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.$\theta$ temperature and $\alpha ,\beta$ are constants.Derive the dimensions of $\beta$

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

Answer 1

The correct option is B $\left[M^0{L}^0{T}^0\right]$

$\frac{\alpha Z}{K_B\theta }=M^0{L}^0{T}^0{K}^0\alpha =\frac{K_B\theta }{Z}=\frac{\left[M{L}^2{T}^{-2}{K}^{-1}\right]\left[K\right]}{\left[L\right]}\alpha =ML{T}^{-2}\left[P\right]=\frac{\left[\alpha \right]}{\left[\beta \right]}\beta =\frac{\alpha }{P}=\frac{ML{T}^{-2}}{M{L}^{-1}{T}^{-2}}=M^0{L}^0{T}^0$

Ans: $M^0{L}^0{T}^0$