`J Kg^(-1) K^(-1)` is the unit of

Options

A. Boltzmann constant
B. PLANCK'S constant
C. Gas constant
D. all the above

Answer

To determine what the unit \( J \, kg^{-1} \, K^{-1} \) represents, we will analyze each of the options provided in the question.

### Step 1: Understand the units involved
The unit \( J \, kg^{-1} \, K^{-1} \) can be broken down as follows:
- \( J \) (joules) is a unit of energy.
- \( kg^{-1} \) indicates that the energy is being divided by mass (in kilograms).
- \( K^{-1} \) indicates that the energy is being divided by temperature (in kelvins).

### Step 2: Analyze the options
1. **Boltzmann constant**: The Boltzmann constant relates the average kinetic energy of particles in a gas to the temperature of the gas. Its unit is \( J \, K^{-1} \), which does not match our unit \( J \, kg^{-1} \, K^{-1} \).

2. **Planck's constant**: Planck's constant relates the energy of a photon to its frequency. Its unit is \( J \, s \) (joules times seconds), which also does not match our unit.

3. **Gas constant**: The gas constant \( R \) is defined as the amount of energy per temperature increment per unit mass. Its unit is indeed \( J \, kg^{-1} \, K^{-1} \), which matches our unit.

4. **All of the above**: Since only the gas constant matches the unit \( J \, kg^{-1} \, K^{-1} \), this option is incorrect.

### Conclusion
The unit \( J \, kg^{-1} \, K^{-1} \) corresponds to the gas constant.

### Final Answer
The unit \( J \, kg^{-1} \, K^{-1} \) is the unit of the gas constant.

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