Question:
The dimensional formula
\[
[ML^2T^{-2}K^{-1}]
\]
corresponds to which physical quantity?
The dimensional formula
\[
[ML^2T^{-2}K^{-1}]
\]
corresponds to which physical quantity?
Show Hint
To identify a physical quantity from its dimensional formula, check which combination of fundamental dimensions (M, L, T, K, etc.) matches the characteristic relation of the quantity, such as energy per temperature for Boltzmann constant.
Updated On: Jul 18, 2026
- Boltzmann constant
- Specific heat capacity
- Thermal expansion coefficient
- Latent heat
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the dimensional formula.
The given dimensional formula is \([ML^2T^{-2}K^{-1}]\). Here, \(M\) represents mass, \(L\) length, \(T\) time, and \(K\) temperature. This combination indicates a quantity involving energy per unit temperature.
Step 2: Analyzing each option.
- Specific heat capacity has dimensions \([L^2T^{-2}\theta^{-1}]\times M^0\) or sometimes \([ML^2T^{-2}\theta^{-1}]\) per unit mass, which is not exactly matching the given formula.
- Thermal expansion coefficient has dimensions \([K^{-1}]\) only, which clearly does not match.
- Latent heat has dimensions of energy per mass, \([L^2T^{-2}]\), also not matching.
Step 3: Boltzmann constant check.
Boltzmann constant \(k_B\) relates energy and temperature: \(E = k_B T\). Energy has dimension \([ML^2T^{-2}]\), temperature has dimension \([K]\), so \(k_B = E/T\). Therefore, the dimension is: \([ML^2T^{-2}K^{-1}]\), which exactly matches the given formula.
Step 4: Verification of other quantities.
Other listed quantities do not involve both energy and temperature in this combination. Specific heat capacity involves energy per mass per temperature, thermal expansion coefficient is per temperature, and latent heat is energy per mass, so none match \([ML^2T^{-2}K^{-1}]\).
Step 5: Conclusion.
From the analysis, only Boltzmann constant has the given dimensional formula.
Step 6: Final statement.
Hence, the dimensional formula \([ML^2T^{-2}K^{-1}]\) corresponds to the Boltzmann constant.
The given dimensional formula is \([ML^2T^{-2}K^{-1}]\). Here, \(M\) represents mass, \(L\) length, \(T\) time, and \(K\) temperature. This combination indicates a quantity involving energy per unit temperature.
Step 2: Analyzing each option.
- Specific heat capacity has dimensions \([L^2T^{-2}\theta^{-1}]\times M^0\) or sometimes \([ML^2T^{-2}\theta^{-1}]\) per unit mass, which is not exactly matching the given formula.
- Thermal expansion coefficient has dimensions \([K^{-1}]\) only, which clearly does not match.
- Latent heat has dimensions of energy per mass, \([L^2T^{-2}]\), also not matching.
Step 3: Boltzmann constant check.
Boltzmann constant \(k_B\) relates energy and temperature: \(E = k_B T\). Energy has dimension \([ML^2T^{-2}]\), temperature has dimension \([K]\), so \(k_B = E/T\). Therefore, the dimension is: \([ML^2T^{-2}K^{-1}]\), which exactly matches the given formula.
Step 4: Verification of other quantities.
Other listed quantities do not involve both energy and temperature in this combination. Specific heat capacity involves energy per mass per temperature, thermal expansion coefficient is per temperature, and latent heat is energy per mass, so none match \([ML^2T^{-2}K^{-1}]\).
Step 5: Conclusion.
From the analysis, only Boltzmann constant has the given dimensional formula.
Step 6: Final statement.
Hence, the dimensional formula \([ML^2T^{-2}K^{-1}]\) corresponds to the Boltzmann constant.
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