An iron sphere has a radius of 10 cm at a temperature of `0^(@)C`. Calculate the change in volume of the sphere if it is heated to `100^(@)C`. Given `alpha_(Fe) = 1.1 xx 10^(-6).^(@)C^(-1)`

Answer

To solve the problem of calculating the change in volume of an iron sphere when heated from `0°C` to `100°C`, we will follow these steps:

### Step 1: Identify the given data
- Radius of the sphere, \( r = 10 \, \text{cm} \)
- Initial temperature, \( T_1 = 0°C \)
- Final temperature, \( T_2 = 100°C \)
- Coefficient of linear expansion for iron, \( \alpha_{Fe} = 1.1 \times 10^{-6} \, °C^{-1} \)

### Step 2: Calculate the change in temperature
\[ \Delta T = T_2 - T_1 = 100°C - 0°C = 100°C \]

### Step 3: Calculate the volume expansion coefficient (γ)
The volume expansion coefficient \( \gamma \) is related to the linear expansion coefficient \( \alpha \) by the formula:
\[ \gamma = 3\alpha \]
Substituting the value of \( \alpha \):
\[ \gamma = 3 \times (1.1 \times 10^{-6}) = 3.3 \times 10^{-6} \, °C^{-1} \]

### Step 4: Calculate the initial volume of the sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting the radius:
\[ V_0 = \frac{4}{3} \pi (10 \, \text{cm})^3 = \frac{4}{3} \pi (1000 \, \text{cm}^3) = \frac{4000}{3} \pi \, \text{cm}^3 \]
Using \( \pi \approx 3.14 \):
\[ V_0 \approx \frac{4000}{3} \times 3.14 \approx 4186.67 \, \text{cm}^3 \]

### Step 5: Calculate the change in volume (ΔV)
The change in volume due to thermal expansion is given by:
\[ \Delta V = V_0 \cdot \gamma \cdot \Delta T \]
Substituting the values:
\[ \Delta V = 4186.67 \, \text{cm}^3 \cdot (3.3 \times 10^{-6} \, °C^{-1}) \cdot (100 \, °C) \]
Calculating:
\[ \Delta V = 4186.67 \cdot 3.3 \times 10^{-6} \cdot 100 \]
\[ \Delta V \approx 4186.67 \cdot 3.3 \times 10^{-4} \approx 1.38 \, \text{cm}^3 \]

### Final Answer
The change in volume of the iron sphere when heated from `0°C` to `100°C` is approximately:
\[ \Delta V \approx 1.38 \, \text{cm}^3 \]

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