The number of particles is given by `n = -D(n_(2) - n_(1))/( x_(2) - x_(1))` crossing a unit area perpendicular to X - axis in unit time , where `n_(1)and n_(2)` are particles per unit volume for the value of `x` meant to `x_(2) and x_(1)` . Find the dimensions of `D` called diffusion constant.

Options

A. `[M^(0) L T^(-2)]`
B. `[M^(0) L^(2) T^(-4)]`
C. `[M^(0) L^(2) T^(-2)]`
D. `[M^(0) L^(2) T^(-1)]`

Explanation

To find the dimensions of the diffusion constant \( D \) from the equation:

\[ n = -D \frac{(n_2 - n_1)}{(x_2 - x_1)} \]

we will analyze each component step by step.

### Step 1: Understand the terms in the equation
- \( n \) represents the number of particles crossing a unit area perpendicular to the X-axis in unit time.
- \( n_1 \) and \( n_2 \) are the number of particles per unit volume.
- \( x_1 \) and \( x_2 \) are positions along the X-axis.

### Step 2: Determine the dimensions of \( n \)
Since \( n \) represents the number of particles crossing a unit area per unit time, it can be expressed dimensionally as:

\[ [n] = \frac{\text{number of particles}}{\text{area} \cdot \text{time}} = \frac{1}{L^2 \cdot T} = L^{-2} T^{-1} \]

### Step 3: Determine the dimensions of \( n_1 \) and \( n_2 \)
Both \( n_1 \) and \( n_2 \) are the number of particles per unit volume, which can be expressed dimensionally as:

\[ [n_1] = [n_2] = \frac{\text{number of particles}}{\text{volume}} = \frac{1}{L^3} = L^{-3} \]

### Step 4: Calculate the dimensions of \( n_2 - n_1 \)
Since \( n_1 \) and \( n_2 \) have the same dimensions, the difference \( n_2 - n_1 \) will also have the same dimensions:

\[ [n_2 - n_1] = L^{-3} \]

### Step 5: Determine the dimensions of \( x_2 - x_1 \)
The difference between two positions \( x_2 - x_1 \) is simply a length, so:

\[ [x_2 - x_1] = L \]

### Step 6: Substitute dimensions into the equation
Now we can substitute the dimensions into the equation for \( D \):

\[ D = -\frac{n (x_2 - x_1)}{(n_2 - n_1)} \]

Substituting the dimensions we have:

\[ [D] = \frac{[n] \cdot [x_2 - x_1]}{[n_2 - n_1]} = \frac{(L^{-2} T^{-1}) \cdot (L)}{L^{-3}} \]

### Step 7: Simplify the dimensions of \( D \)
Now we simplify the right side:

\[ [D] = \frac{L^{-2} T^{-1} \cdot L}{L^{-3}} = \frac{L^{-1} T^{-1}}{L^{-3}} = L^{-1} T^{-1} \cdot L^{3} = L^{2} T^{-1} \]

### Final Result
Thus, the dimensions of the diffusion constant \( D \) are:

\[ [D] = L^{2} T^{-1} \]