An ideal gas sample ( = 1.5 ) is adiabatically compressed from a volume of 150 cm ^3 to 50 cm ^3 . The initial pressure and initial temperature are 150 kPa and 300 K . Determine the work done by the gas during this process.

Step-by-step solution

Given P₁ = 150 kPa = 150 10^3 Pa , V₁ = 150 cm ^3 = 150 10⁻⁶ m ^3 , V₂ = 50 cm ^3 = 50 10⁻⁶ m ^3 , and = 1.5 . The work done by the gas in an adiabatic process is given by: W = P₁ V₁ - P₂ V₂ - 1 Using the adiabatic relation P₁ V₁^ = P₂ V₂^ , we can rewrite P₂ V₂ as P₁ V₁ ( V₁ V₂ )^ - 1 . Substituting this into the work equation: W = P₁ V₁ - 1 [ 1 - ( V₁ V₂ )^ - 1 ] Substituting the given values: W = 150 10^3 150 10⁻⁶ 1.5 - 1 [ 1 - ( 150 50 )^ 1.5 - 1 ] W = 22.5 0.5 [ 1 - (3)^ 0.5 ] W = 45 (1 - 3 ) Using 3 1.732 : W