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Question

$(A)$ Boltzmann's constant $(k)$ is related to the ideal gas constant $(R)$ and Avogadro's number $(N_A)$ by the relation $k = R / N_A$.The ideal gas

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$[M L^2 T^{-2} 	heta^{-1}]$

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$(A)$ Boltzmann's constant $(k)$ is related to the ideal gas constant $(R)$ and Avogadro's number $(N_A)$ by the relation $k = R / N_A$.The ideal gas equation is $PV = nRT$,where $R = PV / (nT)$.The dimensions of pressure $(P)$ are $[M L^{-1} T^{-2}]$,volume $(V)$ are $[L^3]$,amount of substance $(n)$ is $[	ext{mol}]$,and temperature $(T)$ is $[	heta]$.Thus,the dimensions of $R$ are $[M L^{-1} T^{-2}] \cdot [L^3] / ([	ext{mol}] \cdot [	heta]) = [M L^2 T^{-2} 	heta^{-1} 	ext{mol}^{-1}]$.Since Avogadro's number $(N_A)$ is a dimensionless quantity (number of particles per mole),the dimensions of Boltzmann's constant $(k)$ are the same as the gas constant per mole,which is $[M L^2 T^{-2} 	heta^{-1}]$.

The dimensional formula for Boltzmann's constant is

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