One mole of an ideal monatomic gas ( $C_{V} = \frac{3}{2} R$ ) at $305 K$ and $1.0 bar$ is compressed adiabatically by a constant external pressure of $3.0 bar$ until it reaches mechanical equilibrium with that pressure. Find the final temperature in kelvin (nearest integer).

Numerical answer type, answer in K, tolerance 1.

Answer: 549 K

Adiabatic:

Δ U = w

, so

C_{V} (T_{2} - T_{1}) = - P_{e x t} (V_{2} - V_{1})

with

V_{2} = R T_{2} / P_{e x t}

and

V_{1} = R T_{1} / P_{1}

. This gives

1.5 (T_{2} - T_{1}) = - T_{2} + T_{1} (P_{e x t} / P_{1})

, so

2.5 T_{2} = T_{1} (1.5 + 3.0)

and

T_{2} = 305 \times 4.5/2.5 = 549 K