Chapter U3.63, Problem 2E

Interpretation Introduction

Interpretation :

Kinetic theory of gases must be used to explain the relationship between number density and gas pressure.

Concept Introduction :

Number density of gas is the number of gas molecules present per unit volume.

Answer to Problem 2E

Number density increases with increase in gas pressure.

Explanation of Solution

According to kinetic theory of gas

  $\text{P=}\frac{\text{1}}{\text{3}}{\text{ρc}}^{\text{2}}$

Here, $\text{P, ρ, c}$ are pressure, density and root means square velocity of the gas respectively.

  $\begin{array}{l}\text{Or,PV=}\frac{\text{1}}{\text{3}}{\text{ρVc}}^{\text{2}}\text{=}\frac{\text{1}}{\text{3}}{\text{mc}}^{\text{2}}\text{=}\frac{\text{2}}{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{mc}}^{\text{2}}\\ \text{Or,PV=}\frac{\text{2}}{\text{3}}\text{RT}\\ \text{Or,PV=nRT}\end{array}$

  $\text{ρV=m}$

Here, V and m are volume and mass, R is universal gas constant.

Kinetic energy $\frac{\text{1}}{\text{2}}{\text{mc}}^{\text{2}}$ is proportional to temperature T.

Thus, $\text{P=}\frac{\text{nRT}}{\text{V}}$

So at a particular temperature if Pressure is increased then volume will decrease and eventually number of mole n will increase.

Conclusion

Number density increases with increase in pressure.