The dimensional formula for entropy is

Options

A. `[MLT^(-2)K^(-1)]`
B. `[ML^2T^(-2)]`
C. `[ML^2T^(-2)K^(-1)]`
D. `[ML^(-2)T^(-2)K^(-1)]`

Answer

To find the dimensional formula for entropy, we start with its definition and the formula associated with it.

### Step-by-Step Solution:

1. **Understand the Definition of Entropy**:
Entropy (S) is a measure of the randomness or disorder of a system. The mathematical formula for entropy is given by:
\[ S = \frac{Q}{T} \]
where \( Q \) is the heat added to the system and \( T \) is the temperature.

2. **Determine the Dimensional Formula for Heat (Q)**:
Heat is a form of energy, and its dimensional formula is the same as that of work. The dimensional formula for work is derived from the formula:
\[ \text{Work} = \text{Force} \times \text{Displacement} \]
- The dimensional formula for force is:
\[ \text{Force} = \text{mass} \times \text{acceleration} = M L T^{-2} \]
- Displacement has the dimension:
\[ \text{Displacement} = L \]
- Therefore, the dimensional formula for work is:
\[ \text{Work} = M L T^{-2} \times L = M L^2 T^{-2} \]
Thus, the dimensional formula for heat \( Q \) is:
\[ [Q] = M L^2 T^{-2} \]

3. **Determine the Dimensional Formula for Temperature (T)**:
The dimensional formula for temperature is represented as:
\[ [T] = K \]
where \( K \) is the dimension of temperature.

4. **Combine the Dimensional Formulas**:
Now, substituting the dimensions of \( Q \) and \( T \) into the formula for entropy:
\[ [S] = \frac{[Q]}{[T]} = \frac{M L^2 T^{-2}}{K} \]
This simplifies to:
\[ [S] = M L^2 T^{-2} K^{-1} \]

5. **Final Result**:
Therefore, the dimensional formula for entropy is:
\[ [S] = M^1 L^2 T^{-2} K^{-1} \]