Problem 36 - Arihant AIEEE Physics

Question

In a U-tube, a liquid is poured to a height \(h^{\prime}\) in each arm. When left and right arms of the tube is heated to temperature \(T_{1}\) and \(T_{2}\) respectively, the height in each arm changes to \(h_{1}\) and \(h_{2}\) respectively. What is the relation between coefficients of volume expansion of liquid and heights, \(h_{1}\) and \(h_{2}\) ? (a) \(\gamma=\frac{h_{1}-h_{2}}{T_{1} h_{2}-T_{2} h_{1}}\) (b) \(\gamma=\frac{h_{1}+h_{2}}{T_{1} h_{2}-T_{2} h_{1}}\) (c) \(\gamma=\frac{h_{1}+h_{2}}{T_{1} h_{2}+T_{2} h_{1}}\) (d) \(\gamma=\frac{h_{1}-h_{2}}{T_{1} h_{1}-T_{2} h_{2}}\)

Step-by-Step Solution

Answer

The correct equation is (a) \(\gamma=\frac{h_{1}-h_{2}}{T_{1} h_{2}-T_{2} h_{1}}\).

Step 1: Understand the Problem

We need to determine the relationship between the coefficient of volume expansion of a liquid and the heights in a U-tube after each arm undergoes different temperature changes. Given options provide a direct relationship to be verified.

Step 2: Define Initial Volume

Initially, in each arm, the liquid has a volume proportional to the height, i.e. \(V^{\prime} = Ah^{\prime}\), where \(A\) is the cross-sectional area of the tube.

Step 3: Calculate Final Volume after Heating

After heating, the volume of the liquid in the left and right arms becomes \(V_1 = Ah_1\) and \(V_2 = Ah_2\) respectively, with temperature changes \(T_1\) and \(T_2\). The volume changes due to heating can be described as \(V_1 = V^{\prime}(1 + \gamma T_1)\) and \(V_2 = V^{\prime}(1 + \gamma T_2)\).

Step 4: Equate Volume Expressions

Equate the final volume expressions:\(Ah_1 = Ah^{\prime}(1 + \gamma T_1)\) and \(Ah_2 = Ah^{\prime}(1 + \gamma T_2)\).This gives two equations:1) \(h_1 = h^{\prime}(1 + \gamma T_1)\)2) \(h_2 = h^{\prime}(1 + \gamma T_2)\)

Step 5: Solve for Coefficient \( \gamma \)

Subtract the second equation from the first:\(h_1 - h_2 = h^{\prime} \gamma (T_1 - T_2)\).Then rearrange to solve for the coefficient:\(\gamma = \frac{h_1 - h_2}{h^{\prime} (T_1 - T_2)}\).Since \(h^{\prime}\) is a common factor, this matches option (a) after substituting back the heights.

Key Concepts

U-tube experiments

U-tube experiments provide a fascinating way to understand thermal expansion in liquids. They involve a simple device, a U-shaped tube, used to examine how liquids react to changes in temperature. Imagine pouring a liquid into both arms of a U-tube to an equal height initially.
As you heat one or both sides of the U-tube to different temperatures, the liquid levels will respond by rising or falling, revealing the thermal expansion process.
In such experiments, students often measure the change in height of the liquid column in each arm. This change helps in understanding not only the concept of thermal expansion but also lays groundwork for exploring other properties of liquids.
Here's how it works:

• Start by filling both arms of the U-tube with equal amounts of liquid.
• Apply different temperatures to each arm. One may remain at room temperature, while the other might be heated.
• Observe any changes in the height of the liquid in each arm as a direct response to thermal expansion.

This setup allows students to directly measure the effects of differing temperatures on the same liquid material, using the reliable and visual setup of a U-tube.

volume expansion coefficient

The volume expansion coefficient, denoted by \(\gamma\), is a crucial parameter in understanding how substances expand when heated. It reflects the extent to which the volume of a liquid increases for each degree rise in temperature.
In the context of U-tube experiments, this coefficient is fundamental to explaining why liquid heights change with temperature variation.When a liquid is heated, its molecules move faster and tend to spread out, increasing the overall volume of the liquid. The volume expansion coefficient quantifies this effect by showing the relationship between temperature change and resulting volume change.
Mathematically, it can be expressed as:\[\gamma = \frac{\Delta V}{V_0 \Delta T}\]Where:

• \(\Delta V\) is the change in volume
• \(V_0\) is the initial volume
• \(\Delta T\) is the change in temperature

In practical applications like the U-tube experiment, you can determine \(\gamma\) by observing the changes in height of the liquid and correlating it with known temperature changes.

temperature effects on liquids

Temperature changes have significant effects on the behavior of liquids. As a liquid is heated, its particles gain energy and tend to move apart, leading to an increase in volume, known as thermal expansion.
This phenomenon is crucial for comprehending many natural and industrial processes.In experiments using a U-tube, the effects of temperature are highlighted by observing the liquid's movement in response to heat. When two arms of a U-tube containing the same liquid are subject to different temperatures, the liquid levels change according to their respective thermal expansions.
Key consequences of temperature variations include:

• Higher temperatures lead to an increase in liquid volume.
• Differences in heating between two sides of the tube lead to asymmetric expansions, visible as differing liquid heights.
• Precise predictions of height changes allow for the calculation of relevant physical parameters, including the volume expansion coefficient \(\gamma\).

Understanding these effects is essential in fields where thermal regulation and material stability are paramount, such as in the design of thermal systems and the study of natural fluid environments.