Eleven molecules have speeds $15,16,17,\dots ,25m/s$. Calculate

(a) $v_{avg}$and

(b) $v_{rms}$.

Given :

Speeds of eleven molecules : $15,16,17,\dots ,25m/s$

Theory used :

(a)Average of a number is evaluated by :

$\frac{Sumofallobservations}{Numberofobservations}$

(b)$v_{rms}=\sqrt{{v^2}_{avg}}$

(a) To calculate the average speed, add all of the eleven speeds together and divide by $11$(the number of particles) :

$$\begin{gathered} v_{avg}=\underset{i=1}{\overset{25}{\sum v}} \\ =\frac{15m/s+16m/s+17m/s+...+25m/s}{11} \\ =\boxed{\mathbf{20}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}} \end{gathered}$$

(b) Calculate $v_{rms}$by taking root mean square of velocity. We square all of the speeds, average the squares, and calculate the square root at the end. So, let's combine the first and second processes, which are to square all of the speeds and average them.

$$\begin{gathered} {v^2}_{avg}=\underset{i=1}{\overset{25}{\sum v^2}} \\ =\frac{{(15m/s)}^2+{(16m/s)}^2+{(17m/s)}^2+...+{(25m/s)}^2}{11} \\ =\underline{410m^2/s^2} \end{gathered}$$

The last step is to calculate the square root of the number :

localid="1649314027382" $$\begin{gathered} v_{rms}=\sqrt{{v^2}_{avg}} \\ =\sqrt{410m^2/s^2} \\ =\boxed{\mathbf{20}\mathbf{.}\mathbf{2}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}} \end{gathered}$$