Question

The speed distribution for a sample of \(N\) gas particles is shown below. \(P(v) = 0\) for \(v > 2v_0\). How many particles have speeds between \(1.2v_0\) and \(1.8v_0\)?

• A. 0.2 N
• B. 0.4 N
• C. 0.6 N
• D. 0.8 N

Hint:

To calculate the number of particles in a specific speed range, find the area under the speed distribution curve for that range.

Answer 1

The Correct Option is B

Step 1: The problem provides the speed distribution function P(v), which describes the number of particles as a function of speed. The graph indicates that the distribution P(v) is nonzero for speeds between 0 and 2v_0, and zero for speeds higher than 2v_0.

Step 2: To determine how many particles have speeds between 1.2v_0 and 1.8v_0, we calculate the area under the curve from 1.2v_0 to 1.8v_0.

Step 3: From the given graph, the area between these two speeds is 0.4 of the total number of particles N.

Step 4: Therefore, the number of particles with speeds in this range is 0.4N.

Answer 2

To find the number of gas particles with speeds between \(1.2v_0\) and \(1.8v_0\), we look at the area under the speed distribution curve \(P(v)\) in that range.

The total area under the curve from \(0\) to \(2v_0\) represents the entire sample of particles, which is \(N\).

From the graph, the area between \(1.2v_0\) and \(1.8v_0\) accounts for 40% of the total area.

Therefore, the number of particles in this speed range is: 

\[ \text{Number of particles} = 0.4N \]

Final Answer: 0.4 N