If `C_(1), C_(2), C_(3) …..` represent the speeds on `n_(1), n_(2) , n_(3)…..` molecules, then the root mean square speed is

Options

A. `sqrt((n_(1)C_(1)^(2)+n_(2)C_(2)^(2)+n_(3)C_(3)^(2)+...)/(n_(1)+n_(2)+n_(3)+...))`
B. `sqrt(((n_(1)+n_(2)+n_(3)+...)^(2))/(n_(1)C_(1)^(2)+n_(2)C_(2)^(2)+n_(3)C_(3)^(2)+...))`
C. `sqrt(((n_(1)C_(1)))/(n_(1))+((n_(2)C_(2)))/(n_(2))+((n_(3)C_(3)))/(n_(3)))`
D. `sqrt(((n_(1)C_(1)+n_(2)C_(2)+n_(2)C_(3)+...)^(2))/(n_(1)+n_(2)+n_(3)+...))`

Answer

To find the root mean square speed of a collection of molecules with different speeds and quantities, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Root Mean Square Speed**:
The root mean square (RMS) speed is a measure of the average speed of particles in a gas. It is calculated by taking the square of the speeds of the molecules, finding the average of those squares, and then taking the square root of that average.

2. **Square the Speeds**:
For each speed \( c_i \) of the molecules, we first square the speed:
\[ c_1^2, c_2^2, c_3^2, \ldots \]

3. **Weighting by Number of Molecules**:
Since there are \( n_1 \) molecules with speed \( c_1 \), \( n_2 \) molecules with speed \( c_2 \), and so on, we multiply each squared speed by the number of molecules:
\[ n_1 c_1^2 + n_2 c_2^2 + n_3 c_3^2 + \ldots \]

4. **Total Number of Molecules**:
The total number of molecules is the sum of all the molecules:
\[ N = n_1 + n_2 + n_3 + \ldots \]

5. **Calculate the Mean of the Squared Speeds**:
To find the mean of the squared speeds, we divide the total from step 3 by the total number of molecules from step 4:
\[ \text{Mean} = \frac{n_1 c_1^2 + n_2 c_2^2 + n_3 c_3^2 + \ldots}{N} \]

6. **Taking the Square Root**:
Finally, we take the square root of the mean calculated in step 5 to find the root mean square speed:
\[ \text{RMS Speed} = \sqrt{\frac{n_1 c_1^2 + n_2 c_2^2 + n_3 c_3^2 + \ldots}{N}} \]

### Final Formula:
Thus, the root mean square speed \( v_{rms} \) can be expressed as:
\[ v_{rms} = \sqrt{\frac{n_1 c_1^2 + n_2 c_2^2 + n_3 c_3^2 + \ldots}{n_1 + n_2 + n_3 + \ldots}} \]