The dimensional formula for entropy is

Options

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

Answer

To find the dimensional formula for entropy, we start with the definition of entropy in thermodynamics, which is given by the formula:

\[ S = \frac{\Delta Q}{T} \]

where:
- \( S \) is the entropy,
- \( \Delta Q \) is the heat absorbed (or transferred),
- \( T \) is the temperature.

### Step 1: Determine the dimensional formula for heat (\( \Delta Q \))
Heat is a form of energy. The dimensional formula for energy (including heat) is:

\[ [\Delta Q] = [E] = M L^2 T^{-2} \]

where:
- \( M \) is mass,
- \( L \) is length,
- \( T \) is time.

### Step 2: Determine the dimensional formula for temperature (\( T \))
The dimensional formula for temperature is represented as:

\[ [T] = \Theta \]

where \( \Theta \) is the dimension for temperature (Kelvin).

### Step 3: Substitute the dimensions into the entropy formula
Now we can substitute the dimensions of \( \Delta Q \) and \( T \) into the entropy formula:

\[ [S] = \frac{[\Delta Q]}{[T]} = \frac{M L^2 T^{-2}}{\Theta} \]

### Step 4: Simplify the dimensional formula for entropy
This simplifies to:

\[ [S] = M L^2 T^{-2} \Theta^{-1} \]

### Step 5: Write the final dimensional formula for entropy
Thus, the dimensional formula for entropy can be expressed as:

\[ [S] = M^1 L^2 T^{-2} \Theta^{-1} \]

### Conclusion
From the options provided, the dimensional formula for entropy corresponds to the option that includes the inverse of temperature (Kelvin). Therefore, the correct answer is:

**Option 3: Kelvin inverse (K\(^{-1}\))**

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