Chapter 13, Problem 78A

Interpretation Introduction

Interpretation:

The relative number of moles of the gas is to be calculated when pressure, temperature and volume of the gas are given at two different conditions.

Concept introduction:

Ideal gas equation helps to calculate relative number of moles of a gas at given temperature and pressure. According to this law,

$\mathrm{PV}=nRT$

Where

$\mathrm{P}$ is pressure of the gas

$\mathrm{V}$ is volume of the gas

$\mathrm{n}$ is the number of moles of the gas

$\mathrm{R}$ is the ideal gas constant

$\mathrm{T}$ is the temperature of the gas

Answer to Problem 78A

The number of moles of gas will reduce to half of its initial value.

Explanation of Solution

Ideal gas equation helps to calculate volume of a gas at given temperature and pressure. According to this law,

$\mathrm{PV}=nRT$

Or

$\frac{P_1{V}_1}{n_1{T}_1}=\frac{P_2{V}_2}{n_2{T}_2}$

$\frac{{\text{n}}_{\text{2}}}{{\text{n}}_{\text{1}}}\text{=}\frac{{\text{P}}_{\text{2}}{\text{V}}_{\text{2}}{\text{T}}_{\text{1}}}{{\text{P}}_{\text{1}}{\text{V}}_{\text{1}}{\text{T}}_{\text{2}}}$

According to the statement:

$\begin{array}{l}{\text{P}}_{\text{1}}\text{=P}\\ {\text{P}}_{\text{2}}\text{=P}\\ {\text{V}}_{\text{1}}\text{=V}\\ {\text{V}}_{\text{2}}\text{=V}\\ {\text{T}}_{\text{1}}\text{=200}\text{K}\\ {\text{T}}_{\text{2}}\text{=400}\text{K}\end{array}$

Now substituting the values into formula,

$\begin{array}{c}\frac{{\text{n}}_{\text{2}}}{{\text{n}}_{\text{1}}}\text{=}\frac{{\text{P}}_{\text{2}}{\text{V}}_{\text{2}}{\text{T}}_{\text{1}}}{{\text{P}}_{\text{1}}{\text{V}}_{\text{1}}{\text{T}}_{\text{2}}}\\ \text{=}\frac{\text{P×V×200}}{\text{P×V×400}}\\ \text{=}\frac{\text{1}}{\text{2}}\end{array}$

Therefore the number of moles of gas will reduce to half of its initial value.

Conclusion

On doubling the temperature of the gas and keeping the temperature and pressure, the number of moles of gas would reduce to half of its initial value.