The temperature of 1 mole of an ideal monoatomic gas is increased by 50° C at constant pressure. The total heat added and change in internal energy are E₁ and E₂, respectively. If E₁/E₂=x/9 then the value of x is _________.

Solution

Physics sir · 3 steps

Heat at constant pressure uses Cₚ, internal energy always uses Cᵥ, so the ratio is just γ.

1. Heat added. At constant pressure, E₁ = nCₚΔ T. For a monoatomic gas Cₚ = 5/2R, so E₁ = 5/2R × 50 for 1 mole.
2. Internal energy. Internal energy of an ideal gas depends only on temperature, so E₂ = nCᵥΔ T for any process. With Cᵥ = 3/2R, E₂ = 3/2R × 50.

   E₁/E₂ = Cₚ/Cᵥ = (5/2)/(3/2) = 5/3 = 15/9

3. So. Matching 15/9 with x/9 gives x = 15. The 50°C and R cancel and do not affect the answer.

The common slip: 5.4. Writing the ratio upside down, E₁/E₂ = 3/5, gives x = 5.4 instead of 15. Heat added is the bigger quantity because part of it goes into work.

The idea to keep. For an ideal gas, Δ U = nCᵥΔ T in every process. At constant pressure Q = nCₚΔ T, so Q/Δ U = γ, which is 5/3 for a monoatomic gas.

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