The quantity having dimensions only in temperature is

Options

A. Latent heat
B. Entropy
C. Specific heat
D. Coefficient of linear expansion

Answer

To solve the question regarding which quantity has dimensions only in temperature, we will analyze each of the given options step by step.

### Step-by-Step Solution:

1. **Identify the Options**: The options given are:
- Option 1: Latent heat
- Option 2: Entropy
- Option 3: Specific heat
- Option 4: Coefficient of linear expansion

2. **Analyze Latent Heat**:
- The dimensional formula for latent heat is derived from the formula \( Q = mL \), where \( Q \) is heat, \( m \) is mass, and \( L \) is the latent heat per unit mass.
- The dimensions of latent heat are:
\[ [L] = [M^0 L^2 T^{-2}] \]
- Since it contains dimensions of length and time, it does not have dimensions only in temperature.

**Hint**: Check the formula for latent heat and identify the dimensions involved.

3. **Analyze Entropy**:
- The dimensional formula for entropy is derived from the formula \( S = \frac{Q}{T} \), where \( Q \) is heat and \( T \) is temperature.
- The dimensions of entropy are:
\[ [S] = [M^1 L^2 T^{-2} K^{-1}] \]
- Here, we see that entropy includes dimensions of mass, length, time, and temperature, so it does not have dimensions only in temperature.

**Hint**: Recall how entropy is defined in thermodynamics and check its dimensional formula.

4. **Analyze Specific Heat**:
- The dimensional formula for specific heat is derived from the formula \( c = \frac{Q}{m \Delta T} \).
- The dimensions of specific heat are:
\[ [c] = [L^2 T^{-2} K^{-1}] \]
- This also includes dimensions of length and time, thus it does not have dimensions only in temperature.

**Hint**: Consider the definition of specific heat and its relation to heat transfer.

5. **Analyze Coefficient of Linear Expansion**:
- The coefficient of linear expansion \( \alpha \) is defined as:
\[ \alpha = \frac{\Delta L}{L \Delta T} \]
- The dimensions of the coefficient of linear expansion are:
\[ [\alpha] = [K^{-1}] \]
- This shows that the coefficient of linear expansion has dimensions only in temperature.

**Hint**: Look at how the coefficient of linear expansion is defined in terms of length change and temperature change.

6. **Conclusion**:
- After analyzing all the options, we find that the only quantity with dimensions solely in temperature is the **coefficient of linear expansion**.

### Final Answer:
The quantity having dimensions only in temperature is the **coefficient of linear expansion** (Option 4).