A glass flask of volume `200cm^(3)` is completely filled with mercury at `20^(0)C`. The amount of mercury that overflow when the flask is heated to `80^(0)C` (Coefficient of volume expansion of glass is `27 xx 10^(-6 //0)C, gamma` of mercury `0.18 xx 10^(-3) //^(0)C`

Options

A. `2.16cm^(3)`
B. `0.032cm^(3)`
C. `1.84cm^(3)`
D. `2.40cm^(3)`

Explanation

To solve the problem of how much mercury overflows when a glass flask filled with mercury is heated from 20°C to 80°C, we will follow these steps:

### Step 1: Understand the Problem
We have a glass flask with a volume of 200 cm³ filled with mercury. We need to find out how much mercury overflows when both the flask and the mercury are heated from 20°C to 80°C.

### Step 2: Identify the Coefficients of Volume Expansion
The coefficients of volume expansion are given as:
- Coefficient of volume expansion of glass (αg) = 27 × 10^(-6) °C^(-1)
- Coefficient of volume expansion of mercury (αm) = 0.18 × 10^(-3) °C^(-1) = 180 × 10^(-6) °C^(-1)

### Step 3: Calculate the Change in Temperature
The change in temperature (ΔT) is:
ΔT = 80°C - 20°C = 60°C

### Step 4: Calculate the Change in Volume of Mercury
The change in volume of mercury (ΔVm) can be calculated using the formula:
\[ \Delta V_m = \alpha_m \cdot V_0 \cdot \Delta T \]
Where:
- \(V_0\) = initial volume of mercury = 200 cm³
- \(\alpha_m\) = coefficient of volume expansion of mercury

Substituting the values:
\[ \Delta V_m = 180 \times 10^{-6} \cdot 200 \cdot 60 \]

Calculating this gives:
\[ \Delta V_m = 180 \times 10^{-6} \cdot 12000 = 2160 \times 10^{-6} \text{ cm}^3 = 2.16 \text{ cm}^3 \]

### Step 5: Calculate the Change in Volume of Glass
The change in volume of the glass flask (ΔVg) can be calculated using the formula:
\[ \Delta V_g = \alpha_g \cdot V_0 \cdot \Delta T \]
Where:
- \(\alpha_g\) = coefficient of volume expansion of glass

Substituting the values:
\[ \Delta V_g = 27 \times 10^{-6} \cdot 200 \cdot 60 \]

Calculating this gives:
\[ \Delta V_g = 27 \times 10^{-6} \cdot 12000 = 324 \times 10^{-6} \text{ cm}^3 = 0.324 \text{ cm}^3 \]

### Step 6: Calculate the Volume of Mercury that Overflows
The volume of mercury that overflows (ΔV) is the difference between the change in volume of mercury and the change in volume of glass:
\[ \Delta V = \Delta V_m - \Delta V_g \]
Substituting the values:
\[ \Delta V = 2.16 - 0.324 = 1.836 \text{ cm}^3 \]

### Step 7: Final Result
Thus, the amount of mercury that overflows when the flask is heated from 20°C to 80°C is approximately **1.84 cm³**.

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