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

To understand the relationship between most probable speed, average speed, and root mean square (RMS) speed in the context of gas molecules, we can break down each term and see how they relate to one another. These speeds are important in kinetic theory and help us understand the behavior of gases at a molecular level.

Definitions of Speeds

Let's define each type of speed:

• Most Probable Speed (v_mp): This is the speed at which the maximum number of molecules are moving in a gas. It can be calculated using the formula:

v_mp = √(2kT/m)

• Average Speed (v_avg): This is the mean speed of all the molecules in the gas. The formula for average speed is:

v_avg = √(8kT/πm)

• RMS Speed (v_rms): This is the square root of the average of the squares of the speeds of the molecules. It is given by:

v_rms = √(3kT/m)

Relationship Between the Speeds

Now, let's derive the ratios between these speeds. The constants in the formulas involve the Boltzmann constant (k), temperature (T), and mass (m) of the gas molecules. By substituting these into the equations, we can find the relationships:

Calculating Ratios

To find the ratios, we can express each speed in terms of a common variable. For example, if we take the most probable speed as our base (1), we can find the ratios of average speed and RMS speed relative to it:

• From the formulas, we have:
  • v_mp = 1
  • v_avg = √(8/π) * v_mp ≈ 1.128 * v_mp
  • v_rms = √(3) * v_mp ≈ 1.224 * v_mp

Final Ratios

Thus, the ratios of most probable speed, average speed, and RMS speed can be summarized as:

1 : 1.128 : 1.224

Conclusion

Based on the calculations, the correct answer to the question is option A: 1 : 1.128 : 1.224. This relationship is crucial in understanding the distribution of molecular speeds in gases and is derived from the principles of kinetic theory.