Question

The force of interaction between two atoms is given by$\mathrm{F}=\mathrm{\alpha \beta exp}\left[\frac{-{\mathrm{x}}^2}{\mathrm{\alpha kt}}\right]$; where$\mathrm{x}$ is the distance, $\mathrm{k}$ is the Boltzmann constant, and$\mathrm{T}$ is the temperature $\mathrm{\alpha }$and$\mathrm{\beta }$ are two constants. The dimension of $\mathrm{\beta }$is

• A. ${\mathrm{M}}^2{\mathrm{L}}^2{\mathrm{T}}^{-2}$
• B. ${\mathrm{M}}^2{\mathrm{LT}}^{-4}$
• C. ${\mathrm{M}}^0{\mathrm{L}}^2{\mathrm{T}}^{-4}$
• D. ${\mathrm{MLT}}^{-4}$

Answer 1

The correct option is B

${\mathrm{M}}^2{\mathrm{LT}}^{-4}$

The correct answer is (b)

The explanation for the correct option:

(b)

• Explanation: Three-dimensional expression M stands for mass, L stands for length, and T stands for time. As a result, dimensional expression is transformed into unit expression.
• Surface energy and potential energy have the same dimensions.

$$\begin{gathered} \left[\alpha \right]=\left[{\mathrm{M}}^{-1}{\mathrm{T}}^2\right] \\ \left[\mathrm{F}\right]=\left[\mathrm{a}\mathrm{\beta }\right] \\ \left[{\mathrm{MLT}}^{-2}\right]=\left[{\mathrm{M}}^{-1}{\mathrm{T}}^2\right]\left[\mathrm{\beta }\right] \\ \left[\mathrm{\beta }\right]={\mathrm{M}}^2{\mathrm{LT}}^{-4} \end{gathered}$$

The explanation for the incorrect option:

(a)

• Explanation: ${\mathrm{M}}^2{\mathrm{L}}^2{\mathrm{T}}^{-2}$ the dimensional formula is surface energy. In the three-dimensional representation, M represents mass, L represents length, and T represents time.
• The dimensions of surface energy and potential energy are the same.

(c)

• Explanation: ${\mathrm{M}}^0{\mathrm{L}}^2{\mathrm{T}}^{-4}$ the dimensional formula is surface energy. In the three-dimensional representation, M represents mass, L represents length, and T represents time.
• The dimensions of surface energy and potential energy are the same.

(d)

• Explanation: The energy at the surface ${\mathrm{MLT}}^{-4}$dimensional formula is In the three-dimensional representation, M represents mass, L represents length, and T represents time.
• Surface energy and potential energy have the same dimensions.

The correct answer is (b): surface modification and potential energy have the same dimensions.