For a gas sample with `N_(0)` number of molecules, function `N(V)` is given by: `N(V) = (dN)/(dV) = ((3N_(0))/(V_(0)^(3)))V^(2)` for `0 lt V lt V_(0)` and `N(V) = 0` for `V gt V_(0)`. Where `dN` is number of molecules in speed range `V` to `V+ dV`. The rms speed of the molecules is:

Options

A. `sqrt((2)/(5))V_(0)`
B. `sqrt((3)/(5))V_(0)`
C. `sqrt(2)V_(0)`
D. `sqrt(3)V_(0)`

Explanation

To find the root mean square (RMS) speed of the molecules in the given gas sample, we will follow these steps:

### Step 1: Understand the given function
The function \( N(V) \) describes the distribution of molecules in a speed range. It is given by:
\[ N(V) = \frac{dN}{dV} = \frac{3N_0}{V_0^3} V^2 \quad \text{for } 0 < V < V_0 \]
and \( N(V) = 0 \) for \( V > V_0 \).

### Step 2: Write the expression for RMS speed
The RMS speed \( v_{rms} \) is defined as:
\[ v_{rms} = \sqrt{\frac{\int_0^{V_0} v^2 \, dN}{N_0}} \]
where \( dN \) is the number of molecules in the speed range \( V \) to \( V + dV \).

### Step 3: Substitute \( dN \) in the RMS formula
From the given function, we can express \( dN \):
\[ dN = N(V) \, dV = \frac{3N_0}{V_0^3} V^2 \, dV \]
Now substituting this into the RMS speed formula:
\[ v_{rms} = \sqrt{\frac{\int_0^{V_0} v^2 \left(\frac{3N_0}{V_0^3} V^2 \, dV\right)}{N_0}} \]

### Step 4: Simplify the expression
This simplifies to:
\[ v_{rms} = \sqrt{\frac{3N_0}{N_0 V_0^3} \int_0^{V_0} V^4 \, dV} \]
\[ = \sqrt{\frac{3}{V_0^3} \int_0^{V_0} V^4 \, dV} \]

### Step 5: Calculate the integral
Now we need to calculate the integral \( \int_0^{V_0} V^4 \, dV \):
\[ \int_0^{V_0} V^4 \, dV = \left[ \frac{V^5}{5} \right]_0^{V_0} = \frac{V_0^5}{5} \]

### Step 6: Substitute the integral back into the RMS formula
Substituting the result of the integral back:
\[ v_{rms} = \sqrt{\frac{3}{V_0^3} \cdot \frac{V_0^5}{5}} \]
\[ = \sqrt{\frac{3V_0^2}{5}} \]

### Step 7: Final expression for RMS speed
Thus, the final expression for the RMS speed is:
\[ v_{rms} = V_0 \sqrt{\frac{3}{5}} \]

### Summary of Steps
1. Understand the distribution function \( N(V) \).
2. Write the RMS speed formula.
3. Substitute \( dN \) into the RMS formula.
4. Simplify the expression.
5. Calculate the integral \( \int_0^{V_0} V^4 \, dV \).
6. Substitute the integral back into the RMS formula.
7. Derive the final expression for \( v_{rms} \).