You are given the following group of particles \(\left(N_{n}\right.\) represents the number of particles that have a speed \(v_{n}\) ): $$\begin{array}{lc}N_{n} & v_{n}(\mathrm{~km} / \mathrm{s}) \\\\\hline 2 & 1.0 \\\4 & 2.0 \\\6 & 3.0 \\\8 & 4.0 \\\2 & 5.0\end{array}$$ (a) Compute the average speed \(v_{\mathrm{av}} .(b)\) Compute the rootmean- square speed \(v_{\text {rms. }}\) (c) Among the five speeds shown, which is the most probable speed \(v_{\mathrm{p}}\) for the entire group?

Short Answer

Expert verified
The average speed (\(v_{av}\)) is 2.73 km/s. The rootmean-square speed (\(v_{rms}\)) is 3.27 km/s. The most probable speed (\(v_p\)) is 4.0 km/s.

Step by step solution

01

Calculate average speed

The average speed \(v_{av}\) is given by: \[ v_{av} = \frac{\text{total sum of all speeds}}{\text{total sum of all instances}} = \frac{1*2 + 2*4 + 3*6 + 4*8 + 5*2} {2 + 4 + 6 + 8 + 2} = \frac{60}{22} = 2.73 \, \mathrm{km/s}\]
02

Calculate rootmean-square speed

The rootmean-square speed \(v_{rms}\) is given by:\[ v_{rms} = \sqrt{\frac{\text{sum of the squares of all speeds}}{\text{total sum of all instances}}} = \sqrt{\frac{1^2*2 + 2^2*4 + 3^2*6 + 4^2*8 + 5^2*2}{2 + 4 + 6 + 8 + 2}} = \sqrt{\frac{120}{22}} = 3.27 \, \mathrm{km/s}\]
03

Determine the most probable speed

The most probable speed \(v_p\) for the entire group is the one associated with the highest number of instances. Let's look directly at the data: 2 instances of 1.0 km/s. 4 instances of 2.0 km/s. 6 instances of 3.0 km/s. 8 instances of 4.0 km/s. 2 instances of 5.0 km/s. From these counts, 4.0 km/s has the highest number of instances (8), so it is the most probable speed.

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Most popular questions from this chapter

Calculate the root-mean-square speed of ammonia \(\left(\mathrm{NH}_{3}\right)\) molecules at \(56.0^{\circ} \mathrm{C}\). An atom of nitrogen has a mass of \(2.33 \times 10^{-26} \mathrm{~kg}\) and an atom of hydrogen has a mass of \(1.67 \times 10^{-27} \mathrm{~kg}\)

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The speeds of a group of ten molecules are \(2.0,3.0,4.0, \ldots\), \(11 \mathrm{~km} / \mathrm{s}\). ( \(a\) ) Find the average speed of the group. (b) Calculate the root-mean-square speed of the group.

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