A liquid with coefficient of volume expansion `gamma` is filled in a container of a material having coefficient of linear expansion `alpha` . If the liquid overflows on heating, then

Options

A. `gamma gt 3 alpha`
B. `gamma lt 3 alpha`
C. `gamma = 3 alpha`
D. none of these

Answer

To solve the problem, we need to analyze the thermal expansion of both the liquid and the container when heated. Here’s a step-by-step solution:

### Step 1: Understand the initial conditions
Let the initial volume of the liquid and the container be \( V_0 \).

### Step 2: Calculate the new volume of the liquid after heating
The volume expansion of the liquid can be expressed using the coefficient of volume expansion \( \gamma \):
\[ V_L = V_0 (1 + \gamma \Delta T) \]
where \( \Delta T \) is the change in temperature.

### Step 3: Calculate the new volume of the container after heating
Assuming the container is cubical with side length \( L \), the initial volume is:
\[ V_0 = L^3 \]
After heating, the new side length \( L' \) of the container can be expressed as:
\[ L' = L(1 + \alpha \Delta T) \]
Thus, the new volume of the container \( V_C \) becomes:
\[ V_C = (L')^3 = L^3(1 + \alpha \Delta T)^3 \]
Using the binomial expansion, we can approximate this as:
\[ V_C \approx V_0(1 + 3\alpha \Delta T) \quad \text{(neglecting higher order terms)} \]

### Step 4: Set up the inequality for overflow
For the liquid to overflow, the volume of the liquid after heating must be greater than the volume of the container after heating:
\[ V_L > V_C \]
Substituting the expressions from Steps 2 and 3:
\[ V_0(1 + \gamma \Delta T) > V_0(1 + 3\alpha \Delta T) \]

### Step 5: Simplify the inequality
Dividing both sides by \( V_0 \) (assuming \( V_0 > 0 \)):
\[ 1 + \gamma \Delta T > 1 + 3\alpha \Delta T \]
Subtracting 1 from both sides:
\[ \gamma \Delta T > 3\alpha \Delta T \]
Since \( \Delta T \) is positive, we can safely divide both sides by \( \Delta T \):
\[ \gamma > 3\alpha \]

### Conclusion
Thus, the condition for the liquid to overflow when heated is:
\[ \gamma > 3\alpha \]