Question

Calculate the temperature at which the root mean square velocity, the average velocity, and the most probable velocity of oxygen gas are all equal to $ 1500m{{s}^{-1}} $ .

Answer 1

Hint: We need to calculate the various velocities. This could be done by applying the appropriate formula and putting the data given.

Formula used: $ {{U}_{RMS}}=\sqrt{\dfrac{3RT}{M}} $
 $ {{V}_{av}}=\sqrt{\dfrac{8RT}{\pi M}} $
 $ {{U}_{MP}}=\sqrt{\dfrac{2RT}{M}} $
Here $ T $ is the temperature
 $ M $ stands for mass
 $ R $ stands for universal gas constant.

Complete step by step answer
We already know that,
Velocity of oxygen gas are equal for all :
The expression for the RMS speed is $ {U_{RMS}} = \sqrt {\dfrac{{3RT}}{M}} $
Substitute values in the above expression.
 $ 1500 = \sqrt {\dfrac{{3 \times 8.314 \times {T_{RMS}}}}{{32 \times {{10}^{ - 3}}}}} $
 $ \therefore {T_{RMS}} = 2886K $
(b) The expression for the average speed is $ {V_{av}} = \sqrt {\dfrac{{8RT}}{{\pi M}}} $ ​​
Substitute values in the above expression.
 $ 1500 = \sqrt {\dfrac{{8 \times 8.314 \times {T_{av}}}}{{3.1416 \times 32 \times {{10}^{ - 3}}}}} $
 $ \therefore {T_{av}} = 3399K $
(c) The expression for the most probable speed is $ {U_{MP}} = \sqrt {\dfrac{{2RT}}{M}} $ ​​
Substitute values in the above expression.
 $ 1500 = \sqrt {\dfrac{{3 \times 8.314 \times {T_{MP}}}}{{32 \times {{10}^{ - 3}}}}} $
 $ \therefore {T_{MP}} = 4330K $
Hence, we get all the answers.
Note
The root-mean-square speed is the measure of the speed of particles in a gas, defined as the square root of the average velocity-squared of the molecules in a gas. The root-mean-square speed takes into account both molecular weight and temperature, two factors that directly affect the kinetic energy of a material. Solids have the lowest kinetic energy whereas gases have the highest kinetic energy. Also, the compound with the lowest mass has a maximum RMS velocity.
Speed for which the derivative equals zero is the most probable speed. The average speed of molecules is the mean of all magnitudes of velocity at which molecules of the given gas are moving. Gases consist of particles (molecules or atoms) that are in constant random motion. The average kinetic energy of gas particles is proportional to the absolute temperature of the gas, and all gases at the same temperature have the same average kinetic energy.