Question

Pressure depends on distance as , P= α/β exp (-αz/kθ), where α,β are constants, z is distance, k is Boltzmann constant and θ is temperature. Calculate the dimensions of β.

Answer 1

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Answer 2

The dimension of β is $\bold{[L^2]}$

Given:

$P=\frac{\alpha}{\beta} e^{-\frac{\alpha z}{k \theta}}$

Solution:  

Since, the exponentials are devoid of dimensions, the exponential part of the equation is ignored.  

Rest we have, $P=\frac{\alpha}{\beta}$

Where,

Alpha and beta has to be found out.  

Since, $\frac{\alpha z}{k \theta} = Dimensionless$

$\alpha=\frac{k \theta}{z}$

$Kinetic \ energy =\frac{3}{2} k T$

$k=\frac{K . E}{T}$

$\Rightarrow[k]=\left[M^{1} L^{2} T^{-2}\right]\left[K^{-1}\right]$

$\Rightarrow[z]=\left[L^{1}\right]$

$\Rightarrow[\theta]=\left[K^{1}\right]$

From these, we get the value of alpha as,

$[\alpha]=\left[M^{1} L^{1} T^{-2}\right]$

Now, we know the dimension of pressure,

$[P]=\left[M^{1} L^{-1} T^{-2}\right]$

$\beta=\frac{\alpha}{P}$

$\Rightarrow \beta=\frac{\left[M^{1} L^{1} T^{-2}\right]}{\left[M^{1} L^{-1} T^{-2}\right]}$

$\Rightarrow \beta=\left[L^{2}\right]$