An ideal gas is contained in a vessel of volume V₀ at temperature T and pressure p₀ . While maintaining a constant temperature, gas is continuously pumped out of the vessel at a constant volume-rate dV/dt = r . The pressure of the extracted gas is equal to the pressure inside the vessel. Determine the time taken before half the original gas is pumped out.

Step-by-step solution

Let n be the number of moles of gas in the vessel at time t . From the ideal gas equation, p V₀ = n R T p = n R T V₀ . The volume rate of gas pumped out is dV dt = r . The number of moles pumped out in time dt is dn_ ext = p dV R T = p r dt R T . The rate of decrease of moles in the vessel is - dn dt = p r R T . Substituting p = n R T V₀ , we get: - dn dt = ( n R T V₀ ) r R T = n r V₀ Rearranging the variables and integrating: _ n₀ ^ n₀/2 dn n = - ₀^t r V₀ dt ( 1 2 ) = - r V₀ t - 2 = - r V₀ t t = V₀ 2 r