Question

Out of the following relations which one is true in the case of an adiabatic process, where \(\gamma =C_{p}/C_{v}\) ?

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

Hint:

Start from the adiabatic condition with volume and eliminate V using the ideal gas law.

Answer 1

The Correct Option is A

Step 1: Known Relation
Also $TV^{\gamma-1}=$ constant for an adiabatic process.

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

Step 3: Compare the options
Only option (A) has this form.

Final Answer:
The correct relation is P^(1-gamma) T^gamma equal to a constant. \[ \boxed{\text{(A) }P^{(1-\gamma)}T^{\gamma}=\text{constant}} \]