Chapter 5, Problem 5.89QP

Interpretation Introduction

Interpretation:

The rms speed (in m/s) of nitrogen $({\text{N}}_{\text{2}})$ molecules at ${\text{25}}^{\circ }\text{C}$ and at ${\text{125}}^{\circ }\text{C}$ has to be calculated

The approximate curves of molecular speed distributions of   ${\text{N}}_{\text{2}}$ at ${\text{25}}^{\circ }\text{C}$ and at ${\text{125}}^{\circ }\text{C}$ has to be sketched

Concept Introduction:

Root-mean-square (rms):

The root-mean-square molecular speed (u) is given by the below formula.

$\begin{array}{l}u\text{=}\sqrt{\frac{3\text{RT}}{{\text{M}}_{\text{m}}}}\\ ={\left(\frac{3\text{RT}}{{\text{M}}_{\text{m}}}\right)}^{\frac{1}{2}}\end{array}$

Where,

R is the molar gas constant

T is the absolute temperature

${\text{M}}_{\text{m}}$ is the molar mass of the gas

Answer to Problem 5.89QP

The rms speed (in m/s) of nitrogen molecules at ${\text{25}}^{\circ }\text{C}$ is $5.15\times {10}^2\text{ m/s}$

The rms speed (in m/s) of nitrogen molecules at ${\text{125}}^{\circ }\text{C}$ is $5.95\times {10}^2\text{ m/s}$

The approximate curves of molecular speed distributions of  ${\text{N}}_{\text{2}}$ at ${\text{25}}^{\circ }\text{C}$ and at ${\text{125}}^{\circ }\text{C}$ is

Figure 1

Explanation of Solution

To Calculate: The rms speed (in m/s) of nitrogen $({\text{N}}_{\text{2}})$ molecules at ${\text{25}}^{\circ }\text{C}$ and at ${\text{125}}^{\circ }\text{C}$

Use root-mean-square molecular speed formula to calculate rms.

$u={\left(\frac{3\text{RT}}{{\text{M}}_{\text{m}}}\right)}^{\frac{1}{2}}$

rms speed of nitrogen molecules at ${\text{25}}^{\circ }\text{C}$:

Convert temperature to Kelvin:

$\begin{array}{l}\text{T }{\text{= (25}}^{\circ }\text{C+273) }\\ \text{= 298 K}\end{array}$

The molar mass of nitrogen $({\text{N}}_{\text{2}})$ is, ${\text{M}}_{\text{m}}$ = $\text{28}\text{.02}\times {10}^{-3}\text{ kg/mol}$

The absolute temperature is, T = 298 K

The gas constant is, R = $8.31\text{ kg}·{\text{m}}^2/({\text{s}}^{\text{2}}·\text{K}·\text{mol)}$

The root-mean-square is calculated as follows,

$\begin{array}{l}{u}_{25}={\left(\frac{3\text{RT}}{{\text{M}}_{\text{m}}}\right)}^{\frac{1}{2}}\\ ={\left(\frac{3\times 8.31\text{ kg}·{\text{m}}^2/({\text{s}}^{\text{2}}·\text{K}·\text{mol)}\times 298\text{ K}}{28.02\times {\text{10}}^{-3}\text{ kg/mol}}\right)}^{\frac{1}{2}}\\ =\text{5}\text{.149 }\times {10}^2\\ \cong \text{5}\text{.15 }\times {10}^2\text{ m/s}\end{array}$

Therefore, the rms for the nitrogen molecules at ${\text{25}}^{\circ }\text{C}$ is $\text{5}\text{.15 }\times {10}^2\text{ m/s}$

rms speed of nitrogen molecules at ${\text{125}}^{\circ }\text{C}$:

Convert temperature to Kelvin:

$\begin{array}{l}\text{T }{\text{= (125}}^{\circ }\text{C+273) }\\ \text{= 398 K}\end{array}$

The molar mass of nitrogen $({\text{N}}_{\text{2}})$ is, ${\text{M}}_{\text{m}}$ = $\text{28}\text{.02}\times {10}^{-3}\text{ kg/mol}$

The absolute temperature is, T = 398 K

The gas constant is, R = $8.31\text{ kg}·{\text{m}}^2/({\text{s}}^{\text{2}}·\text{K}·\text{mol)}$

The root-mean-square is calculated as follows,

$\begin{array}{l}{u}_{125}={\left(\frac{3\text{RT}}{{\text{M}}_{\text{m}}}\right)}^{\frac{1}{2}}\\ ={\left(\frac{3\times 8.31\text{ kg}·{\text{m}}^2/({\text{s}}^{\text{2}}·\text{K}·\text{mol)}\times 398\text{ K}}{28.02\times {\text{10}}^{-3}\text{ kg/mol}}\right)}^{\frac{1}{2}}\\ =\text{5}\text{.9507 }\times {10}^2\\ \cong \text{5}\text{.95 }\times {10}^2\text{ m/s}\end{array}$

Therefore, the rms for the nitrogen molecules at ${\text{125}}^{\circ }\text{C}$ is $\text{5}\text{.95 }\times {10}^2\text{ m/s}$

To Sketch: The approximate curves of molecular speed distributions of  ${\text{N}}_{\text{2}}$ at ${\text{25}}^{\circ }\text{C}$ and at ${\text{125}}^{\circ }\text{C}$

The approximate curves of molecular speed distributions of distributions of  ${\text{N}}_{\text{2}}$ at ${\text{25}}^{\circ }\text{C}$ and at ${\text{125}}^{\circ }\text{C}$ is drawn as follows,

Figure 1

Conclusion

The rms speed (in m/s) of nitrogen molecules at ${\text{25}}^{\circ }\text{C}$ is calculated as $5.15\times {10}^2\text{ m/s}$

The rms speed (in m/s) of nitrogen molecules at ${\text{125}}^{\circ }\text{C}$ is calculated as $5.95\times {10}^2\text{ m/s}$

The approximate curves of molecular speed distributions of  ${\text{N}}_{\text{2}}$ at ${\text{25}}^{\circ }\text{C}$ and at ${\text{125}}^{\circ }\text{C}$ is sketched as follows,

Figure 1