A gas is confined in a cylindrical can fitted with a piston. Both the piston and the walls of the can are adiabatic. The initial pressure, volume, and temperature of the gas are 100 kPa , 400 cm ^3 , and 300 K respectively. The ratio of the specific heat capacities of the gas is C_p / C_V = 1.5 . Calculate the pressure and the temperature of the gas if it is slowly compressed to a volume of 100 cm ^3 .

Step-by-step solution

For a slow adiabatic compression, the process is reversible and follows the equation P V^ = constant . Given: P₁ = 100 kPa V₁ = 400 cm ^3 T₁ = 300 K = 1.5 V₂ = 100 cm ^3 Using the pressure-volume relation: P₁ V₁^ = P₂ V₂^ P₂ = P₁ ( V₁ V₂ )^ P₂ = 100 ( 400 100 )^ 1.5 P₂ = 100 4^ 1.5 = 100 8 = 800 kPa Using the temperature-volume relation: T₁ V₁^ - 1 = T₂ V₂^ - 1 T₂ = T₁ ( V₁ V₂ )^ - 1 T₂ = 300 ( 400 100 )^ 1.5 - 1 T₂ = 300 4^ 0.5 = 300 2 = 600 K