# PSS 19.1 Turn Down the Volume! Part C

1. Jul 6, 2008

### mantillab

1. The problem statement, all variables and given/known data

A heat engine uses the closed cycle shown in the diagram: (Intro 1 figure) . The working substance is n moles of monoatomic ideal gas. Find the efficiency eta of such a cycle. Use the values for pressure and volume shown in the diagram and assume that the process between points 1 and 3 is isothermal.

Find the efficiency (eta) of the heat engine.

2. Relevant equations

PV = nRT
Efficiency (ETA) = Wout/Qh

3. The attempt at a solution
I think I understand the process to get to the solution:
1. Use the ideal gas law to complete your knowledge of n, p, V, and T at one point in the cycle.
2. Use the ideal gas law and equations for specific gas processes to determine p, V, and T at the beginning and end of each process.
3. Calculate Q, W_s, and delta Eth for each process.
4. Find W_out by adding W_s for each process in the cycle. If the geometry is simple, you can confirm this value by finding the area enclosed within the pV curve.
5. Add just the positive values of Q to find Q_H.
6. Verify that net delta Eth=0.
7. Calculate the thermal efficiency eta and any other quantities you need to complete the solution.

The problem that I'm running into is how to find the temperature (T) or the moles of gas (n), with both variables missing, I can't seem to be able to calculate Q or W for any of the processes without at least one of the variables.

Using pV = nRT: n = [pV]/[RT]

Any suggestions? Thanks!

2. Jul 7, 2008

### Irid

You don't have to find all parameters of the working body everywhere. The efficiency is,

$$\eta = \frac{W}{Q}$$

so you must (a) find total work $$W$$ done during one cycle; (b) total heat $$Q$$ added to the engine during one cycle. Total work is the area enclosed by the loop, I'm sure you can integrate to find that. I'll leave you yourself to decide which part of the loop corresponds to heat added to the system: 1-2, 2-3 or 3-4. If you make a correct decision, the total heat added is also obtained by integrating

$$Q = \int P\, dV+\int V\, dP$$

This is problem is not very complicated and should be done in only several steps.

3. Jul 7, 2008

### dynamicsolo

You won't need to know the number of moles of gas in the system. Just carry the value as n throughout your calculations; when you finally form the quotient for the efficiency, you'll find that n cancels out. (This is physically credible: there is no reason that the efficiency of a process should depend on the mass of gas involved, within reason...)

4. Jul 7, 2008

### mantillab

Thanks everyone!