Thermodynamics Closed System Energy Balance

[SweetBabyLou]

Hi everyone,

I am currently reviewing for a thermodynamics exam and have come across a difficult problem while studying. Here it is:

An insulated tank containing helium, a monatomic gas, at P_1t = 1000kPa and T₁ = 800K is connected to an initially evacuated insulated piston-cylinder device. The mass of the piston is such that an initial pressure in the cylinder of P_1c = 200kPa is required to raise the piston at first. Also at the initial state, a linear spring (F~x) is touching the piston but exerts no force on it at first. Now the valve is opened slightly, and part of the helium flows to the cylinder, pushing the piston up and compressing the spring. This process continues until the colume of the helium has doubled. The final state is in mechanical and thermal equilibrium. Assuming helium behaves as an ideal gas, determine:

a) the final pressure of helium P₂ (in kPa)

b) the final temperature of the helium T₂ (in K)

c) Also, if the initial temperature of T₁ had been doubled to 1600K, explain what effect this would have had on the final pressure P₂ in part a.

Homework Statement

The problem gives us a few values to start out with:
P_1t = 1000kPa
T₁ = 800K
P_1c = 200kPa

Homework Equations

So for this problem, started with the general energy balance equation:
ΔU + ΔKE + ΔPE = Q - W

Since the tank is insulated, I assumed that the process is adiabatic, and thus Q = 0.
I also assumed that ΔKE = ΔPE = 0. With these assumptions, the balance equation becomes:

ΔU = -W → ΔU + W = 0

The Attempt at a Solution

I tried every which way to try and find a relationship between the values given and specific volumes, specific pressures, enthalpy, etc. I also tried using the hint at the ideal gas law to formulate some sort of useful equation. The only thing I could get from this was that:

since volume doubles, V₁ = 1/2 V₂.

With P₁V₁ = mRT₁
and P₂V₂ = mRT₂

I took out the m and R term (since they're constant) and got P₁/P₂ * V₁/V₂ = T₁/T₂

isolating the volume term, I get:
V₁/V₂ = P₂/P₁ * T₁/T₂

Since V2 is double V1, V1/V2 = 1/2

∴1/2 = P₂/P₁ * T₁/T₂

When I plug in the given information, I can say that:

T₂ = 1600K/1000kPa * (P₂)

I wanted to try and substitute this value of T₂ into one of the equations, but by this point, my brain has been wracked with trying to figure all this out. I am currently at an impasse on where I went wrong with this problem (most likely at the beginning) and looking for some advice on where/how I should rethink this problem. It should be an easy problem, save that no masses of helium have been given (and, of course, helium is not represented data in the back appendices of the book).

I'm sorry to bother all of you with this problem, it may even be a trivial question that's just teasing me under my nose. Any help or advice would be greatly appreciated. Attached below is the professor's diagram of the problem.

Thanks

 

[oneomega]

i worked out the problem, and realized that i needed K(spring const) and cross section area of the piston to solve this problem.
i think everything worked out is correct.
the question says everyhting is in equilibrium at the end which means,
Pf*area=kx+200
the final temp can be found out
using first law Q=U+W

 

[Chestermiller]

i worked out the problem, and realized that i needed K(spring const) and cross section area of the piston to solve this problem.
i think everything worked out is correct.
the question says everyhting is in equilibrium at the end which means,
Pf*area=kx+200
the final temp can be found out
using first law Q=U+W

You don't need to know the spring constant, and you don't need to know the cross section area of the piston to solve this problem. Let A be the cross sectional area, and let k be the spring constant (and let m be the mass of the piston). Do a force balance on the piston in the initial state. Its weight is mg and the force acting on it from below is 200A. If the piston rises to a height h, the work done in raising the piston is mgh = 200 Ah. But Ah = V, where V is the volume increase. So the work done to raise the piston is 200 V. Now do a similar kind of analysis for the spring to determine how much work is required to raise the piston to h against against variable spring force. Your result should involve the final pressure P_f, the 200 kPa pressure, and the volume increase V. Now add the two amounts of work to get the total work done by the system.