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Chapter 22: Problem 14

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.

Short Answer

Expert verified
(a) The average speed of the group is 6.0 km/s (b) The root-mean-square speed of the group is approximately 7.10 km/s.

Step by step solution

01

Calculate Average Speed

The average speed can be calculated by adding up all the molecule speeds, and then dividing by the count of molecules. Here are the speeds: \(2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0\) km/s, totaling ten molecules. Summing these up equals 60 km/s. Thus, the average speed equals \(\frac{60 \text{ km/s}}{10} = 6.0 \text{ km/s}\).
02

Calculate Square of Each Speed

In order to calculate the root-mean-square speed, we first need to find the square of each molecule's speed. The squared speeds are: \(4.0, 9.0, 16.0, 25.0, 36.0, 49.0, 64.0, 81.0, 100.0, 121.0\) (km/s)\(^2\).
03

Calculate Root-Mean-Square Speed

The root-mean-square speed is found by taking the square root of the average of these squared speeds. Add up the squared speeds to get 505 (km/s)\(^2\), and divide by the number of molecules (10) to find the average, which equals 50.5 (km/s)\(^2\). Taking the square root of this gives approximately 7.10 km/s.

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Most popular questions from this chapter

Show that the constant \(a\) in the van der Waals equation can be written in units of \(\frac{\text { energy per particle }}{\text { particle density }}\)

Calculate the root-mean-square speed of smoke particles of mass \(5.2 \times 10^{-14} \mathrm{~g}\) in air at \(14^{\circ} \mathrm{C}\) and \(1.07\) atm pressure.

At what frequency would the wavelength of sound be on the order of the mean free path in nitrogen at \(1.02\) atm pressure and \(18.0^{\circ} \mathrm{C} ?\) Take the diameter of the nitrogen molecule to be \(315 \mathrm{pm}\)

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?

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