The physical quantity having the same dimensional formula as that of entropy is

Options

A. Latent heat
B. Thermal capacity
C. Heat
D. Specific heat

Answer

To determine the physical quantity that has the same dimensional formula as entropy, we can follow these steps:

### Step 1: Understand the definition of entropy
Entropy (S) is defined in thermodynamics as a measure of the amount of energy in a physical system that is not available to do work. It is often associated with the degree of disorder or randomness in a system.

### Step 2: Find the dimensional formula of entropy
The dimensional formula of entropy can be derived from its definition. Entropy is defined as the amount of heat (Q) transferred per unit temperature (T). The formula for entropy can be expressed as:
\[ S = \frac{Q}{T} \]

### Step 3: Determine the dimensional formula of heat (Q)
Heat (Q) is a form of energy. The dimensional formula for energy is:
\[ [Q] = [Energy] = [M L^2 T^{-2}] \]
where:
- M = mass
- L = length
- T = time

### Step 4: Determine the dimensional formula of temperature (T)
The dimensional formula for temperature is:
\[ [T] = [K^1] \]
where K represents the dimension of temperature.

### Step 5: Combine the dimensional formulas
Now, substituting the dimensional formulas into the formula for entropy:
\[ [S] = \frac{[Q]}{[T]} = \frac{[M L^2 T^{-2}]}{[K^1]} = [M L^2 T^{-2} K^{-1}] \]

### Step 6: Identify a physical quantity with the same dimensional formula
Now we need to find a physical quantity that has the same dimensional formula as entropy, which is:
\[ [M L^2 T^{-2} K^{-1}] \]

1. **Latent Heat**: The dimensional formula is \( [M L^2 T^{-2}] \) divided by mass \( [M] \), resulting in \( [L^2 T^{-2}] \) which does not match.

2. **Thermal Capacity**: The dimensional formula is \( \frac{[M L^2 T^{-2}]}{[K]} = [M L^2 T^{-2} K^{-1}] \), which matches.

3. **Heat**: The dimensional formula is \( [M L^2 T^{-2}] \), which does not match.

4. **Specific Heat**: The dimensional formula is \( \frac{[M L^2 T^{-2}]}{[M K]} = [L^2 T^{-2} K^{-1}] \), which also matches.

### Conclusion
The physical quantity that has the same dimensional formula as entropy is **Thermal Capacity**.