The workdone by a gas molecule in an isolated system is given by, `W = alpha beta^2 e^( - (x^2)/(alpha k T) )`, where x is the displacement, k is the Boltzmann constant and T is the temperature, `alpha` and `beta` are constants. Then the dimension of B will be :

Options

A. `[ML^2T^(-2)]`
B. `[MLT^(-2)]`
C. `M^2LT^2]`
D. `[M^0LT^0]`

Answer

To find the dimension of \( \beta \) in the given equation for work done by a gas molecule, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Equation**: The work done by a gas molecule is given as:
\[ W = \alpha \beta^2 e^{-\frac{x^2}{\alpha k T}} \]
where \( W \) is work, \( x \) is displacement, \( k \) is the Boltzmann constant, and \( T \) is temperature.

2. **Identify Dimensions**:
- The dimension of work \( W \) is given by:
\[ [W] = [M][L^2][T^{-2}] \]
- The dimension of displacement \( x \) is:
\[ [x] = [L] \]
- The dimension of the Boltzmann constant \( k \) is:
\[ [k] = [M][L^2][T^{-2}][K^{-1}] \]
- The dimension of temperature \( T \) is:
\[ [T] = [K] \]

3. **Exponential Argument Must Be Dimensionless**: The argument of the exponential function must be dimensionless:
\[ \frac{x^2}{\alpha k T} \text{ must be dimensionless.} \]
This implies:
\[ [x^2] = [L^2], \quad [\alpha] = [\alpha], \quad [k] = [M][L^2][T^{-2}][K^{-1}], \quad [T] = [K] \]
Therefore, we can write:
\[ \frac{[L^2]}{[\alpha] \cdot [M][L^2][T^{-2}][K^{-1}] \cdot [K]} = 1 \]

4. **Set Up the Equation**: Rearranging gives:
\[ [\alpha] = \frac{[L^2]}{[M][L^2][T^{-2}][K^{-1}] \cdot [K]} = \frac{[L^2]}{[M][L^2][T^{-2}]} = \frac{1}{[M][T^{-2}]} \]
Therefore:
\[ [\alpha] = [M^{-1}][T^2] \]

5. **Substituting Back into Work Equation**: Now substituting back into the work equation:
\[ [W] = [\alpha][\beta^2] \]
This gives:
\[ [M][L^2][T^{-2}] = [M^{-1}][T^2][\beta^2] \]

6. **Solving for \(\beta^2\)**: Rearranging gives:
\[ [\beta^2] = [M][L^2][T^{-2}] \cdot [M][T^{-2}] = [M^2][L^2][T^{-4}] \]

7. **Finding the Dimension of \(\beta\)**: Taking the square root:
\[ [\beta] = [M][L][T^{-2}] \]

### Final Result:
The dimension of \( \beta \) is:
\[ [M][L][T^{-2}] \]