Understanding Molecular Speed: Finding v avg and v-rms Using a Histogram

[Niki4444]

Hello! I have a question. When I am provided with a histogram, with % of molecules along the y-axis and speed range (m/s) along the x-axis, how do I use this to determine the average speed (v avg) and v-rms (root mean square) of the molecules in the gas? I am very puzzled. Any suggestions or thoughts to point me in the right direction would be hugely appreciated. Thank you so much.

 

[dynamicsolo]

For the average speed, you will perform what is called a "weighted average" on the histogram. You have some number of bins for velocity and some percentage of the total number of molecules in each speed bin. Convert the percentages into decimal fractions (e.g., 6% = .06). Find the center of the velocity range for each bin. You will then add up the terms:

(fraction in first bin) · (speed midpoint for first bin) + (fraction in second bin) · (speed midpoint for second bin) + ... + (fraction in last bin) · (speed midpoint for last bin).

The sum will be the weighted average for molecular speed.

You do something similar for root-mean-square speed, but now the sum is:

(fraction in first bin) · (speed midpoint for first bin)^2 + (fraction in second bin) · (speed midpoint for second bin)^2 + ... + (fraction in last bin) · (speed midpoint for last bin)^2
[that is to say, square each speed midpoint first, multiply that by its corresponding fraction, then sum all the terms].

Finally, take the square root of the sum you've found; this is your root-mean-squared speed. It will be a different result from the weighted average.

 

[Niki4444]

Thank you! That makes much more sense than the confusion I was having over what is v-avg, what is v-rms, and what is v-squared... from a picture.
Thank you, thank you, thank you! I get it now.