Question

The speeds of 10molecules are$\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{,}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{,}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{,}\mathbf{\dots }\mathbf{.}\mathbf{,}\mathbf{11}\frac{\mathbf{k}\mathbf{m}}{\mathbf{s}}$

1. What are their average speeds?
2. What are their rms speeds?

Short answer

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Step-by-step solution

Given data

The speeds of 10 molecules are,$2.0,3.0,4.0,\dots \dots \dots \dots \dots ,11\frac{\text{km}}{\text{s}}.$

Understanding the concept

Average speed is calculated by adding the speed of each molecules and then by dividing the total number of molecules.

The RMS speed of molecules is given by,

${\text{v}}_{rms}=\sqrt{\frac{{\displaystyle {\sum }_{}^{}{\text{v}}^2}}{\text{N}}}$

(a) Calculate the average speeds 

The average speed of molecules is

$\begin{array}{c}\overline{\text{v}}=\frac{{\displaystyle \sum v}}{\text{N}}\\ \overline{\text{v}}=\frac{2+3+4+5+6+7+8+9+10+11}{10}\\ \overline{\text{v}}=6.5\text{\hspace{0.17em}km}/\text{s}\end{array}$

Therefore the average speed of molecules is $6.5\text{\hspace{0.17em}km}/\text{s}$.

(b) Calculate the rms speeds 

The RMS speed of molecules is

${\text{v}}_{rms}=\sqrt{\frac{{\displaystyle {\sum }_{}^{}{\text{v}}^2}}{\text{N}}}$

Here,

$\begin{array}{c}\sum {\text{v}}^2=2^2+3^2+4^2+5^2+6^2+7^2+8^2+9^2+{10}^2+{11}^2\\ =505{\text{km}}^2/{\text{s}}^2\end{array}$

Substitute$505{\text{km}}^2/{\text{s}}^2$for $\sum {\text{v}}^2$and 10 for N into the above equation.

$\begin{array}{c}{\text{v}}_{rms}=\sqrt{\frac{505}{10}}\\ =7.1\text{\hspace{0.17em}km}/\text{s}\end{array}$

Therefore the RMS speed of the molecules is$7.1\text{\hspace{0.17em}km}/\text{s}$ .