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 pressure of the gas as a function of time.

Step-by-step solution

Let the mass of the gas in the vessel at time t be m . The density of the gas inside the vessel is = m V₀ . The rate at which the volume of gas is pumped out is r = dV dt . The corresponding rate at which mass is extracted is given by dm dt = - r = - m V₀ r . Rearranging the variables, we get: dm m = - r V₀ dt Integrating both sides from t = 0 to t = t and m = m₀ to m = m : _ m₀ ^ m dm m = - r V₀ ₀^ t dt ( m m₀ ) = - rt V₀ m = m₀ e^ -rt/V₀ From the ideal gas equation, p V₀ = m M R T , which implies p m at constant