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Question

The planetary model of the atom pictures electrons orbiting the atomic nucleus much as planets orbit the Sun. In this model you can view hydrogen, the simplest atom, ashaving a single electron in a circular orbit $1.06 \times 10^{-10} \textrm{ m}$ in diameter. (a) If the average speed of the electron in this orbit is known to be $2.20 \times 10^6 \textrm{ m/s}$, calculate the number of revolutions per second it makes about the nucleus. (b) What is the electron's average velocity?

Question by OpenStax is licensed under CC BY 4.0.
Final Answer

a) $6.61 \times 10^{15} \textrm{ rev}$

b) $0 \textrm{ m/s}$

Solution Video

OpenStax College Physics Solution, Chapter 2, Problem 15 (Problems & Exercises) (3:24)

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Calculator Screenshots

OpenStax College Physics, Chapter 2, Problem 15 (PE) calculator screenshot 1
Video Transcript

This is College Physics Answers with Shaun Dychko. In this solution, we're going to calculate the number of revolutions that this electron makes in the hydrogen atom around the nucleus in a second. Now, we're told that the diameter of its circular path is 1.06 times 10 to the minus 10 meters, and I've written 2<i>r</i> here instead of diameter just to reduce confusion with the variable letters that I've chosen, because down here when I talk about average speed, I'm saying it's distance traveled over time and I don't want that <i>d</i> to be confused with diameter. So, I wrote 2<i>r</i> over there. We're told that the average speed is 2.2 times 10 to the 6 meters per second. So, notice what I've done at the beginning here, is I've drawn a picture of the situation, and I have written down the information that we've been given. That's a really good way to start every problem really because that helps organize your thoughts and organize the information you begin and then you can find a solution easier that way. So, we're going to figure out what the total distance traveled is first and that'll give us an answer in meters. And then, knowing the size of the circle, we'll figure out how many revolutions make up that many meters. So, we'll find revolutions basically by converting the distance traveled from meters into revolutions by multiplying by one revolution for every so many meters circumference of the circle. So, we switch the two sides around here because we want to isolate <i>d</i> in the left and where you will multiply both sides by <i>t</i> in this line, and the <i>t</i>s cancel on the left making <i>d</i> by itself there. And so, <i>d</i> equals average speed multiplied by time. So, that's 2.2 times 10 to the 6 meters per second times 1 second and that makes 2.2 instead of 6 meters. And then, we'll find the number of revolutions in that distance traveled by multiplying the distance traveled by one revolution for every circumference of the circle, which is 2 Pi <i>r</i>, <i>r</i> being the radius. And... And up here, we have a number representing 2<i>r</i>. And so, let's move these factors around a little bit and just group the 2 beside the <i>r</i>. And then, we can substitute the number we've been given in the question directly for 2<i>r</i>, which by the way is diameter <i>D</i> in the formula for circumference of... Circumference is Pi <i>D</i> by the way. So, we substitute for 2<i>r</i>. We substitute 1.06 times 10 to the minus 10 meters and multiply that by pi in the denominator. All that gets divided into 2.2 times 10 to the 6 meters and we are left with 6.61 times 10 to the 15 revolutions of the electron around the nucleus per second. Now for part B, it says, 'What's the average velocity?' and they like to be tricky with these velocity questions because the electron after one complete circle returns to its starting position. Its displacement is 0 and so the velocity is 0.

Comments

Submitted by Anonymous (not verified) on Mon, 08/27/2018 - 22:09

Why did you divide pi from the 2r value? Shouldn't it be multiplied instead?

Submitted by ShaunDychko on Mon, 08/27/2018 - 22:23

Hi nicholebondurant, thanks for the question. I'm seeing pi multiplied by 2r, as you're correctly suggesting. This is to find the circumference so that we can multiply the distance traveled in one second by 1 revolution per circumference in order to calculate the number of revolutions in a second. Where, specifically, in the video did you notice dividing pi from the 2r? If you could give a video timestamp, that might be helpful.

All the best,

Shaun

In reply to by Anonymous (not verified)

Submitted by Anonymous (not verified) on Thu, 08/30/2018 - 15:47

3:03min shouldnt the 2.20E6 be divided by pi. And pi should be multiplying the 1.06E-10 however the way you entered it in the calculator you are dividing everything. I am missing something? My confusion comes from the calculator snapshots and because if I were to set up that specific calculation from minute 3:03 in the way it is specified in the video they I get another answer.. 

Submitted by ShaunDychko on Thu, 08/30/2018 - 21:14

Ah, I understand where you're coming from now. What you're seeing in the calculator screenshot is a different way of entering what you're expecting. Just to use simpler number for ease of typing, here are the two ways of entering the same calculation:

Version A: 12/(2*3). This is the style that's written in the video, and you're expecting to see this in the calculator screenshot.

Version B: 12/2/3. This is the style that I tend to type in my calculator, for no reason other than personal preference. The calculator always evaluates left to right, meaning that 12 with be divided by 2 first, and that result (namely, 6), will then be divided by 3, resulting in the same answer as for version A, which is 2.

If you have a more modern calculator than my TI-83, and yours displays fractions, I would suspect that when you try entering the same as Version B, that you might in fact be doing the calculation 12/(2/3), which is not the correct calculation.

Hope this helps, and thanks for the question,

Shaun

In reply to by Anonymous (not verified)