Problem 20 - Master Resource Book in JEE Main Physics

Question

A flask of volume \(10^{3} \mathrm{cc}\) is completely filled with mercury at \(0^{\circ} \mathrm{C}\). The coefficient of cubical expansion of mercury is \(1.80 \times 10^{-6}{\underline{\phantom{xx}}}^{\circ} \mathrm{C}^{-1}\) and that of glass is \(1.4 \times 10^{-6} \mathrm{C}^{-1}\). If the flask is now placed in boiling water at \(100^{\circ} \mathrm{C}\), how much mercury will overflow? (a) \(7 \mathrm{cc}\) (b) \(1.4 \mathrm{cc}\) (c) \(21 \mathrm{cc}\) (d) \(28 \mathrm{cc}\)

Step-by-Step Solution

Answer

None of the options is correct; overflow is 0.04 cc.

Step 1: Understanding the Cubical Expansion

Cubical expansion refers to the change in volume of a substance when its temperature changes. The volume change \( \Delta V \) for a change in temperature \( \Delta T \) is given by \( \Delta V = V_0 \beta \Delta T \), where \( V_0 \) is the initial volume, \( \beta \) is the coefficient of cubical expansion, and \( \Delta T \) is the change in temperature.

Step 2: Calculate the Volume Expansion of Mercury

Using the formula \( \Delta V = V_0 \beta \Delta T \) for mercury: \( V_0 = 1000 \, \text{cc} \), \( \beta = 1.80 \times 10^{-6} \, ^{\circ}\text{C}^{-1} \), and \( \Delta T = 100 \, ^{\circ}\text{C} \). Substituting these values, we have: \( \Delta V_{\text{mercury}} = 1000 \times 1.80 \times 10^{-6} \times 100 = 0.18 \, \text{cc}.\) This gives a total expanded volume of 1000 + 0.18 = 1000.18 cc.

Step 3: Calculate the Volume Expansion of the Flask (Glass)

For the glass flask, apply the same formula: \( V_0 = 1000 \, \text{cc} \), \( \beta = 1.4 \times 10^{-6} \, ^{\circ}\text{C}^{-1} \), and \( \Delta T = 100 \, ^{\circ}\text{C} \). So, \( \Delta V_{\text{flask}} = 1000 \times 1.4 \times 10^{-6} \times 100 = 0.14 \, \text{cc}.\) Therefore, the expanded volume of the glass flask is 1000 + 0.14 = 1000.14 cc.

Step 4: Determine the Overflow Volume

The amount of mercury that overflows is the difference between the expanded volume of mercury and the expanded volume of the flask: \( \text{Overflow} = 1000.18 \, \text{cc} - 1000.14 \, \text{cc} = 0.04 \, \text{cc}.\)

Key Concepts

Coefficient of Cubical Expansion

The coefficient of cubical expansion, represented as \( \beta \), is a key factor when studying thermal expansion of materials. It helps us understand how much a substance's volume will change with temperature.

• It is defined as the fractional change in volume per degree of temperature change.
• Essentially, \( \beta \) tells us how sensitive a material is to temperature increase.For example, if \( \beta = 1.80 \times 10^{-6} \, ^{\circ}\text{C}^{-1} \) for mercury, this means that for each degree Celsius increase in temperature, the volume of mercury will increase by \( 1.80 \times 10^{-6} \times \text{its initial volume} \).
• This property differs among materials, meaning mercury and glass have different responses to temperature changes, as shown in their coefficients.

Volume Expansion

Volume expansion describes how the volume of a substance changes when its temperature changes. This happens because increased temperature provides energy to particles, allowing them to move more freely and occupy more space.

• To calculate the change in volume, the formula \( \Delta V = V_0 \beta \Delta T \) can be utilized, where \( V_0 \) is the initial volume, \( \beta \) is the coefficient of cubical expansion, and \( \Delta T \) is the change in temperature.
• For mercury with an initial volume of 1000 cc, its volume at a 100 degrees Celsius increase is calculated as: \( \Delta V=1000 \times 1.80 \times 10^{-6} \times 100 = 0.18 \, \text{cc} \).
• Similarly, the glass flask also expands, but at a slower rate due to having a lower \( \beta \).

Temperature Change Effects in Physics

Temperature changes significantly impact physical substances, leading to changes in state, volume, and density, among others.

• The materials we encounter daily expand when heated and contract when cooled, though the extent varies.
• Temperature increase gives kinetic energy to particles, causing expansion. For a temperature decrease, the reverse happens. In our example, the glass flask and the mercury respond to heating when placed in boiling water, expanding at rates proportional to their coefficients of cubical expansion.
• Understanding these concepts is essential as they affect everything from building materials to precision instruments, where even a small change in temperature can yield significant results.