A student asked me today (thanks, Austin) why there are two different expressions given on the AP exam for calculating the root mean square speed of a gas, and how this can be explained in terms of the use of k, R, m and M. Here’s the answer that I gave.

You will find two expressions on the data packet that comes with the AP chemistry exam that deal with the calculation of urms (root mean square speed).

urms = SQRT (3kT/m)

AND

urms = SQRT (3RT/M)

In the 2nd expression, R is the universal gas constant. The universal gas constant = 8.31 J mol-1 K-1 (i.e., one of the values that you see on the data packet).

In the 1st expression, k is the Boltzmann constant. The Boltzmann constant  = the universal gas constant / 6.022 x 1023 = 1.38 x 10-23 J K-1 (i.e., the value that you see on the data packet).

So, since in the first expression, we have replaced a term in the numerator of the 2nd expression, R, by dividing it by Avogadro’s number, in order to keep the expression consistent, i.e., in order to STILL calculate urms, we have to change a term in the denominator of the 2nd expression in the same way. Since the only term in the denominator in the 2nd expression is M, dividing it by 6.022 x 1023 and substituting it in the 1st expression, does the job.

So, M = mass of 1 mole of the gas in kg, and in the other case, m = mass of a SINGLE particle of the gas, also in kg.

As an example, if we are calculating the urms for say, O2 at 30oC, then we can either use;

urms = SQRT [(3)(8.314)(303)/(0.032)] = 486 ms-1

(this calculation is based on a mole of molecules)

OR

urms = SQRT [(3)(1.381 x 10-23)(303)/(0.032/6.022 x 1023)] = 486 ms-1

(this calculation is based on a single molecule)

I don’t EVER recall seeing a calculation on an AP exam that could NOT be dealt with by the R based expression, rather than the k based expression, so I wonder why it’s there. I hope that this helps.