A vertical column 50 cm long at `50^(@)C` balances another column of same liquid 60 cm long at `100^(@)C` . The coefficient of absolute expansion of the liquid is

Options

A. `0.005//^(@)C`
B. `0.0005//^(@)C`
C. `0.002//^(@)C`
D. `0.0002//^(@)C`

Answer

To find the coefficient of absolute expansion of the liquid, we can follow these steps:

### Step 1: Understand the relationship between density, volume, and temperature
The density of a liquid changes with temperature. The volume \( V \) of a liquid at a temperature \( T \) can be expressed as:
\[ V = V_0 (1 + \gamma T) \]
where \( \gamma \) is the coefficient of volumetric expansion, and \( V_0 \) is the volume at 0°C.

### Step 2: Set up the relationship between the two columns of liquid
Given that the first column of liquid is 50 cm long at 50°C and balances with a 60 cm long column at 100°C, we can express the densities of the two columns:
- For the first column at 50°C:
\[ \rho_1 = \frac{m}{V_1} = \frac{m}{V_0(1 + \gamma \cdot 50)} \]
- For the second column at 100°C:
\[ \rho_2 = \frac{m}{V_2} = \frac{m}{V_0(1 + \gamma \cdot 100)} \]

### Step 3: Use the balance condition
Since the two columns balance each other, we can set up the equation:
\[ \rho_1 \cdot g \cdot 50 = \rho_2 \cdot g \cdot 60 \]
Cancelling \( g \) from both sides gives:
\[ \rho_1 \cdot 50 = \rho_2 \cdot 60 \]
This can be rearranged to:
\[ \frac{\rho_1}{\rho_2} = \frac{60}{50} = \frac{6}{5} \]

### Step 4: Substitute the expressions for densities
Substituting the expressions for \( \rho_1 \) and \( \rho_2 \):
\[ \frac{\frac{m}{V_0(1 + \gamma \cdot 50)}}{\frac{m}{V_0(1 + \gamma \cdot 100)}} = \frac{6}{5} \]
This simplifies to:
\[ \frac{1 + \gamma \cdot 100}{1 + \gamma \cdot 50} = \frac{6}{5} \]

### Step 5: Cross-multiply and solve for \( \gamma \)
Cross-multiplying gives:
\[ 5(1 + \gamma \cdot 100) = 6(1 + \gamma \cdot 50) \]
Expanding both sides:
\[ 5 + 500\gamma = 6 + 300\gamma \]
Rearranging terms:
\[ 500\gamma - 300\gamma = 6 - 5 \]
This simplifies to:
\[ 200\gamma = 1 \]
Thus:
\[ \gamma = \frac{1}{200} \]

### Step 6: Convert to the required units
The coefficient of volumetric expansion \( \gamma \) is:
\[ \gamma = 0.005 \, \text{per degree Celsius} \]

### Final Answer
The coefficient of absolute expansion of the liquid is \( \gamma = 0.005 \, \text{per degree Celsius} \).

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