In the relation P =(a/β) e -(a z/ k θ), p is the pressure, z the distance, k is Boltzmann constant and θ is the temperature, the dimensional formula of β will be

Question

In the relation P =(a/β) e -(a z/ k θ), p is the pressure, z the distance, k is Boltzmann constant and θ is the temperature, the dimensional formula of β will be (A) M0 L2 T0 (B) ML2 T (C) ML0 T-1 (D) ML2 T-1

Tardigrade Admin · Accepted Answer

In given equation, ( az / k θ) should be dimensionless ∴ a =( k θ/ z ) ⇒ a =( ML 2 T -2 K -1 × K / L )= MLT -2 And p=(a/β) ⇒ β=(a/p)=(M L T-2/M L-1 T-2) =M0 L2 T0

Q. In the relation $P = \frac{a}{β} e^{- \frac{a z}{k θ}}, p$ is the pressure, $z$ the distance, $k$ is Boltzmann constant and $θ$ is the temperature, the dimensional formula of $β$ will be

$M^{0} L^{2} T^{0}$

$M L^{2} T$

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

$M L^{2} T^{- 1}$

In given equation, $\frac{a z}{k θ}$ should be dimensionless
$∴ a = \frac{k θ}{z}$
$\Rightarrow a = \frac{M L ^{2} T ^{- 2} K ^{- 1} \times K}{L} = M L T^{- 2}$
And $p = \frac{a}{β}$
$\Rightarrow β = \frac{a}{p} = \frac{M L T ^{- 2}}{M L ^{- 1} T ^{- 2}}$
$= M^{0} L^{2} T^{0}$

Q. In the relation P=βa​e−kθaz​,p is the pressure, z the distance, k is Boltzmann constant and θ is the temperature, the dimensional formula of β will be

M0L2T0

ML2T

ML0T−1

ML2T−1