Question

A physical quantity that has the same dimensions as those of Boltzmann constant is

• A. energy
• B. bulk modulus
• C. power
• D. gas constant $R$
• E. thermal conductivity

Hint:

Boltzmann constant and gas constant: - Both relate energy and temperature - $R = N_A k$, so they have same dimensions

Answer 1

The Correct Option is D

Concept: Boltzmann constant $k$ relates energy to temperature: \[ k = \frac{\text{energy}}{\text{temperature}} \] So its dimensions are: \[ [k] = \frac{ML^2T^{-2}}{K} = ML^2T^{-2}K^{-1} \]

Step 1: Find dimensions of given quantities.
Energy: \[ ML^2T^{-2} \] Bulk modulus (pressure): \[ ML^{-1}T^{-2} \] Power: \[ ML^2T^{-3} \] Gas constant $R$: \[ R = \frac{\text{energy}}{\text{temperature}} \Rightarrow ML^2T^{-2}K^{-1} \] Thermal conductivity: \[ MLT^{-3}K^{-1} \]

Step 2: Compare with Boltzmann constant.
\[ [k] = ML^2T^{-2}K^{-1} \] Only gas constant $R$ has the same dimensions.