A monatomic ideal gas undergoes a process satisfying Uᵝ ∝ V, where U is its internal energy and V its volume. If dU/dQ = ⅓ throughout the process, find β.
Physics · Mixed Chapter
JEE Mains

Question

A monatomic ideal gas undergoes a process satisfying Uᵝ ∝ V, where U is its internal energy and V its volume. If dU/dQ = ⅓ throughout the process, find β.

G1 Physics · Accepted Answer

Step 1 — For a monatomic ideal gas, internal energy U = (3/2)PV. Thus P = 2U/(3V). Step 2 — The source specifies Uᵝ ∝ V. Take logarithmic differentials: β(dU/U) = dV/V, so dV = (βV/U)dU. Step 3 — The mechanical work done is dW = P dV = [2U/(3V)] × [(βV/U)dU] = (2β/3)dU. Step 4 — First law: dQ = dU + dW = [1 + (2β/3)]dU. Step 5 — Given dU/dQ = 1/3, so 1/[1 + (2β/3)] = 1/3. Therefore 1 + 2β/3 = 3 ⇒ β = 3. Answer: 3.