The number density of molecules of a gas depends on their distance `r` from the origin as, `n(r)=n_(0) e^(- alpha r^(4))`. Then the total number of molecules is proportional to :

Options

A. `n_0 alpha^(-3//4)`
B. `sqrt(n_0) alpha^(1//2)`
C. `n_0 alpha^(1//4)`
D. `n_0 alpha^(-3)`

Explanation

To solve the problem, we need to find the total number of molecules based on the given number density function \( n(r) = n_0 e^{-\alpha r^4} \). The total number of molecules can be found by integrating the number density over the entire volume of the gas.

### Step-by-Step Solution:

1. **Understanding Number Density**:
The number density of the gas is given by:
\[ n(r) = n_0 e^{-\alpha r^4} \]
where \( n_0 \) is a constant and \( \alpha \) is a positive constant.

2. **Volume Element in Spherical Coordinates**:
To find the total number of molecules, we need to integrate the number density over the volume. In spherical coordinates, the volume element \( dv \) is given by:
\[ dv = 4\pi r^2 dr \]

3. **Setting Up the Integral**:
The total number of molecules \( N \) is given by the integral of the number density times the volume element:
\[ N = \int_0^\infty n(r) \, dv = \int_0^\infty n_0 e^{-\alpha r^4} \cdot 4\pi r^2 \, dr \]
Thus, we can rewrite the integral as:
\[ N = 4\pi n_0 \int_0^\infty r^2 e^{-\alpha r^4} \, dr \]

4. **Changing Variables**:
To evaluate the integral, we can use a substitution. Let:
\[ u = \alpha r^4 \implies du = 4\alpha r^3 dr \implies dr = \frac{du}{4\alpha r^3} \]
We also need to express \( r^2 \) in terms of \( u \):
\[ r^2 = \left(\frac{u}{\alpha}\right)^{1/2} \cdot \left(\frac{1}{\alpha}\right)^{1/4} \]
The limits of integration remain from \( 0 \) to \( \infty \).

5. **Evaluating the Integral**:
The integral becomes:
\[ \int_0^\infty r^2 e^{-\alpha r^4} \, dr = \int_0^\infty \left(\frac{u}{\alpha}\right)^{1/2} e^{-u} \cdot \frac{du}{4\alpha \left(\frac{u}{\alpha}\right)^{3/4}} = \frac{1}{4\alpha^{3/4}} \int_0^\infty u^{1/2} e^{-u} \, du \]
The integral \( \int_0^\infty u^{1/2} e^{-u} \, du \) is a known result and equals \( \Gamma(3/2) = \frac{\sqrt{\pi}}{2} \).

6. **Final Expression**:
Substituting back, we have:
\[ N = 4\pi n_0 \cdot \frac{1}{4\alpha^{3/4}} \cdot \frac{\sqrt{\pi}}{2} = n_0 \cdot \frac{\pi^{3/2}}{2\alpha^{3/4}} \]
Therefore, the total number of molecules \( N \) is proportional to:
\[ N \propto n_0 \alpha^{-3/4} \]

### Conclusion:
The total number of molecules is proportional to \( n_0 \alpha^{-3/4} \).