Question

An engine using 1 mol of an ideal gas initially at 22.3 L and 353 K performs a cycle consisting of four steps:

1) an isothermal expansion at 353 K from 22.3 L to 41.5 L;

2) cooling at constant volume to 170 K;

3) an isothermal compression to its original volume of 22.3 L; and

4) heating at constant volume to its original temperature of 353 K.

Find its efficiency. Assume that the heat capacity is 21 J/K and the universal gas constant is 0.08206 L atm/mol/K = 8.314 J/mol/K.

Answer in units of %.

Answer 1

1 )

n = 1 mole ,

H = 21 J/K

R = 8.314 J/mole.K

W₁ = Q₁

= 2.303 n R T log( V₂/V₁ )

= 2.303 x 1 x 8.314 x 353 x log( 41.5 / 22.3 )

W₁ = Q₁ = 1823.178 J

2 )

Q₂ = - dU

so at constant volume W₂ and Q₂ is equal to zero.

3 )

W₃ = Q₃

= 2.303 n R T log( V₁/V₂ )

= 2.303 x 1 x 8.314 x 170 x log( 22.3 / 41.5 )

W₃ = Q₃ = - 878.018 J

4 )

Q₄ = m C dT

= 21 x ( 353 - 170 )

Q₄ = 3843 J

the efficiency is = ( 1823.178 - 878.018 / 1823.178 + 3843 ) x 100 %

= 16.68 %