Question

The work done by a gas molecule in an isolated system is given by, $W = \alpha \beta^2 e^{-\frac{x^2}{akT}}$, where $x$ is the displacement, $k$ is the Boltzmann constant and $T$ is the temperature. $\alpha$ and $\beta$ are constants. Then the dimensions of $\beta$ will be :

• A. $[M^2 L T^{-2}]$
• B. $[M^0 L T^0]$
• C. $[M L T^{-2}]$
• D. $[M L^2 T^{-2}]$

Hint:

Arguments of exponential, logarithmic, and trigonometric functions are always dimensionless. Use this to find unknown dimensions in complex formulas.

Answer 1

The Correct Option is C

Step 1: The exponent in $e^{-\frac{x^2}{\alpha k T}}$ must be dimensionless. So, $[\alpha] = [\frac{x^2}{kT}]$.
Step 2: Dimensions of $kT$ are the same as Energy ($[ML^2T^{-2}]$). Thus, $[\alpha] = \frac{[L^2]}{[ML^2T^{-2}]} = [M^{-1}T^2]$.
Step 3: Work $W$ has dimensions $[ML^2T^{-2}]$. In the equation $W = \alpha \beta^2$, the exponential part is dimensionless.
Step 4: $[ML^2T^{-2}] = [M^{-1}T^2] \cdot [\beta^2]$.
Step 5: $[\beta^2] = [M^2L^2T^{-4}] \Rightarrow [\beta] = [MLT^{-2}]$.