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Chapter 9: Problem 1288

If three molecules have velocities \(0.5,1\) and 2 the ratio of rms speed and average speed is (The velocities are in \(\mathrm{km} / \mathrm{s}\) ) (A) \(0.134\) (B) \(1.34\) (C) \(1.134\) (D) \(13.4\)

Short Answer

Expert verified
The ratio of rms speed to average speed for the given velocities is approximately 1.134.

Step by step solution

01

Calculate the average speed

To calculate the average speed, we simply add the velocities of all three molecules, and then divide the sum by the number of molecules (3). Average speed = (0.5 + 1 + 2) / 3
02

Solve for the average speed

Now, we can substitute the values and calculate the average speed: Average speed = (0.5 + 1 + 2) / 3 = 3.5 / 3 = 1.16667 km/s
03

Calculate the rms speed

To calculate the root-mean-square speed, we have to first square each molecule's velocity, add those squares together, divide the sum by the number of molecules (3), and then take the square root of the result. Rms speed = \(\sqrt{\frac{(0.5^2 + 1^2 + 2^2)}{3}}\)
04

Solve for the rms speed

Now, we can substitute the values and calculate the rms speed: Rms speed = \(\sqrt{\frac{(0.5^2 + 1^2 + 2^2)}{3}}\) = \(\sqrt{\frac{(0.25 + 1 + 4)}{3}}\) = \(\sqrt{\frac{5.25}{3}}\) = 1.31529 km/s
05

Calculate the ratio of rms speed to average speed

Finally, we will find the ratio of the rms speed and the average speed: Ratio = Rms speed / Average speed = 1.31529 / 1.16667
06

Solve for the ratio

Now, we can substitute the values we found in the previous steps to get the ratio: Ratio = 1.31529 / 1.16667 ≈ 1.127 Looking at the answer choices, the closest answer is (C) 1.134, hence the ratio of rms speed to average speed for the given velocities is approximately 1.134.

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