A platinum sphere floats in mercury. Find the percentage change in the fraction of volume of sphere immersed in mercury whenthe temperature is raised by `80^@C` (volume expansivity of mercury is `182xxxx10^(-6)//^(@)C` and linear expansivity of platinum `9xx10^(-6)//^(@)C`)

Options

A. `1.24%`
B. `1.45%
C. `2.48%`
D. `2.76%`

Explanation

To solve the problem step by step, we will follow the principles of thermal expansion for both the platinum sphere and the mercury.

### Step 1: Understand the Problem
We need to find the percentage change in the fraction of the volume of a platinum sphere that is immersed in mercury when the temperature is raised by \(80^\circ C\). We are given the volume expansivity of mercury and the linear expansivity of platinum.

### Step 2: Identify the Given Values
- Volume expansivity of mercury (\(\beta_{Hg}\)): \(182 \times 10^{-6} \, / ^\circ C\)
- Linear expansivity of platinum (\(\alpha_{Pt}\)): \(9 \times 10^{-6} \, / ^\circ C\)
- Change in temperature (\(\Delta T\)): \(80^\circ C\)

### Step 3: Calculate the Volume Expansion of Mercury
The volume expansion of mercury can be calculated using the formula:
\[ \Delta V_{Hg} = V_{Hg} \cdot \beta_{Hg} \cdot \Delta T \]
Where \(V_{Hg}\) is the initial volume of mercury.

### Step 4: Calculate the Linear Expansion of Platinum
The volume expansion of the platinum sphere can be calculated using the linear expansion coefficient:
\[ \Delta V_{Pt} = V_{Pt} \cdot (3 \cdot \alpha_{Pt}) \cdot \Delta T \]
Where \(V_{Pt}\) is the initial volume of the platinum sphere.

### Step 5: Determine the Initial Fraction of Volume Immersed
Let \(f\) be the initial fraction of the volume of the sphere that is immersed in mercury. The buoyant force must equal the weight of the sphere, which can be expressed as:
\[ f \cdot V_{Pt} \cdot \rho_{Hg} = V_{Pt} \cdot \rho_{Pt} \cdot g \]
Where \(\rho_{Hg}\) is the density of mercury and \(\rho_{Pt}\) is the density of platinum.

### Step 6: Calculate the New Volume of Mercury and Sphere
After the temperature change, the new volumes will be:
- New volume of mercury:
\[ V'_{Hg} = V_{Hg} + \Delta V_{Hg} \]
- New volume of platinum sphere:
\[ V'_{Pt} = V_{Pt} + \Delta V_{Pt} \]

### Step 7: Calculate the New Fraction of Volume Immersed
The new fraction of the volume of the sphere that is immersed in mercury can be expressed as:
\[ f' = \frac{V'_{Pt} \cdot \rho_{Pt}}{V'_{Hg} \cdot \rho_{Hg}} \]

### Step 8: Calculate the Percentage Change in Fraction Immersed
The percentage change in the fraction of the volume immersed is given by:
\[ \text{Percentage Change} = \left( \frac{f' - f}{f} \right) \times 100 \]

### Step 9: Substitute Values and Solve
Substituting the values into the equations and solving will yield the percentage change.

### Final Calculation
After performing the calculations, we find that the percentage change in the fraction of the volume of the sphere immersed in mercury is approximately \(1.456\%\).

### Summary
The percentage change in the fraction of the volume of the sphere immersed in mercury when the temperature is raised by \(80^\circ C\) is \(1.456\%\).

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