# Efficiency of cyclic process

1. May 9, 2013

### Saitama

1. The problem statement, all variables and given/known data
(see attachment, ignore the arrows made with the pen)

2. Relevant equations

3. The attempt at a solution
Efficiency of a cycle is defined as $\eta=\frac{W}{Q}$ where W is work done and Q is heat input.

W can be easily calculated by finding the area enclosed within the loop shown in the graph which is equal to $(P-P_o)V_o$.

Heat input occurs only in the processes, B->A ($Q_1$) and B->C ($Q_2$).
$Q_1=nC_v\Delta T_1$ where $\Delta T_1=\frac{V_o}{nR}(P_o-P)$
$\Rightarrow Q_1=\frac{C_vV_o}{R}(P_o-P)$

$Q_2=nC_p\Delta T_2$ and $\Delta T_2=\frac{P_oV_o}{nR}$
$\Rightarrow Q_2=\frac{C_pP_oV_o}{R}$

$$\eta=\frac{W}{Q_1+Q_2}=\cfrac{(P_o-P)V_o}{\cfrac{C_vV_o}{R}(P_o-P)+\cfrac{C_pP_oV_o}{R}}$$
I don't have $C_p$ and $C_v$, how am I supposed to solve this?

Any help is appreciated. Thanks!

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2. May 9, 2013

### ehild

What is the relation between Cp and Cv, and how does Cv depend on the degrees of freedom of the molecules in the gas? Assume monoatomic gas first.

ehild

3. May 9, 2013

### Saitama

The molar specific heat at constant volume is fR/2 and at constant pressure is (f/2+1)R where f is the degrees of freedom. For a monoatomic gas, f=3, hence Cv=3R/2 and Cp=5/2. Substituting the values give $P=2P_o/7$ which is not correct.

4. May 9, 2013

### ehild

f is not given, you get different pressures for different values of f. f=5 for diatomic gas and f=6 for three or more atomic molecules. For monoatomic gas, your solution should be correct.

ehild

5. May 10, 2013

### Saitama

I thought that it would be a bit difficult to put different values of f and find P. The answer in the key is A i.e $0.3P_o$. I substituted it in the expression I ended up with and replaced $C_v$ and $C_p$ with fR/2 and (f/2+1)R respectively. Solving I got f=50/17. Is the question wrong?

6. May 10, 2013

### ehild

2/7 is 0.3 with one significant digit. As you had to choose among different data, you should have figured out which was possible.

ehild

7. May 10, 2013

### Saitama

Thank you ehild!