The dimensional formula for specific heat is

Options

A. (a)`M^(0)L^(2)T^(-2)`
B. (b)`M^(1)L^(2)T^(-2)K^(-1)`
C. (c)`M^(0)L^(2)T^(-2)K^(-1)`
D. (d)`M^(1)L^(2)T^(-2)`

Answer

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

### Step 1: Understand the definition of specific heat
Specific heat (denoted as \( s \)) is defined as the amount of heat \( Q \) required to raise the temperature of a unit mass \( m \) of a substance by 1 degree Celsius (or 1 Kelvin). Mathematically, it can be expressed as:

\[ s = \frac{Q}{m \cdot \Delta T} \]

where \( \Delta T \) is the change in temperature.

### Step 2: Identify the dimensions of heat \( Q \)
The amount of heat \( Q \) can be expressed in terms of its dimensional formula. The dimensional formula for heat is derived from the formula for energy, which is:

\[ Q = m \cdot a \cdot d \]

where \( m \) is mass, \( a \) is acceleration, and \( d \) is distance. The dimensional formula for heat is:

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

### Step 3: Identify the dimensions of mass \( m \)
The dimensional formula for mass \( m \) is simply:

\[ [m] = M^1 \]

### Step 4: Identify the dimensions of temperature change \( \Delta T \)
The change in temperature \( \Delta T \) is measured in Kelvin (K), and its dimensional formula is:

\[ [\Delta T] = K^1 \]

### Step 5: Substitute the dimensions into the specific heat formula
Now, substituting the dimensions into the specific heat formula:

\[ s = \frac{[Q]}{[m] \cdot [\Delta T]} = \frac{M^1 L^2 T^{-2}}{M^1 \cdot K^1} \]

### Step 6: Simplify the expression
Now, simplifying the expression:

\[ s = \frac{M^1 L^2 T^{-2}}{M^1 K^1} = M^{1-1} L^2 T^{-2} K^{-1} = M^0 L^2 T^{-2} K^{-1} \]

### Step 7: Write the final dimensional formula
Thus, the final dimensional formula for specific heat is:

\[ [M^0 L^2 T^{-2} K^{-1}] \]

### Step 8: Identify the correct option
Now, we can check the options provided in the question. The correct option corresponding to the dimensional formula \( M^0 L^2 T^{-2} K^{-1} \) is option C.

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