The volume of a metal sphere is increased by 1% of its original volume when it is heated from 320 K to 522 K. calculate the coefficients of linear, superficial and cubical expansion of the metal.

Answer

To solve the problem step by step, we will calculate the coefficients of linear, superficial, and cubical expansion of the metal sphere based on the given information.

### Step 1: Understand the given data
- The volume of the metal sphere increases by 1% of its original volume when heated from 320 K to 522 K.
- We need to find:
- Coefficient of linear expansion (α)
- Coefficient of superficial expansion (β)
- Coefficient of volumetric expansion (γ)

### Step 2: Calculate the change in temperature (ΔT)
\[ \Delta T = T_{\text{final}} - T_{\text{initial}} = 522 \, \text{K} - 320 \, \text{K} = 202 \, \text{K} \]

### Step 3: Express the change in volume (ΔV)
The change in volume is given as 1% of the original volume \( V_0 \):
\[ \Delta V = 0.01 V_0 \]

### Step 4: Use the formula for volumetric expansion
The formula for volumetric expansion is:
\[ \Delta V = V_0 \cdot \gamma \cdot \Delta T \]
Substituting the expression for ΔV:
\[ 0.01 V_0 = V_0 \cdot \gamma \cdot 202 \]

### Step 5: Simplify to find γ
Dividing both sides by \( V_0 \) (assuming \( V_0 \neq 0 \)):
\[ 0.01 = \gamma \cdot 202 \]
Now, solve for γ:
\[ \gamma = \frac{0.01}{202} \approx 4.95 \times 10^{-5} \, \text{K}^{-1} \]

### Step 6: Relate γ to α and β
The relationships between the coefficients of expansion are:
- \( \beta = 2\alpha \)
- \( \gamma = 3\alpha \)

From the value of γ, we can find α:
\[ \alpha = \frac{\gamma}{3} = \frac{4.95 \times 10^{-5}}{3} \approx 1.65 \times 10^{-5} \, \text{K}^{-1} \]

### Step 7: Calculate β using α
Using the relationship \( \beta = 2\alpha \):
\[ \beta = 2 \times 1.65 \times 10^{-5} \approx 3.30 \times 10^{-5} \, \text{K}^{-1} \]

### Final Results
- Coefficient of linear expansion (α): \( 1.65 \times 10^{-5} \, \text{K}^{-1} \)
- Coefficient of superficial expansion (β): \( 3.30 \times 10^{-5} \, \text{K}^{-1} \)
- Coefficient of volumetric expansion (γ): \( 4.95 \times 10^{-5} \, \text{K}^{-1} \)