2018 USA Physics Olympiad Exam Part A 10 b.Find an expression for the temperature of the gas as it is emitted from the pump cylinder into the atmosphere.Your answer may depend on time. Solution When the piston is all the way to the left,the pressure is P(t)and the temperature is Ta.As the piston moves to the right,the gas is adiabatically compressed until its pressure reaches Pa and the outlet valve opens.Since PVy is constant during adiabatic compression and PV/T is constant by the ideal gas law, (品) 1-1/ Tout(t)=T -Ta Pa 2/5 P(t) =Tae2Rt/5V where we used y=5/3 for a monatomic ideal gas. For the remainder of this problem 0<Vo <AV<V. c.Find an expression for the minimum possible pressure in the chamber,Pmin. Solution Since Vo >0,the inlet valve will not open immediately when the piston begins moving to the left;instead it will open once the pressure in the cylinder equals the pressure in the chamber. Since the expansion of the cylinder is adiabatic,PVy is constant,so Pmin =Pa Vo Copyright C2018 American Association of Physics Teachers2018 USA Physics Olympiad Exam Part A 10 b. Find an expression for the temperature of the gas as it is emitted from the pump cylinder into the atmosphere. Your answer may depend on time. Solution When the piston is all the way to the left, the pressure is P(t) and the temperature is Ta. As the piston moves to the right, the gas is adiabatically compressed until its pressure reaches Pa and the outlet valve opens. Since P V γ is constant during adiabatic compression and P V /T is constant by the ideal gas law, Tout(t) = Ta  Pa P(t) 1−1/γ = Ta  Pa P(t) 2/5 = Tae 2Rt/5V where we used γ = 5/3 for a monatomic ideal gas. For the remainder of this problem 0 < V0 < ∆V  V . c. Find an expression for the minimum possible pressure in the chamber, Pmin. Solution Since V0 > 0, the inlet valve will not open immediately when the piston begins moving to the left; instead it will open once the pressure in the cylinder equals the pressure in the chamber. Since the expansion of the cylinder is adiabatic, P V γ is constant, so Pmin = Pa  V0 V0 + ∆V γ = Pa  1 + ∆V V0 −γ . Copyright c 2018 American Association of Physics Teachers