Question

Calculate the root mean square, average and most probable speed of oxygen at ${27^0}C$.

Answer 1

Hint: Using the Maxwell distribution of molecular velocities, we have three types of molecular speeds. They are most probable speed, average speed and root mean square speed.

Complete step by step answer:
-Most probable speed is the speed possessed by the maximum number of molecules of a gas at a given temperature. The formula to find out most probable speed is,
Most probable speed, ${V_{mp}} = \sqrt {\dfrac{{2RT}}{M}} $
Where, R is the universal gas constant. Its value is $8.314$ $J{K^{ - 1}}mo{l^{ - 1}}$.
T is the temperature in Kelvin.
M is the molar mass of a gas molecule in kg.
-Average speed is the arithmetic mean of different speeds possessed by the molecules of a gas at a given temperature. The formula for average speed is,
Average speed, ${V_{av}} = \sqrt {\dfrac{{8RT}}{{\pi M}}} $
-Root mean square speed is the square root of the mean of the squares of different speeds possessed by molecules of a gas at a given temperature. The formula for root mean square speed is,
Root mean square speed, ${V_{rms}} = \sqrt {\dfrac{{3RT}}{M}} $
-We need to calculate the root mean square, average and most probable speed of oxygen at .
We know, molar mass of oxygen $ = 32g = 32 \times {10^{ - 3}}kg$
Given, temperature = $ = {27^0}C = 300K$\[\]
Now let us find out the speeds by substituting the values to the above equations.
 Most probable speed, ${V_{mp}} = \sqrt {\dfrac{{2RT}}{M}} $$ = \sqrt {\dfrac{{2 \times 8.314 \times 300}}{{32 \times {{10}^{ - 3}}}}} = 394.83m/s$
Average speed, ${V_{av}} = \sqrt {\dfrac{{8RT}}{{\pi M}}} $$ = \sqrt {\dfrac{{8 \times 8.314 \times 300}}{{3.14 \times 32 \times {{10}^{ - 3}}}}} = 445.63m/s$
-Root mean square speed, ${V_{rms}} = \sqrt {\dfrac{{3RT}}{M}} $\[ = \sqrt {\dfrac{{3 \times 8.314 \times 300}}{{32 \times {{10}^{ - 3}}}}} = 483.56m/s\]
-Hence the values of the root mean square, average and most probable speed of oxygen are $483.56m/s,445.63m/s$ and $394.83m/s$ respectively.

Note:
We can use a simple formula for finding the three velocities. First we need to calculate the most probable speed using the above equation. Then we can calculate average speed and root mean square speed using the ratio, ${V_{mp}}:{V_{av}}:{V_{rms}} = 1:1.128:1.225$.