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Question

The work done by a gas molecule in an isolated system is given by,

W=αβ2ex2/αkT

Where x is the displacement, k is the Boltzmann constant, T is the temperature, α and β are constants, then the dimension of β will be :

A
[M0LT0]
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B
[M2LT2]
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C
[MLT2]
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D
[ML2T2]
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Solution

The correct option is C [MLT2]
Given:
Work done, W=αβ2ex2/αkT

Where,
x= Displacement
k= Boltzmann constant
T= Temperature
α,β= Constants

We know that,
x2αkT = dimensionless

[x2αkT]=[M0L0T0]

[kT]=[PV]=[ML2T2]
[x2]=[L2]

[α]=[M1T2]

Also,
[W]=[αβ2]
[ML2T2]=[M1T2][β2]
[β2]=[M2L2T4]
[β]=[MLT2]

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