An expression for a dimensionless quantity P is given by P =(α/β) log e(( kt /β x )); where α and β are constants, x is distance ; k is Boltzmann constant and t is the temperature. Then the dimensions of α will be :

Question

An expression for a dimensionless quantity P is given by P =(α/β) log e(( kt /β x )); where α and β are constants, x is distance ; k is Boltzmann constant and t is the temperature. Then the dimensions of α will be : (A) [ M 0 L -1 T 0] (B) [ ML 0 T -2] (C) [ MLT -2] (D) [ ML 2 T -2]

Tardigrade Admin · Accepted Answer

P =(α/β) log e (( kt /β x )) ( kt /β x )=1 ⇒ β=( kt / x )=( ML 2 T -2/ L ) (∵ E =(1/2) kt ) As P is dimensionless ⇒[α]=[β]=[ MLT -2]

Q. An expression for a dimensionless quantity $P$ is given by $P = \frac{α}{β} lo g_{e} (\frac{k t}{β x})$ ; where $α$ and $β$ are constants, $x$ is distance ; $k$ is Boltzmann constant and $t$ is the temperature. Then the dimensions of $α$ will be :

$[M^{0} L^{- 1} T^{0}]$

$[M L^{0} T^{- 2}]$

$[M L T^{- 2}]$

$[M L^{2} T^{- 2}]$

$P = \frac{α}{β} lo g_{e} (\frac{k t}{β x})$
$\frac{k t}{β x} = 1$
$\Rightarrow β = \frac{k t}{x} = \frac{M L ^{2} T ^{- 2}}{L}$
$(∵ E = \frac{1}{2} k t)$
As $P$ is dimensionless
$\Rightarrow [α] = [β] = [M L T^{- 2}]$

Q. An expression for a dimensionless quantity P is given by P=βα​loge​(βxkt​); where α and β are constants, x is distance ; k is Boltzmann constant and t is the temperature. Then the dimensions of α will be :

[M0L−1T0]

[ML0T−2]

[MLT−2]

[ML2T−2]