Question

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

(a) $v_{avg}$and

(b) $v_{rms}$.

Short answer

(a) $v_{avg}=\mathbf{20}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$

(b)$v_{rms}=\mathbf{20}\mathbf{.}\mathbf{2}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$

Step-by-step solution

Given information and formula used

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

Calculating vavg

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

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

Calculating vrms

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