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 change in internal energy of the gas.

Step-by-step solution

For an adiabatic process, the relation between temperature and volume is given by T₁ V₁^ - 1 = T₂ V₂^ - 1 . Substituting the given values: T₂ = T₁ ( V₁ V₂ )^ - 1 = 300 ( 150 50 )^ 1.5 - 1 = 300(3)^ 0.5 = 300 3 K The number of moles of the gas is given by n = P₁ V₁ R T₁ . The change in internal energy is U = n C_v T = n R (T₂ - T₁) - 1 . Substituting n R = P₁ V₁ T₁ : U = P₁ V₁ T₁ (T₂ - T₁) - 1 = P₁ V₁ - 1 ( T₂ T₁ - 1 ) Given P₁ = 150 10^3 Pa and V₁ = 150 10⁻⁶ m ^3 : P₁ V₁ = 150 10^3 150 10⁻⁶ = 22.5 J U = 22.5 1.5 -