides temperature and volume?

Tip:

Always remember that the logarithmic dependency of entropy on temperature comes from the mathematical integration of the thermodynamic definition of entropy for a system.