Question

For an adiabatic process, choose the correct relation

• A. \(PT^{\gamma }=\text{constant}\)
• B. \(P^{1-\gamma }T^{\gamma }=\text{constant}\)
• C. \(P^{\gamma }T^{\gamma }=\text{constant}\)
• D. \(P^{\gamma }T^{1-\gamma }=\text{constant}\)

Hint:

Start from PV to the power gamma equals a constant and replace V using the ideal gas equation.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
For an adiabatic process, \(PV^{\gamma}=\text{constant}\). The ideal gas law gives \(V\propto\frac{T}{P}\).

Step 2: Calculation
Substitute \(V=\frac{nRT}{P}\): \(P\left(\frac{T}{P}\right)^{\gamma}=\text{constant}\), i.e. \(P^{1-\gamma}T^{\gamma}=\text{constant}\).

Step 3: Compare the options
This matches the second option.

Final Answer:
The relation is P to the power (1 minus gamma) times T to the power gamma equals a constant. \[ \boxed{\text{(B) }P^{1-\gamma}T^{\gamma}=\text{constant}} \]