Problem 39 - Master Resource Book in JEE Main Physics

Question

The coefficient of apparent expansion of mercury in a glass vessel is \(153 \times 10^{-6} /{ }^{\circ} \mathrm{C}\) and in a steel vessel is \(144 \times 10^{6} /{ }^{\circ} \mathrm{C}\). If \(\alpha\) for steel is \(12 \times 10^{-6} /{ }^{\circ} \mathrm{C}\), then that of glass is (a) \(9 \times 10^{-6} /^{-} \mathrm{C}\) (b) \(6 \times 10^{-6} /{ }^{\circ} \mathrm{C}\) (c) \(36 \times 10^{-6} /{ }^{\prime} \mathrm{C}\) (d) \(27 \times 10^{-6} /{ }^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

The coefficient of expansion for glass, calculated as \(3 \times 10^{-6}/{ }^{\circ} \mathrm{C}\), does not match any provided answer choice.

Step 1: Understand the Apparent Expansion Concept

The apparent expansion coefficient of a liquid in a vessel is the observed increase in volume per degree Celsius, taking into account the expansion of the container. It is the difference between the actual expansion coefficient of the liquid and the linear expansion coefficient of the material of the container.

Step 2: Define Formula for Apparent Expansion Coefficient

The formula to find the coefficient of apparent expansion of a liquid in a vessel is: \[\beta_{apparent} = \beta_{liquid} - \alpha_{vessel}\] where \( \beta_{apparent} \) is the apparent expansion coefficient, \( \beta_{liquid} \) is the real coefficient of expansion of the liquid, and \( \alpha_{vessel} \) is the linear expansion coefficient of the vessel.

Step 3: Relate Data for Mercury in Glass Vessel

For mercury in a glass vessel, we have \(\beta_{apparent} = 153 \times 10^{-6} /{}^{\circ} \mathrm{C}\) and we need to find \(\alpha_{glass}\). We can represent \(\beta_{liquid}\) for mercury as \(\beta_{mercury}\). So,\[153 \times 10^{-6} = \beta_{mercury} - \alpha_{glass}\]

Step 4: Relate Data for Mercury in Steel Vessel

For mercury in a steel vessel, \(\beta_{apparent} = 144 \times 10^{-6} /{}^{\circ} \mathrm{C}\) and \(\alpha_{steel} = 12 \times 10^{-6} /{}^{\circ} \mathrm{C}\). Therefore,\[144 \times 10^{-6} = \beta_{mercury} - 12 \times 10^{-6}\]

Step 5: Calculate Real Coefficient of Mercury

From Step 4, solve for \(\beta_{mercury}\): \[\beta_{mercury} = 144 \times 10^{-6} + 12 \times 10^{-6}\]\[\beta_{mercury} = 156 \times 10^{-6} /{}^{\circ} \mathrm{C}\]

Step 6: Calculate Coefficient of Expansion for Glass

Substitute the value of \(\beta_{mercury}\) into the equation from Step 3: \[153 \times 10^{-6} = 156 \times 10^{-6} - \alpha_{glass}\]Solve for \(\alpha_{glass}\): \[\alpha_{glass} = 156 \times 10^{-6} - 153 \times 10^{-6}\]\[\alpha_{glass} = 3 \times 10^{-6} /{}^{\circ} \mathrm{C}\]

Step 7: Conclusion

The coefficient of expansion for glass is \(3 \times 10^{-6}/{ }^{\circ} \mathrm{C}\), which does not match any given option. Therefore, there may be an error in the options or given data.

Key Concepts

Apparent Expansion

In thermal physics, the concept of apparent expansion is critical when working with liquids in containers. Apparent expansion refers to the observed increase in the volume of a liquid when it is heated. It considers two factors: the actual volume increase of the liquid and the expansion of the container housing the liquid. This is important because when a liquid is heated, both the liquid and its container expand, meaning the observed change is not just due to the liquid itself. This can be mathematically described by the formula:

• \( \beta_{apparent} = \beta_{liquid} - \alpha_{vessel} \)

Here, \( \beta_{apparent} \) is the coefficient of apparent expansion, \( \beta_{liquid} \) is the real coefficient of expansion of the liquid, and \( \alpha_{vessel} \) is the linear expansion coefficient of the container. Understanding this relationship is vital for correctly determining how much a liquid appears to expand in various materials like glass or steel due to heating.

Thermal Expansion of Liquids

Thermal expansion is a natural behavior of liquids when they are subjected to an increase in temperature. This phenomenon happens because heating increases the kinetic energy of the molecules, causing them to move apart, and thereby increasing the volume the liquid occupies. When examining the thermal expansion of liquids, it is important to measure the coefficient of volumetric expansion, which tells how much the liquid expands per degree of temperature increase. This property is material-specific, meaning different liquids will expand differently under the same temperature change. In the case of mercury, as discussed in the original exercise, the thermal expansion is affected by both the properties of mercury and its containing vessel, like glass or steel. Therefore, knowing the thermal characteristics of the container is equally essential when computing the apparent expansion.

Linear Expansion Coefficient

The linear expansion coefficient, symbolized as \( \alpha \), is an indicator of how much a material (typically a solid) will expand per degree change in temperature. This value is crucial when calculating the apparent expansion of a liquid inside a container made of that material. The linear expansion coefficient is particularly important for materials like glass or metal, which often serve as containers for liquids in thermal expansion studies. In the original exercise, the coefficient for steel was noted as \(12 \times 10^{-6}/^{\circ}\mathrm{C}\), demonstrating how the steel container expands linearly with temperature.Understanding the linear expansion coefficient allows us to determine how changes in the temperature will influence not just the liquid but also the vessel, affecting the apparent expansion measurements. This precise calculation ensures that observed changes in liquid volume are accurately interpreted, which is crucial for scientific and industrial applications.