Question

A monoatomic gas undergoes adiabatic change. Its pressure \(P\) and temperature \(T\) are related as \(P\propto T^{x}\) where \(x\) is

• A. \(3.5\)
• B. \(2.5\)
• C. \(4.5\)
• D. \(1.5\)

Hint:

Use the adiabatic relation between P and T, which gives the exponent gamma divided by gamma minus one, with gamma 5/3 for a monoatomic gas.

Answer 1

The Correct Option is B

Step 1: Start from PV
Using $PV^{\gamma}=\text{const}$ and $V\propto\frac{T}{P}$, we get $P\left(\frac{T}{P}\right)^{\gamma}=\text{const}$.

Step 2: Simplify
$P^{1-\gamma}T^{\gamma}=\text{const}$, so $P\propto T^{\gamma/(\gamma-1)}$.

Step 3: Substitute
With $\gamma=\frac{5}{3}$, $x=\frac{5}{2}=2.5$, option (B).

Final Answer:
The exponent x is 2.5. \[ \boxed{\text{(B) }2.5} \]