Chapter 20, Problem 10STP

Interpretation Introduction

Interpretation:

The ratio of speed of nitrogen gas to that of chlorine gas is to be explained.

Concept introduction:

Root mean square speed of the gas particles is used to determine the speed of the gas molecules.

$\begin{array}{l}{\text{µ}}_{\text{rms}}\text{=}\sqrt{\frac{\text{3RT}}{\text{M}}}\\ {\text{µ}}_{\text{rms}}\text{=Root}\text{mean}\text{square}\text{speed}\\ \text{R=gas}\text{constant}\\ \text{T=Temperature}\\ \text{M=Molecular}\text{Mass}\end{array}$

Answer to Problem 10STP

The root mean square speed of nitrogen gas and chlorine gas are given as:

$\begin{array}{l}\frac{{\text{µ}}_{\text{rms}}\left({\text{N}}_{\text{2}}\right)}{{\text{µ}}_{\text{rms}}\left({\text{Cl}}_{\text{2}}\right)}\text{=}\frac{\sqrt{\frac{\text{3RT}}{{\text{M}}_{{\text{N}}_{\text{2}}}}}}{\sqrt{\frac{\text{3RT}}{{\text{M}}_{{\text{Cl}}_{\text{2}}}}}}\\ \text{=}\sqrt{\frac{{\text{M}}_{{\text{Cl}}_{\text{2}}}}{{\text{M}}_{{\text{N}}_{\text{2}}}}}\\ \text{=}\sqrt{\frac{\text{71}}{\text{28}}}\\ \text{=}\frac{\text{8}\text{.4}}{\text{5}\text{.3}}\end{array}$

The ratio of speed of nitrogen gas to the chlorine gas is $\frac{\text{8}\text{.4}}{\text{5}\text{.3}}$.

Explanation of Solution

The root mean square speed of nitrogen gas and chlorine gas are given as:

$\begin{array}{l}{\text{µ}}_{\text{rms}}\left({\text{N}}_{\text{2}}\right)\text{=}\sqrt{\frac{\text{3RT}}{{\text{M}}_{{\text{N}}_{\text{2}}}}}\\ {\text{µ}}_{\text{rms}}\left({\text{Cl}}_{\text{2}}\right)\text{=}\sqrt{\frac{\text{3RT}}{{\text{M}}_{{\text{Cl}}_{\text{2}}}}}\end{array}$

$\begin{array}{l}\frac{{\text{µ}}_{\text{rms}}\left({\text{N}}_{\text{2}}\right)}{{\text{µ}}_{\text{rms}}\left({\text{Cl}}_{\text{2}}\right)}\text{=}\frac{\sqrt{\frac{\text{3RT}}{{\text{M}}_{{\text{N}}_{\text{2}}}}}}{\sqrt{\frac{\text{3RT}}{{\text{M}}_{{\text{Cl}}_{\text{2}}}}}}\\ \text{=}\sqrt{\frac{{\text{M}}_{{\text{Cl}}_{\text{2}}}}{{\text{M}}_{{\text{N}}_{\text{2}}}}}\\ \text{=}\sqrt{\frac{\text{71}}{\text{28}}}\\ \text{=}\frac{\text{8}\text{.4}}{\text{5}\text{.3}}\end{array}$

Conclusion

The root mean square speed of nitrogen gas and chlorine gas are given as:

$\begin{array}{l}\frac{{\text{µ}}_{\text{rms}}\left({\text{N}}_{\text{2}}\right)}{{\text{µ}}_{\text{rms}}\left({\text{Cl}}_{\text{2}}\right)}\text{=}\frac{\sqrt{\frac{\text{3RT}}{{\text{M}}_{{\text{N}}_{\text{2}}}}}}{\sqrt{\frac{\text{3RT}}{{\text{M}}_{{\text{Cl}}_{\text{2}}}}}}\\ \text{=}\sqrt{\frac{{\text{M}}_{{\text{Cl}}_{\text{2}}}}{{\text{M}}_{{\text{N}}_{\text{2}}}}}\\ \text{=}\sqrt{\frac{\text{71}}{\text{28}}}\\ \text{=}\frac{\text{8}\text{.4}}{\text{5}\text{.3}}\end{array}$

The ratio of speed of nitrogen gas to the chlorine gas is $\frac{\text{8}\text{.4}}{\text{5}\text{.3}}$.