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Pressure ($P$): Force per unit area.

$$[P] = \frac{[F]}{[A]} = \frac{[MLT^{-2}]}{[L^2]} = [ML^{-1}T^{-2}]$$

Distance ($z$):

$$[z] = [L]$$

Boltzmann's constant ($k$): Energy per unit temperature. Energy is force times distance.

$$[k] = \frac{[Energy]}{[Temperature]} = \frac{[MLT^{-2} \cdot L]}{[\theta]} = [ML^2T^{-2}\theta^{-1}]$$

Temperature ($\theta$):

$$[\theta] = [\theta]$$

The exponent of $e$ must be dimensionless.

$$\left[\frac{\alpha z}{k\theta}\right] = [M^0L^0T^0]$$

Therefore, the dimensions of the numerator must equal the dimensions of the denominator:

$$[\alpha][z] = [k][\theta]$$

Substitute the known dimensions:

$$[\alpha][L] = [ML^2T^{-2}\theta^{-1}][\theta]$$ $$[\alpha][L] = [ML^2T^{-2}]$$

Solve for $[\alpha]$:

$$[\alpha] = \frac{[ML^2T^{-2}]}{[L]} = [MLT^{-2}]$$

The term $\frac{\alpha}{\beta}$ must have the same dimensions as $P$, because $e^{-\frac{\alpha z}{k\theta}}$ is a dimensionless quantity.

$$\left[\frac{\alpha}{\beta}\right] = [P]$$

Substitute the dimensions of $[\alpha]$ and $[P]$:

$$\frac{[MLT^{-2}]}{[\beta]} = [ML^{-1}T^{-2}]$$

Solve for $[\beta]$:

$$[\beta] = \frac{[MLT^{-2}]}{[ML^{-1}T^{-2}]}$$ $$[\beta] = [M^{1-1}L^{1-(-1)}T^{-2-(-2)}]$$ $$[\beta] = [M^0L^2T^0]$$

The dimensions of $\beta$ are $[L^2]$.