For an ideal gas the instantaneous change in pressure 'p' with volume 'v' is given by the equation dp/dv = - ap. If p = p₀ at v =0 is the given boundary condition, then the maximum temperature one mole of gas can attain is : (Here R is the gas constant)

Solution

Physics sir · 4 steps

Integrate to get p(v), write T = pv/R, then find where T stops rising and starts falling.

1. Find p(v). Separate the variables: dp/p = -a dv, so ln p = -av + C. With p = p₀ at v = 0, this gives p = p₀ e⁻ᵃᵛ.
2. Write T(v). For one mole, pv = RT, so T = p₀/R v e⁻ᵃᵛ. The factor v pushes T up and e⁻ᵃᵛ pulls it down, so T has a peak in between.

   dT/dv = p₀/R e⁻ᵃᵛ(1 - av) = 0 ⇒ v = 1/a

3. Check the peak. For v < 1/a the slope is positive, and for v > 1/a it is negative. So v = 1/a is a maximum.
4. So. Tₘₐₓ = p₀/R × 1/a × e⁻¹ = p₀/aeR, which is option (a).

The idea to keep. When a question asks for the maximum temperature on a path, write T as a function of one variable using pV = nRT, set dT/dV = 0, and check that the slope changes sign.

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