Velocities of three molecules are 2,3 and 4 m/s. Ratio of mean velocity to R.M.S velocity is

Options

A. lt1
B. 1
C. gt1
D. gt2

Answer

To solve the problem of finding the ratio of mean velocity to root mean square (RMS) velocity for three molecules with velocities of 2 m/s, 3 m/s, and 4 m/s, we can follow these steps:

### Step 1: Calculate the Mean Velocity (V_mean)
The mean velocity is calculated using the formula:
\[ V_{\text{mean}} = \frac{V_1 + V_2 + V_3}{n} \]
where \( V_1, V_2, V_3 \) are the velocities of the molecules and \( n \) is the number of molecules.

Substituting the values:
\[ V_{\text{mean}} = \frac{2 + 3 + 4}{3} = \frac{9}{3} = 3 \, \text{m/s} \]

### Step 2: Calculate the RMS Velocity (V_rms)
The RMS velocity is calculated using the formula:
\[ V_{\text{rms}} = \sqrt{\frac{V_1^2 + V_2^2 + V_3^2}{n}} \]
First, we need to calculate the squares of the velocities:
\[ V_1^2 = 2^2 = 4, \quad V_2^2 = 3^2 = 9, \quad V_3^2 = 4^2 = 16 \]
Now, sum these squares:
\[ V_1^2 + V_2^2 + V_3^2 = 4 + 9 + 16 = 29 \]
Now, substitute this into the RMS formula:
\[ V_{\text{rms}} = \sqrt{\frac{29}{3}} \]

### Step 3: Calculate the Ratio of Mean Velocity to RMS Velocity
Now, we can find the ratio of mean velocity to RMS velocity:
\[ \text{Ratio} = \frac{V_{\text{mean}}}{V_{\text{rms}}} = \frac{3}{\sqrt{\frac{29}{3}}} \]
To simplify this, we can multiply the numerator and denominator by \(\sqrt{3}\):
\[ \text{Ratio} = \frac{3\sqrt{3}}{\sqrt{29}} \]

### Step 4: Final Result
The ratio of mean velocity to RMS velocity is:
\[ \text{Ratio} = \frac{3\sqrt{3}}{\sqrt{29}} \]