The amount of heat energy Q, used to heat up a substance depends on its mass m its specific heat capacity (s) and the change in temperature `DeltaT` of the substance. Using dimensional method, find the expression for s is ( Given that `[S]=[L^2 T^(-2)K^(-1)]`) is

Options

A. `QmDeltaT`
B. `Q/(mDeltaT)`
C. `"Qm"/(DeltaT)`
D. `m/(QDeltaT)`

Answer

To find the expression for specific heat capacity \( s \) using dimensional analysis, we start by recognizing that the amount of heat energy \( Q \) used to heat a substance depends on its mass \( m \), its specific heat capacity \( s \), and the change in temperature \( \Delta T \).

### Step-by-Step Solution:

1. **Identify the relationship**:
We can express the relationship as:
\[ Q = k \cdot m^b \cdot (\Delta T)^c \]
where \( k \) is a dimensionless constant, and \( a \), \( b \), and \( c \) are the powers we need to determine.

2. **Write down the dimensions**:
- The dimension of heat energy \( Q \) is given by:
\[ [Q] = [M][L^2][T^{-2}] \]
- The dimension of mass \( m \) is:
\[ [m] = [M] \]
- The dimension of change in temperature \( \Delta T \) is given as:
\[ [\Delta T] = [K] \]
- The dimension of specific heat capacity \( s \) is given as:
\[ [s] = [L^2][T^{-2}][K^{-1}] \]

3. **Set up the dimensional equation**:
We can equate the dimensions from both sides:
\[ [s] = [Q]^{a} [m]^{b} [\Delta T]^{c} \]
Substituting the dimensions:
\[ [L^2][T^{-2}][K^{-1}] = ([M][L^2][T^{-2}])^{a} \cdot ([M])^{b} \cdot ([K])^{c} \]

4. **Expand the right-hand side**:
This gives us:
\[ [L^2][T^{-2}][K^{-1}] = [M^{a+b}][L^{2a}][T^{-2a}][K^{c}] \]

5. **Equate the dimensions**:
Now we can equate the powers of each dimension:
- For mass \( M \):
\[ a + b = 0 \quad \text{(1)} \]
- For length \( L \):
\[ 2a = 2 \quad \text{(2)} \]
- For time \( T \):
\[ -2a = -2 \quad \text{(3)} \]
- For temperature \( K \):
\[ c = -1 \quad \text{(4)} \]

6. **Solve the equations**:
From equation (2):
\[ a = 1 \]
Substitute \( a = 1 \) into equation (1):
\[ 1 + b = 0 \implies b = -1 \]
From equation (4):
\[ c = -1 \]

7. **Write the expression for specific heat capacity**:
Now substituting \( a \), \( b \), and \( c \) back into the relationship:
\[ s = k \cdot Q^{1} \cdot m^{-1} \cdot (\Delta T)^{-1} \]
Assuming \( k = 1 \):
\[ s = \frac{Q}{m \cdot \Delta T} \]

### Final Expression:
Thus, the expression for specific heat capacity \( s \) is:
\[ s = \frac{Q}{m \cdot \Delta T} \]