Question

For a certain thermodynamic system, the internal energy U = PV and P is proportional to T². The entropy of the system is proportional to

• A. UV
• B. \(\sqrt{\frac{U}{V}}\)
• C. \(\sqrt{\frac{V}{U}}\)
• D. \(\sqrt{UV}\)

Answer 1

The Correct Option is D

To determine the entropy of the thermodynamic system, we start by interpreting the given information:

1. The internal energy \( U \) is given as \( U = PV \), suggesting the relationship between pressure (\( P \)), volume (\( V \)), and internal energy.
2. It is also provided that \( P \) is proportional to \( T^2 \), i.e., \( P = kT^2 \), where \( k \) is a proportionality constant.

Hence, we can express the internal energy \( U \) as:

\(U = P \times V = kT^2 \times V\)

This indicates that \( U \) is proportional to \( T^2V \).

We need to find the entropy \( S \) of the system, for which we consider the relationship involving temperature (\( T \)) and internal energy (\( U \)). The basic thermodynamic identity in terms of internal energy, entropy, volume, and temperature is:

\(dU = TdS - PdV\)

Given \( P = kT^2 \), we substitute in the above equation:

\(dU = TdS - kT^2dV\)

Simplifying the expression and considering small changes, it can be derived that entropy \( S \) that follows these conditions should depend on \( UV \). However, we notice the problem requests an expression for \( S \), which leads to exploring the functional relationship further with given options.

Given the options and upon derivations involving standard thermodynamic identities or common proportionality in thermodynamics, the most suitable entropy expression in terms of \( U \) and \( V \) is:

\(S \propto \sqrt{UV}\)

Therefore, the entropy of the system is proportional to \(\sqrt{UV}\).