In a certain thermodynamical process, the pressure of a gas depends on its volume as kV³. The work done when the temperature changes from 100°C to 300°C will be ___________ nR, where n denotes number of moles of a gas.

Solution

Physics sir · 4 steps

Integrate P dV with P = kV³, then swap PV for nRT so only the temperature change is left.

1. Set up the work. Work done by the gas is W = ∫ P dV. With P = kV³ the integral is direct.

   W = ∫_(V₁)^(V₂) kV³ dV = k(V₂⁴ - V₁⁴)/4 = (P₂V₂ - P₁V₁)/4

2. Bring in temperature. The last step uses kV⁴ = (kV³)V = PV. For an ideal gas PV = nRT, so W = nR(T₂ - T₁)/4.
3. Put in numbers. The temperature change is 300 - 100 = 200 degrees, and a change in °C equals the same change in K. So W = (200 nR)/4 = 50 nR.
4. So. The work done is 50 nR, so the answer is 50.

The common slip: -100. Reading P = kV³ as PV³ = constant makes x = 3, and (200 nR)/(1-3) gives -100. The power of V changes sign when you move it to the left side.

The idea to keep. For PVˣ = constant, the work done is W = (nR Δ T)/(1 - x). Here P = kV³ means PV⁻³ is constant, so x = -3.

Answer from the NTA final answer key. Solution by PrepWiser.