Dimensional formula for latent heat is________

Options

A. `M^(0)L^(2)T^(-2)`
B. `ML^(2)T^(-1)`
C. `MLT^(-2)`
D. `ML^(-2)T^(-2)`

Answer

To find the dimensional formula for latent heat, we can follow these steps:

### Step 1: Understand the Definition of Latent Heat
Latent heat (L) is defined as the amount of heat (Q) required to change the state of a unit mass (m) of a substance without changing its temperature. Thus, we can express this relationship mathematically as:
\[ L = \frac{Q}{m} \]

### Step 2: Identify the Dimensional Formula for Mass
The dimensional formula for mass (m) is:
\[ [m] = M^1 L^0 T^0 \]
This indicates that mass has a dimension of M (mass) and does not depend on length (L) or time (T).

### Step 3: Identify the Dimensional Formula for Heat
The dimension of heat is equivalent to the dimension of energy. Energy can be expressed in terms of work done, which is defined as:
\[ \text{Work} = \text{Force} \times \text{Displacement} \]
The dimensional formula for force (F) is given by:
\[ F = m \cdot a \]
Where acceleration (a) has the dimensional formula:
\[ [a] = L^1 T^{-2} \]
Thus, the dimensional formula for force becomes:
\[ [F] = M^1 L^1 T^{-2} \]

Now, since displacement has the dimensional formula:
\[ [d] = L^1 \]
We can substitute this into the work formula:
\[ [\text{Work}] = [F] \times [d] = (M^1 L^1 T^{-2}) \times (L^1) = M^1 L^2 T^{-2} \]
Therefore, the dimensional formula for heat (Q) is:
\[ [Q] = M^1 L^2 T^{-2} \]

### Step 4: Substitute the Dimensional Formulas into the Latent Heat Equation
Now we can substitute the dimensional formulas for heat and mass into the equation for latent heat:
\[ L = \frac{Q}{m} \]
Substituting the dimensions:
\[ [L] = \frac{[Q]}{[m]} = \frac{M^1 L^2 T^{-2}}{M^1 L^0 T^0} \]

### Step 5: Simplify the Expression
When we simplify the expression, we get:
\[ [L] = M^{1-1} L^{2-0} T^{-2-0} = M^0 L^2 T^{-2} \]
Thus, the dimensional formula for latent heat is:
\[ [L] = L^2 T^{-2} \]

### Final Answer
The dimensional formula for latent heat is:
\[ \text{Dimensional formula for latent heat} = L^2 T^{-2} \]

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