A crystal has a coefficient of expansion `13xx10^(-7)` in one direction and `231xx10^(-7)` in every direction at right angles to it. Then the cubical coefficient of expansion is

Options

A. `462xx10^(-7)`
B. `244xx10^(-7)`
C. `475xx10^(-7)`
D. `257xx10^(-7)`

Explanation

To find the cubical coefficient of expansion for the given crystal, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coefficients of Linear Expansion**:
- The coefficient of linear expansion in one direction (let's call it \( \alpha_1 \)) is given as:
\[ \alpha_1 = 13 \times 10^{-7} \, \text{°C}^{-1} \]
- The coefficient of linear expansion in every direction at right angles to it (let's call it \( \alpha_2 \) and \( \alpha_3 \)) is given as:
\[ \alpha_2 = \alpha_3 = 231 \times 10^{-7} \, \text{°C}^{-1} \]

2. **Calculate the Cubical Coefficient of Expansion**:
- The cubical coefficient of expansion \( \gamma \) is the sum of the linear coefficients of expansion in all three dimensions:
\[ \gamma = \alpha_1 + \alpha_2 + \alpha_3 \]
- Substituting the values we have:
\[ \gamma = 13 \times 10^{-7} + 231 \times 10^{-7} + 231 \times 10^{-7} \]

3. **Perform the Addition**:
- First, calculate \( \alpha_2 + \alpha_3 \):
\[ \alpha_2 + \alpha_3 = 231 \times 10^{-7} + 231 \times 10^{-7} = 462 \times 10^{-7} \]
- Now add \( \alpha_1 \):
\[ \gamma = 13 \times 10^{-7} + 462 \times 10^{-7} = (13 + 462) \times 10^{-7} = 475 \times 10^{-7} \]

4. **Final Result**:
- Therefore, the cubical coefficient of expansion is:
\[ \gamma = 475 \times 10^{-7} \, \text{°C}^{-1} \]

### Conclusion:
The cubical coefficient of expansion for the crystal is \( 475 \times 10^{-7} \, \text{°C}^{-1} \).

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