The speeds of 25 molecules are distributed as follows: 5 in the range from 2 to 3 m/s, 10 in the range from 3 to 4 m/s, 5 in the range from 4 to 5 m/s, 3 in the range from 5 to 6 m/s, 1 in the range from 6 to 7 m/s, and 1 in the range from 7 to 8 m/s. Their average speed is about

Options

A. 2m/s
B. 3m/s
C. 4m/s
D. 5m/s

Answer

To find the average speed of the 25 molecules based on the given speed distribution, we can follow these steps:

### Step 1: Determine the Average Speed for Each Range
We need to calculate the average speed for each range of speeds given in the problem.

1. **For the range 2 to 3 m/s (5 molecules)**:
\[ V_{avg1} = \frac{2 + 3}{2} = 2.5 \, \text{m/s} \]

2. **For the range 3 to 4 m/s (10 molecules)**:
\[ V_{avg2} = \frac{3 + 4}{2} = 3.5 \, \text{m/s} \]

3. **For the range 4 to 5 m/s (5 molecules)**:
\[ V_{avg3} = \frac{4 + 5}{2} = 4.5 \, \text{m/s} \]

4. **For the range 5 to 6 m/s (3 molecules)**:
\[ V_{avg4} = \frac{5 + 6}{2} = 5.5 \, \text{m/s} \]

5. **For the range 6 to 7 m/s (1 molecule)**:
\[ V_{avg5} = \frac{6 + 7}{2} = 6.5 \, \text{m/s} \]

6. **For the range 7 to 8 m/s (1 molecule)**:
\[ V_{avg6} = \frac{7 + 8}{2} = 7.5 \, \text{m/s} \]

### Step 2: Calculate the Total Speed Contribution
Next, we multiply the average speed of each range by the number of molecules in that range to get the total speed contribution.

1. Contribution from 2 to 3 m/s:
\[ \text{Total Speed}_1 = 5 \times 2.5 = 12.5 \, \text{m/s} \]

2. Contribution from 3 to 4 m/s:
\[ \text{Total Speed}_2 = 10 \times 3.5 = 35.0 \, \text{m/s} \]

3. Contribution from 4 to 5 m/s:
\[ \text{Total Speed}_3 = 5 \times 4.5 = 22.5 \, \text{m/s} \]

4. Contribution from 5 to 6 m/s:
\[ \text{Total Speed}_4 = 3 \times 5.5 = 16.5 \, \text{m/s} \]

5. Contribution from 6 to 7 m/s:
\[ \text{Total Speed}_5 = 1 \times 6.5 = 6.5 \, \text{m/s} \]

6. Contribution from 7 to 8 m/s:
\[ \text{Total Speed}_6 = 1 \times 7.5 = 7.5 \, \text{m/s} \]

### Step 3: Sum All Speed Contributions
Now we sum all the contributions to get the total speed:

\[ \text{Total Speed} = 12.5 + 35.0 + 22.5 + 16.5 + 6.5 + 7.5 = 100.5 \, \text{m/s} \]

### Step 4: Calculate the Average Speed
Finally, we divide the total speed by the total number of molecules to find the average speed:

\[ V_{avg} = \frac{\text{Total Speed}}{\text{Total Number of Molecules}} = \frac{100.5}{25} = 4.02 \, \text{m/s} \]

### Final Answer
The average speed of the molecules is approximately **4.02 m/s**.

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