Explain, what is meant by the coefficients of linear `(alpha)`, superficial `(beta)` and cubical expansion `(gamma)` of a solid. Given their units. Find the relationship between them.

Answer

To explain the coefficients of linear, superficial, and cubical expansion of a solid, we will break down the definitions, units, and relationships between these coefficients step by step.

### Step 1: Definition of Coefficients of Expansion
1. **Coefficient of Linear Expansion (α)**: This coefficient measures how much a material expands in one dimension (length) when the temperature changes. It is defined as:
\[ \Delta L = \alpha L \Delta T \]
where:
- \(\Delta L\) = change in length
- \(L\) = original length
- \(\Delta T\) = change in temperature

2. **Coefficient of Superficial Expansion (β)**: This coefficient measures how much a material expands in two dimensions (area) when the temperature changes. It is defined as:
\[ \Delta A = \beta A \Delta T \]
where:
- \(\Delta A\) = change in area
- \(A\) = original area

3. **Coefficient of Cubical Expansion (γ)**: This coefficient measures how much a material expands in three dimensions (volume) when the temperature changes. It is defined as:
\[ \Delta V = \gamma V \Delta T \]
where:
- \(\Delta V\) = change in volume
- \(V\) = original volume

### Step 2: Units of the Coefficients
- The units for all three coefficients are the same:
- **Units of α, β, and γ**: Since they relate change in length, area, or volume to temperature change, their units are:
\[ \text{Units} = \text{Temperature}^{-1} \quad (\text{e.g., } \text{°C}^{-1} \text{ or } \text{K}^{-1}) \]

### Step 3: Relationship Between the Coefficients
To find the relationship between the coefficients, we can analyze how they relate to each other mathematically.

1. From the definitions, we know:
- For area expansion:
\[ A' = A(1 + \alpha \Delta T)^2 \approx A(1 + 2\alpha \Delta T) \quad (\text{neglecting higher order terms}) \]
- Thus, the change in area can be expressed as:
\[ \Delta A = A' - A \approx 2\alpha A \Delta T \]
- Therefore, we can relate β to α:
\[ \beta = 2\alpha \]

2. For volume expansion:
- For a cube, the volume expansion can be expressed as:
\[ V' = V(1 + \alpha \Delta T)^3 \approx V(1 + 3\alpha \Delta T) \quad (\text{again neglecting higher order terms}) \]
- Thus, the change in volume can be expressed as:
\[ \Delta V = V' - V \approx 3\alpha V \Delta T \]
- Therefore, we can relate γ to α:
\[ \gamma = 3\alpha \]

### Final Relationship
Combining these results, we find the relationship between the coefficients:
\[ \alpha : \beta : \gamma = 1 : 2 : 3 \]

### Summary
- **Coefficient of Linear Expansion (α)**: Measures length change per temperature change.
- **Coefficient of Superficial Expansion (β)**: Measures area change per temperature change, related to α by β = 2α.
- **Coefficient of Cubical Expansion (γ)**: Measures volume change per temperature change, related to α by γ = 3α.