Question

Consider a thermodynamic process where internal energy $U \equiv A p^2 V$ (A = Constant). If the process is performed adiabatically, then

• A. AP2 ( V+1= Constant)
• B. (AP+1)2 V= Constant
• C. ( AP+1 V)2= Constant
• D. V/( AP+1)2 = Constant

Answer 1

The Correct Option is B

Given: The internal energy is defined as \( U = A p^2 V \), where \( A \) is a constant. The process is performed adiabatically, and we are tasked with finding the relationship between pressure \( p \) and volume \( V \) during this process.

Approach: For an adiabatic process, the first law of thermodynamics tells us that the change in internal energy is equal to the work done: \[ dU = -PdV. \] This equation implies that: \[ A p^2 dV + 2A p V dp = -p dV. \] From this, we can derive a relationship between pressure and volume for an adiabatic process. Since we know the form of the internal energy \( U = A p^2 V \), we can proceed to find the specific relationship between \( p \) and \( V \) under the condition that the process is adiabatic. 

Solution: Rearranging terms and solving the equation for an adiabatic process, we find that the relationship between \( p \) and \( V \) is: \[ (AP + 1)^2 V = \text{Constant}. \] 

Final Answer: The correct relationship is \( (AP + 1)^2 V = \text{Constant} \).