Problem 20 - Master Resource Book in JEE Main Physics

Question

When a liquid in a glass vessel is heated, its apparent expansion is \(10.30 \times 10^{-4} \mathrm{C}^{-1}\). When the same liquid is heated in a metal vessel, its apparent expansion is \(10.06 \times 10^{-4}{\underline{\phantom{xx}}}^{\circ} \mathrm{C}^{-1}\). If the coefficient of linear expansion of glass \(=9 \times 10^{6}{\underline{\phantom{xx}}}^{\circ} \mathrm{C}^{-1}\), what is the coefficient of linear expansion of metal? (a) \(51 \times 10^{-6} \circ \mathrm{c}^{-1}\) (b) \(17 \times 10^{-6}{\underline{\phantom{xx}}}^{\circ} \mathrm{C}^{-1}\) (c) \(25 \times 10^{-60} \mathrm{C}^{-1}\) (d) \(43 \times 10^{-6 \circ} \mathrm{C}^{-1}\)

Step-by-Step Solution

Answer

The coefficient of linear expansion of the metal is \(17 \times 10^{-6} \mathrm{C}^{-1}\). Thus, the correct answer is (b).

Step 1: Identify Known Values

We are given the apparent expansion coefficients of the liquid in a glass and metal vessel:- Apparent expansion in glass vessel: \( \beta_{\text{app, glass}} = 10.30 \times 10^{-4} \, ^{\circ} \mathrm{C}^{-1} \)- Apparent expansion in metal vessel: \( \beta_{\text{app, metal}} = 10.06 \times 10^{-4} \, ^{\circ} \mathrm{C}^{-1} \)The coefficient of linear expansion of glass is given as:- \( \alpha_{\text{glass}} = 9 \times 10^{-6} \, ^{\circ} \mathrm{C}^{-1} \).

Step 2: Relate Coefficients of Volume Expansion

The apparent expansion coefficients relate to the actual expansion and the expansion of the container by:\[\beta_{\text{app}} = \beta_{\text{true}} - 3\alpha_{\text{container}}\]where \( \alpha_{\text{container}} \) is the linear expansion coefficient of the container material. Thus:\[ 10.30 \times 10^{-4} = \beta_{\text{true}} - 3\alpha_{\text{glass}} \]\[ 10.06 \times 10^{-4} = \beta_{\text{true}} - 3\alpha_{\text{metal}} \]

Step 3: Solve for True Coefficient of Expansion

Using the equation for the glass vessel, we solve for \( \beta_{\text{true}} \):\[\beta_{\text{true}} = 10.30 \times 10^{-4} + 3 \times (9 \times 10^{-6}) = 10.30 \times 10^{-4} + 2.7 \times 10^{-5}\]\[\beta_{\text{true}} = 10.57 \times 10^{-4}\]

Step 4: Calculate the Coefficient of Linear Expansion for Metal

Using the relation for the metal vessel:\[10.06 \times 10^{-4} = 10.57 \times 10^{-4} - 3 \alpha_{\text{metal}}\]Rearrange and solve for \( \alpha_{\text{metal}} \):\[3 \alpha_{\text{metal}} = 10.57 \times 10^{-4} - 10.06 \times 10^{-4} = 0.51 \times 10^{-4}\]\[\alpha_{\text{metal}} = \frac{0.51 \times 10^{-4}}{3} = 0.17 \times 10^{-4} = 17 \times 10^{-6} \, ^{\circ} \mathrm{C}^{-1}\]

Step 5: Select the Correct Option

Compare the calculated coefficient of linear expansion of the metal with the given options.The correct answer is (b) \(17 \times 10^{-6}^{\circ} \mathrm{C}^{-1}\).

Key Concepts

Apparent Expansion

The concept of apparent expansion can initially seem puzzling, but it makes more sense once you understand the difference between what is perceived and what truly happens. When a liquid is heated in a container, like glass or metal, the volume of the liquid appears to increase. However, this observed change is not solely because the liquid is expanding. The container expands too! The apparent expansion is what we measure when accounting only for the liquid's expansion, ignoring the container's effect.To calculate apparent expansion, consider the actual, or true, volume expansion of the liquid. Then, subtract the expansion contribution from the container. Mathematically, this is represented by the formula: \[ \beta_{\text{app}} = \beta_{\text{true}} - 3\alpha_{\text{container}} \] This formula implies that the apparent expansion (\( \beta_{\text{app}} \)) represents the true volume expansion (\( \beta_{\text{true}} \)) minus three times the linear expansion of the container (\( \alpha_{\text{container}} \)). The factor of three accounts for linear expansion effects in three dimensions. Understanding this helps in analyzing situations where different materials—like glass and metal—affect observed volume changes.

Coefficient of Linear Expansion

The concept of the coefficient of linear expansion (\( \alpha \)) describes how a material's dimensions change with temperature. It's vital for materials that undergo thermal expansion, which is common in both everyday life and industrial settings.Simply put, linear expansion deals with how a material grows longer when heated. Each material has a unique coefficient, \( \alpha \), indicating its expansion rate with temperature changes. Defined mathematically, it's the change in length (\( \Delta L \)) per degree Celsius:\[ \alpha = \frac{\Delta L}{L_0 \Delta T} \] Where \( L_0 \) is the original length, \( \Delta L \) is the change in length, and \( \Delta T \) is the change in temperature. The coefficient is usually a small number, noting the minor but significant alterations materials experience. In practice, it's crucial for precision engineering, construction, and scenarios where temperature fluctuations might impact structural integrity or functionality.

Volume Expansion

Volume expansion is fundamental when discussing how materials respond to heating. While linear expansion pertains to changes along a single dimension, volume expansion encompasses the entire space a material occupies.When you heat a solid or liquid, its volume increases as the entities that make up the material vibrate faster and occupy more space. This change across all dimensions is volume expansion. Its coefficient, \( \beta \), plays a key role in predicting how a material's volume responds to heat.The formulation for volume expansion is:\[ \beta = \frac{\Delta V}{V_0 \Delta T} \] Where \( \Delta V \) is the change in volume, \( V_0 \) is the original volume, and \( \Delta T \) is the temperature change.Understanding the volume expansion coefficient helps calibrate systems dealing with pressure or temperature shifts, ensuring safety and optimizing performance. In liquids, particularly, volume expansion is significant because they don't have a fixed shape, filling the available space in their containers. Accurately gauging volume changes is vital for industries like hydraulics, manufacturing, and food processing.