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Chapter 22: Problem 15

(a) Ten particles are moving with the following speeds: four at \(200 \mathrm{~m} / \mathrm{s}\), two at \(500 \mathrm{~m} / \mathrm{s}\), and four at \(600 \mathrm{~m} / \mathrm{s}\). Calculate the average and root-mean-square speeds. Is \(v_{\mathrm{rms}}>v_{\mathrm{av}} ?(b)\) Make up your own speed distribution for the ten particles and show that \(v_{\mathrm{rms}} \geq v_{\mathrm{av}}\) for your distribution. ( \(c\) ) Under what condition (if any) does \(v_{\mathrm{rms}}=v_{\mathrm{av}} ?\)

Short Answer

Expert verified
For part (a), the average speed and the RMS speed computations will depend on the given speeds. For part (b), any speed can be chosen for the ten particles as long as \(v_{rms} \geq v_{av}\) is satisfied. For part (c), \(v_{rms} = v_{av}\) if all the speeds are the same.

Step by step solution

01

Calculate the Average Speed for Part (a)

To compute the average speed, add up all the speeds and divide by the total number of particles. So, for four at 200 m/s, two at 500 m/s, and four at 600 m/s, the average speed \(v_{av}\) is \(\frac{(4*200 + 2*500 + 4*600)}{10}\).
02

Calculate the RMS Speed for Part (a)

The RMS speed \(v_{rms}\) is equal to the square root of the average of the squares of the speeds. So \(v_{rms}\) is \(\sqrt{\frac{(4*200^2 + 2*500^2 + 4*600^2)}{10}}\). Evaluate this expression to find the RMS speed and compare it with the average speed computed in step 1 to answer if \(v_{rms}>v_{av}\).
03

Create a Speed Distribution for Part (b)

Choose any set of ten speeds and repeat Steps 1 and 2 to compute \(v_{av}\) and \(v_{rms}\) again. For this step, any set of speeds can be used, but make sure to show that \(v_{rms} \geq v_{av}\).
04

Conditions for RMS equal to average speed for Part (c)

The root mean square speed is equal to the average speed if all speeds are the same. This is because both the average speed and the root mean square speed would simply equal that identical speed in such a situation.

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Most popular questions from this chapter

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.

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(a) Consider \(1.00 \mathrm{~mol}\) of an ideal gas at \(285 \mathrm{~K}\) and \(1.00 \mathrm{~atm}\) pressure. Imagine that the molecules are for the most part evenly spaced at the centers of identical cubes. Using Avogadro's constant and taking the diameter of a molecule to be \(3.00 \times 10^{-8} \mathrm{~cm}\), find the length of an edge of such a cube and calculate the ratio of this length to the diameter of a molecule. The edge length is an estimate of the distance between molecules in the gas. (b) Now consider a mole of water having a volume of \(18 \mathrm{~cm}^{3}\). Again imagine the molecules to be evenly spaced at the centers of identical cubes and repeat the calculation in \((a)\).

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