An ideal gas undergoes a reversible adiabatic compression from an initial volume V_i and initial pressure p_i to a final pressure p_f. Determine the work W_{if} performed on the gas during this process.
Numerical Application:
V_i = 1 l, p_i = 5 · 10 ^5 Pa, p_f = 2p_i, c = 5/2 (see definition (5.62)).
U = c N R T (5.62)
For an adiabatic compression, the work performed on the gas is expressed as,
W_{ if} = ΔU_{if} = c N_0 R (T_f − T_i) = c \Bigl(p_f V_f − p_i V_i\Bigr) .
where
p_f V^γ_f = p_i V^γ_i thus V_f = \Bigl(\frac{p_i}{p_f}\Bigr)^{\frac{1}{\gamma } } V_i = \Bigl(\frac{p_i}{p_f}\Bigr)^{\frac{c}{c+1 } } V_i.
Thus,
W_{ if} = c V_i \biggl(p_f \Bigl(\frac{p_i}{p_f}\Bigr)^{\frac{c}{c+1 } } – p_i \biggr) = 28.5 J.