Pressure depends on distance as, where are constants, is distance, is Boltzmann's constant and is temperature. The dimension of are [2004]

Question

Pressure depends on distance as, $P = \frac{α}{β} \exp (- \frac{α z}{k θ}),$ where $α, β$ are constants, $z$ is distance, $k$ is Boltzmann's constant and $θ$ is temperature. The dimension of $β$ are [2004]

Smart Achievers · Accepted Answer

(3)

The power of an exponent is a number or constant.

Therefore, dimensionally $\frac{α z}{k θ} = [M^{0} L^{0} T^{0}]$

$∴$ $α = \frac{k θ}{z}$ $∴$ $α = \frac{[M L^{2} T^{- 2} θ^{- 1}] [θ]}{[L]} = [M L T^{- 2}]$

$As k = \frac{joule}{kelvin} = [M L^{2} T^{- 2} θ^{- 1}] and z = [L]$

$And, dimensionally P = \frac{α}{β} \Rightarrow β = \frac{α}{P}$

$∴$ $[β] = \frac{M L T^{- 2}}{M L^{- 1} T^{- 2}} = [M^{0} L^{2} T^{0}]$ $[∵ P = \frac{F}{A} = [M L^{- 1} T^{- 2}]]$

Solution

Q.
Pressure depends on distance as, $P = \frac{α}{β} \exp (- \frac{α z}{k θ}),$ where $α, β$ are constants, $z$ is distance, $k$ is Boltzmann's constant and $θ$ is temperature. The dimension of $β$ are [2004]

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

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

3 $M^{0} L^{2} T^{0}$

4 $M^{- 1} L^{1} T^{2}$

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Solution

Q. Pressure depends on distance as, P=αβexp(-αzkθ), where α,β are constants, z is distance, k is Boltzmann's constant and θ is temperature. The dimension of β are [2004]

1 M0L0T0

2 M-1L-1T-1

3 M0L2T0

4 M-1L1T2