A container is filled with a simple of gas having molecules with speed `alpha, 2 alpha, 3alpha, ….nalpha.` The ratio of average speed to root mean square speed is

Options

A. `sqrt((3 (n+1))/(2 (2n +1))`
B. `sqrt(((n+1))/(2 (2n+1))`
C. `sqrt((5 (2n+2))/(7(3n+1))`
D. `sqrt((3(n+2))/(5(3n+2))`

Answer

To solve the problem of finding the ratio of average speed to root mean square speed for a gas with molecular speeds of \( \alpha, 2\alpha, 3\alpha, \ldots, n\alpha \), we will follow these steps:

### Step 1: Calculate the Average Speed
The average speed \( V_{\text{avg}} \) is given by the formula:
\[ V_{\text{avg}} = \frac{V_1 + V_2 + V_3 + \ldots + V_n}{n} \]
Substituting the speeds:
\[ V_{\text{avg}} = \frac{\alpha + 2\alpha + 3\alpha + \ldots + n\alpha}{n} \]
Factoring out \( \alpha \):
\[ V_{\text{avg}} = \frac{\alpha(1 + 2 + 3 + \ldots + n)}{n} \]
Using the formula for the sum of the first \( n \) natural numbers:
\[ 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]
Thus, we have:
\[ V_{\text{avg}} = \frac{\alpha \cdot \frac{n(n + 1)}{2}}{n} = \frac{\alpha(n + 1)}{2} \]

### Step 2: Calculate the Root Mean Square Speed
The root mean square speed \( V_{\text{rms}} \) is given by:
\[ V_{\text{rms}} = \sqrt{\frac{V_1^2 + V_2^2 + V_3^2 + \ldots + V_n^2}{n}} \]
Substituting the speeds:
\[ V_{\text{rms}} = \sqrt{\frac{(\alpha^2) + (2\alpha)^2 + (3\alpha)^2 + \ldots + (n\alpha)^2}{n}} \]
This simplifies to:
\[ V_{\text{rms}} = \sqrt{\frac{\alpha^2(1^2 + 2^2 + 3^2 + \ldots + n^2)}{n}} \]
Factoring out \( \alpha^2 \):
\[ V_{\text{rms}} = \alpha \sqrt{\frac{1^2 + 2^2 + 3^2 + \ldots + n^2}{n}} \]
Using the formula for the sum of squares of the first \( n \) natural numbers:
\[ 1^2 + 2^2 + 3^2 + \ldots + n^2 = \frac{n(n + 1)(2n + 1)}{6} \]
Thus, we have:
\[ V_{\text{rms}} = \alpha \sqrt{\frac{\frac{n(n + 1)(2n + 1)}{6}}{n}} = \alpha \sqrt{\frac{(n + 1)(2n + 1)}{6}} \]

### Step 3: Calculate the Ratio of Average Speed to Root Mean Square Speed
Now we find the ratio:
\[ \frac{V_{\text{avg}}}{V_{\text{rms}}} = \frac{\frac{\alpha(n + 1)}{2}}{\alpha \sqrt{\frac{(n + 1)(2n + 1)}{6}}} \]
The \( \alpha \) cancels out:
\[ \frac{V_{\text{avg}}}{V_{\text{rms}}} = \frac{(n + 1)}{2 \sqrt{\frac{(n + 1)(2n + 1)}{6}}} \]
Simplifying further:
\[ = \frac{(n + 1) \cdot \sqrt{6}}{2 \sqrt{(n + 1)(2n + 1)}} \]
\[ = \frac{\sqrt{6(n + 1)}}{2 \sqrt{2n + 1}} \]

### Final Result
Thus, the ratio of average speed to root mean square speed is:
\[ \frac{V_{\text{avg}}}{V_{\text{rms}}} = \frac{\sqrt{6(n + 1)}}{2\sqrt{2n + 1}} \]