Question

Pressure P varies as $P=\frac{\alpha}{\beta}exp(-\frac{\alpha x}{k_{B}T})$ where x denotes the distance, $k_{B}$ is the Boltzmann's constant, T is the absolute temperature and $\alpha$ and $\beta$ are constant. The dimension of $\beta$ is ________

• A. $[MLT^{-2}]$
• B. $[ML^{-1}T^{-2}]$
• C. $[M^{0}L^{2}T^{0}]$
• D. $[M^{0}L^{0}T^{0}]$

Hint:

Exponential and trigonometric arguments are always dimensionless.

Answer 1

The Correct Option is C

Step 1: Concept
Dimensional Homogeneity. The exponent of $e$ must be dimensionless.
Step 2: Analysis
- Exponent term: $\frac{\alpha x}{k_B T}$ is dimensionless ($M^0 L^0 T^0$). - $[k_B T]$ has dimensions of energy: $[ML^2 T^{-2}]$. - $[\alpha] = \frac{[k_B T]}{[x]} = \frac{ML^2 T^{-2}}{L} = [ML T^{-2}]$.
Step 3: Calculation
- The term $\frac{\alpha}{\beta}$ must have dimensions of Pressure ($P$). - $[P] = [ML^{-1} T^{-2}]$. - $[\beta] = \frac{[\alpha]}{[P]} = \frac{ML T^{-2}}{ML^{-1} T^{-2}} = [L^2]$.
Step 4: Conclusion
Hence, the dimension of $\beta$ is $[M^0 L^2 T^0]$.
Final Answer:(C)