In the relation P = β α e k θ α Z P is pressure, Z is distance, k is Boltzmannconstant and θ is the temperature. The dimensional formula of β will be

Question

In the relation P = β α ​ e k θ α Z ​ P is pressure, Z is distance, k is Boltzmannconstant and θ is the temperature. The dimensional formula of β will be

TestHub · Accepted Answer

The argument of the exponential function must be dimensionless. Therefore, $\frac{α Z}{k θ}$ is dimensionless, which means the dimensions of $α$ are the same as $k θ$ .

Since $P = \frac{α}{β} e^{\frac{α Z}{k θ}}$ , $\frac{α}{β}$ must have the dimensions of pressure.

Thus, $[β] = \frac{[ α ]}{[ P ]} = \frac{[ k θ ]}{[ P ]}$ .

The dimensions of $k θ$ are energy, i.e., $[M L^{2} T^{- 2}]$ .

Pressure is force per area, so $[P] = [M L^{- 1} T^{- 2}]$ .

Therefore, $[β] = \frac{[ M L ^{2} T ^{- 2} ]}{[ M L ^{- 1} T ^{- 2} ]} = [L^{3}] = [M^{0} L^{3} T^{0}]$ .

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