Question

$p \: = \: \frac{ \alpha }{ \beta } {e}^{ \frac{ \alpha z}{k \infty } } \\ \\ here \: p \: = \: pressure \\ z = \: length \\ \infty \: = \: temperature \\ k = \: boltzmann \: constant \\ \\ \\ find \: the \: dimensions \: o f \: \alpha \: and \: \beta$

Answer 1

Hello mate.

Thanks for asking this question.

Your answer is =>

As you know , the index of any number or any quantity is a dimensionless quantity.

so ,

$=>\frac {\alpha z}{k \infty } = {{[M^{0} L^{0} T^{0}]}}$

here is

M = mass

L = length

T = time

but we know that ,

$a^{0}\:=\:1\:$

so ,

$=>\frac { \alpha z }{k \infty}\:=\:1$

$=> \alpha\:=\: \frac {k \infty }{z}$

but , k = Boltzman constant

$k = \: [ M^{1}L^{2}T^{-2} \infty^{-1}]$

z = length

$z\:=\: [L^{1}]$

$\infty = temperature$

$\infty \:= [ \infty^{1}]$

so ,

$\alpha \: = \:\frac{[M^{1}L^{\cancel {2}}T^{-2} \cancel {\infty{-1}}][\cancel {\infty^{+1}}]}{[ \cancel {L^{1}}]}$

$\boxed {\alpha \:=\: [ M^{1}L^{1}T^{-2}]}$

in the given equation ,

$\boxed {e^{\frac {\alpha z}{k \infty}}\:=\:constant}$

so , we do not consider the dimensions of this equation as it becomes a dimensionless quantity.

hence ,

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

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

P = pressure
$P \:=\: [M^{1}L^{-1}T^{-2}]$

$=>\:\beta =\:\frac {[M^{1}L^{1} T^{-2}]}{[M^{1}L^{-1}T^{-2}]}$

$=>\: \boxed {\beta\:=\:[M^{0}L^{2}T^{0}]}$



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