Thermodynamics physics homework help

Question

An engine using 1 mol of an ideal gas ini- tially at 21.7 L and 326 K performs a cycle consisting of four steps:

1) an isothermal expansion at 326 K from 21.7 L to 42.1 L;

2) cooling at constant volume to 186 K ;

3) an isothermal compression to its original volume of 21.7 L; and

4) heating at constant volume to its original

temperature of 326 K .

Find its efficiency. Assume that the

heat capacity is 21 J/K and the univer- sal gas constant is 0.08206 L · atm/mol/K = 8.314 J/mol/K.

Answer in units of %.

Answer 1

In this problem you have a cycle that consists of four processes. In the expansion step, the one mole of gas will be generating work, and in the compression step work is done on the gas. So in order to compute the net work out, you will have to calculate the work done during those two volume changing steps.

You calculate the work by integrating P dv order the processes. Since we are working with an ideal gas and an isothermal process we can use the ideal gas law to express pressure as nRT/V. Thus we can find the work by integrating nRT/V * dv over the range of volumes. The result is that the work is

nRT * ln (V1/V2). For the expansion, Temp = 326K and V1 = 42.1 and V2 = 21.7. For the compression, T = 186. V1 = 21.7 and V2 = 42.1. Add the two calculated works to get the network. Notice that the compression work is negative. Also, since our units are Joules and degrees K, the correct value of R to use is 8.314 J/mole/K

Once we have the net work, we can find the efficiency by dividing the work by the net heat added. We do not need to calculate the heat rejected during the constant volume cooling as it is irrelevant.

The heat added during the constant volume process is Cv * dt = 21 J/K * (326 - 186)K.

At this point, it is a simple matter of grabbing the TI-84 or whatever and calculating the final result. I won't do that for you, but hopefully my remarks illustrate the method of attack.