Question

An ideal gas is taken through a complete cycle in three steps: adiabatic expansion with work equal to I25 J, isothermal contraction at 325 K, and increase in pressure at constant volume.

(a) Draw a p-V diagram for the three steps.

(b) How much energy is transferred as heat in step 3, and

(c) is it transferred to or from the gas?

Answer 1

(a)

During the first step, the gas expands adiabatically, which means the heat Q remains constant. The first law of thermodynamics says that , where is the change in internal energy and W is the work the gas does on its surroundings. Since , or . At the beginning of the first step, the system is at temperature and associated energy (because internal energy only depends on temperature). After the adiabatic process is done, it is in a state with temperature   and associated energy . Then it begins the isothermic contraction where it stays at , but the volume decreases and therefore the pressure increases. Finally for the third step, the system leaves the temperature and energy to return to the first state of temperature and associated energy , and therefore the pressure increases, because the volume must remain constant.

(b)

To figure out how much energy is transferred as heat in step 3 (the isochoric/constant volume process), recall that during the first process (adiabatic expansion) there was zero heat exchange and therefore we found that . We were given though that the work the gas did on its surroundings by expanding during this process was 125 J, so .

Keep that in mind while now looking at step 3 (isochoric process). Using the 1st law of thermodynamics, since there is no volume expansion or contraction, the gas does no work. Therefore since in this step, becomes or . Since in the third process the gas is going from to , the associated change in energy is .

But recall that . So the heat exchange in process 3 is .

The heat exchanged in process 3 is 125 J.

(c)

Since the heat is a positive 125 J, this means heat is flowing INTO the system (ideal gas). This makes sense because during process 3, the temperature increases from 325 K to whatever temperature the adiabatic process started at. Since the gas is not allowed to expand, that added heat must be transformed into raising the temperature rather than doing work (expanding its volume), according to the first law of thermodynamics.