The dimensional formula for molar thermal capacity is same as that of

Options

A. gas constant
B. specific heat
C. Boltzmann's constant
D. Stefan's constant

Answer

To find the dimensional formula for molar thermal capacity and determine what it is equivalent to, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Molar Thermal Capacity**:
Molar thermal capacity (C) is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin).

2. **Formula for Molar Thermal Capacity**:
Molar thermal capacity can be expressed in terms of work done (W) and temperature (T). The formula is:
\[ C = \frac{W}{T \cdot N_A} \]
where \( N_A \) is Avogadro's number.

3. **Work Done by a Gas**:
The work done (W) by a gas can be expressed using the ideal gas equation:
\[ W = P \cdot V \]
From the ideal gas equation, we know:
\[ PV = nRT \]
where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

4. **Substituting Work Done**:
We can substitute \( PV \) in the formula for molar thermal capacity:
\[ C = \frac{nRT}{T \cdot N_A} \]

5. **Simplifying the Expression**:
The temperature \( T \) in the numerator and denominator cancels out:
\[ C = \frac{nR}{N_A} \]

6. **Relating to Boltzmann Constant**:
The term \( \frac{R}{N_A} \) is known as the Boltzmann constant (\( k \)):
\[ C = n \cdot k \]
Thus, the dimensional formula for molar thermal capacity is equivalent to that of the Boltzmann constant.

### Final Answer:
The dimensional formula for molar thermal capacity is the same as that of the Boltzmann constant.

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