`1 kg` of iron (specific heat `120 Cal Kg^(-1) C^(-1)`) is heated by `1000^(@)C`. The increases in its mass is

Options

A. zero
B. `5.6xx10^(-8) Kg`
C. `5.6xx10^(-16)Kg`
D. `5.6xx10^(-12) Kg`

Explanation

To solve the problem of finding the increase in mass when `1 kg` of iron is heated by `1000°C`, we will use the concept of mass-energy equivalence from Einstein's theory, which states that energy (E) is related to mass (m) by the equation:

\[ E = mc^2 \]

Where:
- \( E \) is the energy in joules,
- \( m \) is the mass in kilograms,
- \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \) m/s.

### Step 1: Calculate the heat energy (E) absorbed by the iron
The heat energy absorbed by the iron can be calculated using the formula:

\[ E = m \cdot s \cdot \Delta T \]

Where:
- \( m = 1 \, \text{kg} \) (mass of iron),
- \( s = 120 \, \text{Cal/kg°C} \) (specific heat of iron),
- \( \Delta T = 1000 \, \text{°C} \) (change in temperature).

First, we need to convert the specific heat from calories to joules. We know that:

\[ 1 \, \text{Cal} = 4.2 \, \text{J} \]

Thus,

\[ s = 120 \, \text{Cal/kg°C} \times 4.2 \, \text{J/Cal} = 504 \, \text{J/kg°C} \]

Now, substituting the values into the heat energy formula:

\[ E = 1 \, \text{kg} \cdot 504 \, \text{J/kg°C} \cdot 1000 \, \text{°C} \]

Calculating this gives:

\[ E = 504000 \, \text{J} \]

### Step 2: Calculate the increase in mass (Δm)
Using the mass-energy equivalence formula, we can find the increase in mass:

\[ \Delta m = \frac{E}{c^2} \]

Substituting the values:

- \( E = 504000 \, \text{J} \)
- \( c = 3 \times 10^8 \, \text{m/s} \)

Calculating \( c^2 \):

\[ c^2 = (3 \times 10^8 \, \text{m/s})^2 = 9 \times 10^{16} \, \text{m}^2/\text{s}^2 \]

Now substituting into the mass increase formula:

\[ \Delta m = \frac{504000 \, \text{J}}{9 \times 10^{16} \, \text{m}^2/\text{s}^2} \]

Calculating \( \Delta m \):

\[ \Delta m = \frac{504000}{9 \times 10^{16}} \]

\[ \Delta m = 5.6 \times 10^{-12} \, \text{kg} \]

### Final Answer
The increase in mass when `1 kg` of iron is heated by `1000°C` is:

\[ \Delta m = 5.6 \times 10^{-12} \, \text{kg} \]

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