Question

In a typical combustion engine the work done by a gas molecule is given by\[W = {\alpha ^2}\beta {e^{\dfrac{{ - \beta {x^2}}}{{KT}}}}\], where x is the displacement, k is the Boltzmann constant and T is the temperature. If \[\alpha \]and \[\beta \]are constants, find the dimension of\[\alpha \].
A. \[\left[ {{M^0}L{T^0}} \right]\]
B. \[\left[ {{M^2}L{T^{ - 2}}} \right]\]
C. \[\left[ {ML{T^{ - 2}}} \right]\]
D. \[\left[ {ML{T^{ - 1}}} \right]\]

Answer 1

Hint:Before going to solve this question we need to understand the work done and dimensional formula. Work is nothing but a force needed to move an object from one place to another. Work done can be positive, negative, or zero depending on the angle between the force applied and the displacement. The dimensional formula is the expression of a physical quantity in terms of their basic unit with proper dimensions.

Complete step by step solution:
Now, consider the given equation,
\[W = {\alpha ^2}\beta {e^{\dfrac{{ - \beta {x^2}}}{{KT}}}}\]
The dimension of exponential or any power to the exponential is dimensionless. Therefore, we can write it as 1. That is,
\[\left[ W \right] = {\left[ \alpha \right]^2}\left[ \beta \right]\] …….. (1)
Here, we need to find the dimension of \[\alpha \]. if we had known the value of \[\beta \] we could have found the value of \[\alpha \], since the work done is given. But, now in order to find the value dimension of \[\beta \]we consider, the term \[ - \dfrac{{\beta {x^2}}}{{KT}}\] which is dimensionless

So, we can write this term as,
\[\dfrac{{\left[ \beta \right]{{\left[ x \right]}^2}}}{{\left[ {{K_B}} \right]\left[ T \right]}} = 1\]
\[\left[ \beta \right] = \dfrac{{\left[ T \right]\left[ {{K_B}} \right]}}{{{{\left[ x \right]}^2}}}\]
Now, substituting the dimensional formula for temperature and other terms, we get
\[\left[ \beta \right] = \dfrac{{\left[ K \right]\left[ {M{L^2}{T^{ - 2}}{K^{ - 1}}} \right]}}{{{{\left[ L \right]}^2}}}\]
\[ \Rightarrow \left[ \beta \right] = \left[ {M{T^{ - 2}}} \right]\]

Now, write the dimensional formula for work done and substitute the dimensional formula of \[\beta \] in equation (1) we get,
\[\left[ {M{L^2}{T^{ - 2}}} \right] = {\left[ \alpha \right]^2}\left[ {M{T^{ - 2}}} \right]\]
\[\Rightarrow \left[ {{L^2}} \right] = {\left[ \alpha \right]^2}\]
\[\Rightarrow \left[ \alpha \right] = \left[ L \right]\]
\[\therefore \left[ \alpha \right] = \left[ {{M^0}L{T^0}} \right]\]
Therefore, the dimensional formula for \[\alpha \] is \[\left[ {{M^0}L{T^0}} \right]\].

Hence, Option A is the correct answer

Note:Remember that, any function or an equation that contains the exponential term is dimensionless. Therefore, when you are going to find the dimensional formula for the given equation check with the exponential terms carefully and then solve it.