Two marks on a glass rod `10 cm` apart are found to increase their distance by `0.08 mm` when the rod is heated from `0^@ C "to" 100^@ C`. A flask made of the same glass as that of rod measures a volume of `100 cc at 0^@ C`. The volume it measures at `100^@ C` in (cc) is.

Options

A. `100.24`
B. `100.12`
C. `100.36`
D. `100.48`

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the problem
We have a glass rod with two marks that are 10 cm apart. When heated from 0°C to 100°C, the distance between the marks increases by 0.08 mm. We need to find the volume of a flask made of the same glass at 100°C, given that its volume is 100 cc at 0°C.

### Step 2: Convert the change in distance to the same unit
The change in distance is given as 0.08 mm. We need to convert this to centimeters for consistency:
\[ 0.08 \text{ mm} = 0.08 \div 10 = 0.008 \text{ cm} \]

### Step 3: Calculate the coefficient of linear expansion (α)
The formula for linear expansion is given by:
\[ \Delta L = L \cdot \alpha \cdot \Delta \theta \]
Where:
- \(\Delta L\) = change in length = 0.008 cm
- \(L\) = original length = 10 cm
- \(\Delta \theta\) = change in temperature = 100°C - 0°C = 100°C

Rearranging the formula to find α:
\[ \alpha = \frac{\Delta L}{L \cdot \Delta \theta} \]
Substituting the values:
\[ \alpha = \frac{0.008 \text{ cm}}{10 \text{ cm} \cdot 100} = \frac{0.008}{1000} = 8 \times 10^{-6} \text{ °C}^{-1} \]

### Step 4: Calculate the coefficient of volume expansion (γ)
The coefficient of volume expansion is related to the coefficient of linear expansion by:
\[ \gamma = 3\alpha \]
Substituting the value of α:
\[ \gamma = 3 \cdot (8 \times 10^{-6}) = 24 \times 10^{-6} \text{ °C}^{-1} \]

### Step 5: Calculate the change in volume (ΔV)
Using the formula for volume expansion:
\[ \Delta V = V \cdot \gamma \cdot \Delta \theta \]
Where:
- \(V\) = original volume = 100 cc
- \(\Delta \theta\) = change in temperature = 100°C

Substituting the values:
\[ \Delta V = 100 \text{ cc} \cdot (24 \times 10^{-6}) \cdot 100 = 100 \cdot 24 \times 10^{-4} = 2.4 \text{ cc} \]

### Step 6: Calculate the final volume (V')
The final volume is given by:
\[ V' = V + \Delta V \]
Substituting the values:
\[ V' = 100 \text{ cc} + 2.4 \text{ cc} = 102.4 \text{ cc} \]

### Final Answer
The volume of the flask at 100°C is **102.4 cc**.

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