a question about position of piston after achieving thermal equilibrium

jianghan

A vertical cylindrical container contains some helium gas that is in thermodynamic equilibrium with the surroundings. The gas is confined by a movable heavy piston. the piston is slowly elevated a distance H from its equilibrium position and then kept in the elevated position long enough for the thermodynamic equilibrium to be reestablished. after that,the container is insulated and then the piston is released. after the piston comes to rest,what is the new equilibrium position of the piston?

This question is quite complex to me and i am totally at a loss.
i know the second process is an adiabatic process but not sure about the first,the answer is the piston dropped by 0.6H to 0.4H at the new equilibrium,but who can show me how to arrive at it?

Andrew Mason

You are correct that the second part (after insulating the cylinder) is adiabatic.

Assume an initial temperature T₀. So what is the temperature of the gas after the first part? (Hint: easy). How do you determine the state of the gas (P, V, T) after the second part? Hint: this involves the [itex]\gamma[/itex] for Helium.

AM

jianghan

during the first part,is the pressure constant? if it is,T1 would be T0(h+H)/h.

jianghan

After the adiabatic process,P1V1^γ=P₂V₂^γ,for monoatomic helium,γ is 1.67,but still not sure what process the first is,i think i am stuck at this point

jianghan

Thank you for the help,i presume the first part is an isothermal process hence T₁=T₀ and After the adiabatic process,P1V1γ=P2V2γ,for monoatomic helium,γ is 1.67,but still not sure how to calculate the volume change

Andrew Mason

What is the initial pressure in terms of T0 and H (ie. after the isothermal expansion but before the adiabatic compression)? What is the condition that causes the piston to stop after it is released? (what are the forces on the piston? if the piston is stopped - not accelerating, what does that tell you about the sum of all the forces?) How is that related to pressure? How is volume related to H?

Once you figure that out, all you have to do is apply [tex]P_1V_1^\gamma = P_2V_2^\gamma[/tex] to determine what the new volume will be in terms of H.

AM

jianghan

assume the initial length of the container to be h,
by p₀v₀=p₁v₁,i can get
p₁=p₀h/h+H
as p₀=p₂=p_atm,
i have p₀h/(h+H)*(h+H)^1.67=p₀*(h+kH)^1.67,assuming new equilibrium is kH away from the old equilibrium position
simplify the above eqn i have h³(h+H)²=(h+kH)⁵
i think i would be able to get k=0.4 from this equation,but actually i am not very sure about how to solve it,could you give me hints on this final step? thank you

Andrew Mason

In looking at this problem again, the adiabatic condition does not apply here as this is not a quasi-static adiabatic compression.

Since the cylinder is insulated, no energy is lost from the system. The work done by the atmosphere and the piston ( by the change in gravitational potential energy of the piston at height H to its final position ) is converted into heat flow into the gas. I'll take a closer look at this and see if this is solvable.

AM

ehild

The final pressure is equal to the original one: p_i=p_o+mg/A, the atmospheric pressure + the pressure due to the weight of the piston. (A is the cross section area of the piston).

The force F=Ap_o+mg does the work when the piston is released from the elevated position. The work done by the external force when the piston moves from height H to height h is W=(Ap_o+mg)(H-h)=Ap_i(H-h). This work is converted entirely into the internal energy of the helium gas. ΔU=W.

The internal energy of n mol He is 3/2 nRT at temperature T. The change of internal energy is ΔU=3/2nR(T_f-T_i)

Initially the gas had volume V_i, temperature T_i and pressure p_i: T_i=P_iV_i/(nR). In the final equilibrium state, the volume is V_f=V_i+Ah, the pressure is equal to p_i, so T_f=p_i(V_i+Ah)/(nR).

Write up the equation ΔU=W in terms of H, h, V_i, and p_i and solve for h.

ehild

jianghan

Thank you for suggesting a new method,i tried it,at first the equation looked promising but it turned out to be hugely complex and almost unsolvable as there were to many unknowns...

ehild

What have you done? It is very simple. ΔU=3/2nR(Tf-Ti), W=Api(H-h)
Ti=PiVi/(nR), Tf=pi(Vi+Ah)/(nR).

Plug in and all unknowns cancel, except H and h.

ehild

jianghan

oh,ok,i always thought i needed to assume the initial length of the cylinder to be h_o or something...

so actually the information about the first isothermal expansion is not useful except telling me pressure is not changed?

why for the adiabatic process p_i=p_f,i thought p_i should be lower than p_f?

ehild

The pressure changes in the first isothermal process, but it does not matter.
The initial volume of the cylinder is assumed to be Vi, but it cancels at the end.

ehild

jianghan

sorry one more question,if pressure before the isothermal expansion is the same as pressure after the adiabatic compression,and pressure changes in the first isothermal process,shouldn't pressure before and after the adiabatic process be different? why p_i=p_f ?

ehild

The pressure is different before and after the final compression step, but the process is not quasi-static. The gas is not in equilibrium when the piston moves downward by its own weight, and as pressure is defined only for equilibrium, you can not speak about pressure of the gas during the process. The work on the gas is done by external forces, those caused by the atmospheric pressure and gravity.
pi means the pressure before anything happened to the gas. pf is the final pressure after the piston has been released.

ehild

jianghan

Oh,so your initial state actually refers to the state before the isothermal process,not the state just before the adiabatic process,i misunderstood it. now i understand your method.

Thank you for answering so many questions, i think i learned a lot.

ehild

You are welcome.

Remember the change of internal energy is equal to the heat added and the work done on the gas: ΔU=Q+W. The work W is done by the external force, and it is -∫PdV only for quasi-static processes, when the pressure of the gas balances the external force exerted on the piston.

ehild