Chapter 10, Problem 10.59QA

Interpretation Introduction

To find:

The effect of the following conditions on the volume of the gas in a cylinder with a movable piston:

a) Both the absolute temperature and the external pressure on the piston are doubled.

b) The absolute temperature is halved, and the external pressure on the piston is doubled.

c) The absolute temperature is increased by $75 \%,$ and the external pressure is increased by $50 \%$.

Answer to Problem 10.59QA

Solution:

a) There would be no change in the volume when both the absolute temperature and the external pressure on the piston are doubled.

b) When the absolute temperature is halved and the external pressure on the piston doubled, then the volume of the gas decreases to the ${\frac{1}{4}}^{th}$ of the initial volume.

c) When the absolute temperature increased by $75 \%$ and the external pressure increased by $50 \%,$ the volume would be increased by $17 \%$.

Explanation of Solution

1) Concept:

To find the change in the volume of gas, we have to apply the combined gas law.

2)  Formula:

The combined gas law is

$\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$

3)  Given:

i) Absolute temperature $T_2=2 T_1$ and external pressure $P_2=2 P_1$

ii) Absolute temperature $T_2=\frac{1}{2} T_1$ and external pressure $P_2=2 P_1$

iii) $75 \%$ increase in absolute temperature and $50 \%$ increase in  external pressure.

4)  Calculations:

a) The change in the volume by doubling the absolute temperature and the external pressure,

$\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$

Inserting, $T_2=2 T_1$ and $P_2=2 P_1$

$\frac{P_1{V}_1}{T_1}=\frac{2P_1{V}_2}{2T_1}$

$V_2=V_1$

The final volume is the same as initial volume. Therefore, there is no change in volume.

b) When the absolute temperature is halved and the external pressure on the piston is doubled, the change in the volume would be

$\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$

Inserting $T_2=\frac{1}{2} T_1$ and $P_2=2 P_1$

$\frac{P_1{V}_1}{T_1}=\frac{2 P_1{V}_2}{\frac{1}{2} T_1}$

$V_2=\frac{1}{4}{V}_1$

The volume of gas decreases to the ${\frac{1}{4}}^{th}$ of the initial volume.

c) The absolute temperature is increased by $75 \%$ and the external pressure is increased by $50 \%$.

We can assume the $75 \%$ increase in the temperature as

$T_2=T_1+0.75 T_1=1.75 T_1$

and the external pressure by $50 \%$ increase $P_2=P_1+0.50 P_1=1.50 P_1$

Applying the combined gas law,

$\frac{P_1{V}_1}{T_1}=\frac{1.50 P_1{V}_2}{1.75 T_1}$

$V_2=\frac{1.75}{1.50}{V}_1$

$V_2=1.17 V_1$

We can write this equation by considering the initial volume as $V_1$. The increase in the volume would be

$V_2=V_1+0.17 V_1$

So, the percent increase in the volume would be $17 \%$.

Conclusion:

We have estimated the change in the volume of the gas in a cylinder with the movable piston for the three conditions by using the combined gas law.