Question

Pressure P varies as $P=\frac{\alpha }{\beta }$ exp$\left(\frac{-\alpha }{K_B\theta }Z\right)$ where Z denotes distance,$K_B$ Boltzman's constant,$\theta$ absolute temperature and $\alpha ,\beta$ are constants.Derive the dimensions of $\beta$

• A. $\left[M^0{L}^0{T}^0\right]$
• B. $\left[M^0{L}^2{T}^0\right]$
• C. $\left[M^{-1}L{T}^{-2}\right]$
• D. $\left[M^0{L}^{-1}{T}^{-2}\right]$

Answer 1

Answer: (B)

$P=\frac{\alpha }{\beta }$ exp$\left(\frac{-\alpha }{k_B\theta }Z\right)$

Since the terms inside the exponential do not have any dimension,we can find out the dimension of $\alpha$

$\frac{\alpha \times L^{\prime }\times k}{J\times k}$

$\alpha$ has dimension of $\frac{J}{L^{\prime }}=\frac{M{L}^2{T}^{-2}}{L^{\prime }}=ML{T}^{-2}$

For $\frac{\alpha }{\beta }$ to have dimension of pressure ,i.e $M{L}^{-1}{T}^{-2}$

$\beta$ should have the dimension $L^2$ i.e,$M^0{L}^2{T}^0$