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: Check by the exponent
Writing the result as $\frac{T^{\gamma}}{P^{\gamma-1}}=\text{constant}$ shows that temperature rises when pressure rises in adiabatic compression.

Step 2: Why the other options fail
$PT^{\gamma}$ and $P^{\gamma}T^{\gamma}$ make both quantities increase together in a way that is not consistent with $PV^{\gamma}$. $P^{\gamma}T^{1-\gamma}$ has the exponents swapped.

Step 3: Result
Only $P^{1-\gamma}T^{\gamma}$ follows from $PV^{\gamma}$ and $PV=nRT$.

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