Which of the following is greater for identical conditions and the same gas?a)most probable speedb)average speedc)root mean square speedd)most probable and average speed have the same valueCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

According to the formula, the root mean square speed is greater than the average speed and the average speed is greater than the most probable speed at given identical conditions and for the same gas.

Answer

Explanation:

To understand why the root mean square speed is greater than the most probable and average speed, we need to first understand what each of these terms represents.

Most Probable Speed:

The most probable speed is the speed at which the majority of gas particles in a sample are moving. It represents the peak of the Maxwell-Boltzmann speed distribution curve, which shows the distribution of speeds for gas particles at a given temperature.

Average Speed:

The average speed is the average of all the individual speeds of gas particles in a sample. It is calculated by summing up all the speeds and dividing by the total number of particles.

Root Mean Square Speed:

The root mean square speed is the square root of the average of the squares of all the individual speeds of gas particles in a sample. It is calculated by taking the square root of the average of the squares of the speeds.

Comparison:

For identical conditions and the same gas, the root mean square speed is always greater than the most probable and average speed. This can be explained by the mathematical relationship between these speeds.

The most probable speed represents the peak of the speed distribution curve, which means it is the speed at which the highest number of particles are moving. However, there are still particles moving at higher speeds, although in smaller numbers.

The average speed takes into account all the individual speeds of gas particles in the sample. It includes both the slower and faster-moving particles, but it doesn't give us a representative value that characterizes the majority of the particles.

On the other hand, the root mean square speed takes into account the squares of all the individual speeds. Since the squares of higher speeds are greater than the squares of lower speeds, the root mean square speed will be larger than both the most probable and average speed.

In summary, the root mean square speed is greater because it considers the squares of individual speeds, which increases the contribution of higher speeds to the overall value.

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