Question

Most probable speed, average speed and RMS speed are related as
A. 1 : 1.128 : 1.224
B. 1 : 1.128 : 1.424
C. 1 : 2.128 : 1.224
D. 1 : 1.428 : 1.442

Answer 1

Hint: Think about the formulae for all the three terms given, root mean square velocity, average velocity, and the most probable speed and how they are related to the universal gas constant, temperature, and the molar mass.

Complete answer:
For the ratio of root mean square speed, average speed and most probable speed, we use the formulae $V_{rms}=\sqrt{{\displaystyle \frac{3RT}{M}}},V_{av}=\sqrt{{\displaystyle \frac{8RT}{\pi M}}}$and $V_{mps}=\sqrt{{\displaystyle \frac{2RT}{M}}}$.
There are common terms in all the equations, they are Universal gas constant (R), temperature (T) and Molar mass (M). So the ratio according to the order given in the question is:

$$\sqrt{{\displaystyle \frac{2RT}{M}}}:\sqrt{{\displaystyle \frac{8RT}{\pi M}}}:\sqrt{{\displaystyle \frac{3RT}{M}}}$$
By cancelling out all the common terms from these three equations, we get the ratio as:

$$\sqrt{2}:\sqrt{{\displaystyle \frac{8}{\pi }}}:\sqrt{3}$$
Now finding the values of these terms we get:

$$1.414:1.595:1.732$$
In each option, the value of the first term is 1, Thus we will divide all values by the first term to get the appropriate answer.
$$1:1.128:1.224$$

Thus, we get the ratio for most probable speed, average speed and root mean square speed as: 1 : 1.128 : 1.224

So, the correct answer is “Option A”.

Additional Information:
The kinetic molecular theory of gases should be understood to know about origins of ideal gas laws.
- According to the kinetic molecular theory, gaseous particles are in a state of constant random motion; individual particles move at different speeds, constantly colliding and changing directions.
- The most probable speed is the one which informs about the speed possessed by the maximum number of molecules of the gas. The most probable speed is the maximum value on Maxwell's distribution plot.
- The average speed is the sum of the speeds of all the molecules divided by the number of molecules.
- 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.

Note: The R, T and M used in the formulas of most probable speed, average speed and root mean square speed are the universal gas constant, temperature in Kelvin scale and the molar mass of gases respectively.
Remember to take the ratios in the order mentioned in the question, if you do not do this, you may go wrong in the step where we have to find the ratios relative to the first term given.