Question

The fundamental thermodynamic relation for a rubber band is given by 
\[ dU = TdS + \tau dL \] where \( T \) is the absolute temperature, \( S \) is the entropy, \( \tau \) is the tension in the rubber band, and \( L \) is the length of the rubber band.
Which one of the following relations is CORRECT: 
 

• A. \( \tau = \left( \frac{\partial U}{\partial S} \right)_L \)
• B. \( \left( \frac{\partial T}{\partial L} \right)_S = \left( \frac{\partial \tau}{\partial S} \right)_L \)
• C. \( \left( \frac{\partial T}{\partial S} \right)_L = \left( \frac{\partial \tau}{\partial L} \right)_S \)
• D. \( T = \left( \frac{\partial U}{\partial S} \right)_\tau \)

Hint:

In thermodynamics, partial derivatives with respect to different variables describe how a system's properties change with respect to those variables while holding others constant.

Answer 1

The Correct Option is A

From the fundamental thermodynamic relation for the rubber band: \[ dU = TdS + \tau dL \] This equation expresses the changes in internal energy \( U \) in terms of changes in entropy \( S \) and the length \( L \) of the rubber band. The terms \( T \) and \( \tau \) are the partial derivatives of \( U \) with respect to \( S \) and \( L \), respectively. The first law suggests: \[ \tau = \left( \frac{\partial U}{\partial S} \right)_L \] This means that the tension \( \tau \) is the partial derivative of the internal energy \( U \) with respect to entropy \( S \), keeping the length \( L \) constant. Hence, option (A) is the correct answer. Final Answer: (A)