The ratio among coefficient of volume expansion, superficial expansion and linear expansion i.e.,
`gamma` : `beta` : `alpha` is

Options

A. `1:2:3`
B. `3:2:1`
C. `4:3:2`
D. All of these

Answer

To find the ratio among the coefficients of volume expansion (γ), superficial expansion (β), and linear expansion (α), we can follow these steps:

### Step 1: Understand the Definitions
- **Linear Expansion (α)**: This is the coefficient of linear expansion, which describes how the length of an object changes with temperature.
- **Superficial Expansion (β)**: This is the coefficient of superficial (area) expansion, which describes how the area of an object changes with temperature.
- **Volume Expansion (γ)**: This is the coefficient of volume expansion, which describes how the volume of an object changes with temperature.

### Step 2: Establish Relationships
From thermodynamics, we know the following relationships between these coefficients:
- The coefficient of superficial expansion (β) is related to the linear expansion (α) by the equation:
\[ \beta = 2\alpha \]
- The coefficient of volume expansion (γ) is related to the linear expansion (α) by the equation:
\[ \gamma = 3\alpha \]

### Step 3: Set Up the Ratios
Now we can express the coefficients in terms of α:
- γ = 3α
- β = 2α
- α = α

### Step 4: Write the Ratio
The ratio of γ : β : α can now be expressed as:
\[ \text{Ratio} = \gamma : \beta : \alpha = 3\alpha : 2\alpha : \alpha \]

### Step 5: Simplify the Ratio
To simplify this ratio, we can divide each term by α:
\[ \text{Ratio} = 3 : 2 : 1 \]

### Conclusion
Thus, the ratio among the coefficients of volume expansion, superficial expansion, and linear expansion is:
\[ \gamma : \beta : \alpha = 3 : 2 : 1 \]

### Final Answer
The answer is 3 : 2 : 1.

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