One mole of an ideal gas is taken through an adiabatic process where the temperature rises from 27° C to 37° C. If the ideal gas is composed of polyatomic molecule that has 4 vibrational modes, which of the following is true? [R = 8.314 J mol⁻¹ k⁻¹]

Solution

Physics sir · 4 steps

Adiabatic means no heat, so the whole rise in internal energy is paid for by work done on the gas.

1. Count degrees of freedom. A polyatomic molecule has 3 translational and 3 rotational degrees of freedom. Each vibrational mode adds 2 (kinetic and potential), so 4 modes add 8. In total f = 3 + 3 + 8 = 14.
2. Molar heat capacity. By equipartition, each degree of freedom gives 1/2R per mole, so Cᵥ = f/2R = 7R.

   Δ U = nCᵥΔ T = 1 × 7 × 8.314 × 10 ≈ 582 J

3. Sign of the work. In an adiabatic process Q = 0, so Δ U = Q - W_(by) gives W_(by) = -Δ U = -582 J. A negative work by the gas means 582 J of work is done on the gas. That is why its temperature rose.
4. So. The work done on the gas is close to 582 J, option (b).

The idea to keep. Each vibrational mode carries 2 degrees of freedom, not 1. In an adiabatic process, a temperature rise always means work was done on the gas, equal to nCᵥΔ T.

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