Question

The Stirling engine, a heat engine invented by a Scottish minister, has been considered for use in automobile engines because of its efficiency. In such an engine, a gas goes through a four-step cycle: (1) expansion at constant T, (2) cooling at constant V, (3) compression at constant T to its original volume, and (4) heating at constant V to its original temperature. Suppose the gas starts at a pressure of 1.23 atm and the volume of the gas changes from 0.350 to 1.31 L during its expansion at constant T. Calculate the pressure of the gas at the end of this step in the cycle.

Short answer

Therefore, the pressure of the gas at the end of the cycle is 0.329 atm.

Step-by-step solution

Boyle’s Law

The gas law that comes into account when the temperature is kept constant by varying pressure and volume is referred to as Boyle's law. It states that the pressure is in reverse related to the volume at a constant temperature.

Explanation

As per the thermodynamic principle, pressure and volume are thermodynamic state functions. So, they will not depend on the path that is going to the end of the process they will not depend on the initial and final conditions only. Due to this, the calculation involves the entire change of the pressure and volume instead of steps that are taken to reach the final state.

Calculation

As the temperature and amount of the gas are constant, so we need to apply Boyles law for the given system. The formula used to calculate the final pressure at the end of the cycle is as:

$P_1\times V_1=P_2\times V_2$

Here,

*$P_1$is the initial pressure of the gas, i.e., 1.23 atm.

*$P_2$is the final pressure of the gas.

*$V_1$is the initial volume of the gas, i.e.,0.350 L.

*localid="1663696337980" $V_2$is the final volume of the gas, i.e., 1.31 L.

Rearranging and substituting in the given equation, we get

$$\begin{gathered} P_2=\frac{P_1\times V_1}{V_2} \\ =\frac{1.23atm\times 0.350L}{1.31L} \\ =\frac{0.4305}{1.31} \\ =0.329atm \end{gathered}$$

Thus, the pressure of the gas at the end of the cycle is 0.329 atm.