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Question #43540

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The Question

A glass capillary tube with a coefficient of linear thermal expansion $\alpha \text{ (in K}^{-1}\text{)}$ is dipped vertically into a liquid with a coefficient of volume thermal expansion $\gamma \text{ (in K}^{-1}\text{)}$ . The surface tension of the liquid varies with temperature $\theta$ as $S = S_0(1 - \beta \Delta \theta)$ , where $\Delta \theta \text{ (in K)}$ is a small temperature increase and $\beta \text{ (in K}^{-1}\text{)}$ is a positive constant. The density of the liquid at the initial temperature is $\rho_0 \text{ (in kg m}^{-3}\text{)}$ . Assuming the angle of contact is zero and remains constant, which of the following expressions gives the fractional change in the capillary rise $\left(\frac{\Delta h}{h_0}\right)$ ?

Options

A.
$-(\alpha + \beta - \gamma)\Delta \theta$
B.
$-(\beta - \alpha - \gamma)\Delta \theta$
C.
$-(\alpha + \beta + \gamma)\Delta \theta$
D.
$-(\beta - \gamma)\Delta \theta$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Thermal Expansion of Area and Volume|Capillary Rise Mechanics|Linear Regression Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize thermal expansion mechanics with fluid statics, specifically targeting the competing effects of geometric expansion and thermodynamic property decay.

A common trap is the misconception that a cavity (the capillary bore) shrinks upon heating, leading students to incorrectly assign a negative sign to the radius expansion ( $dr/r = -\alpha \Delta \theta$ ).

Furthermore, students frequently misapply the density variation by either ignoring it or using the wrong sign ( $\rho = \rho_0(1+\gamma\Delta\theta)$ ).

The correct approach requires careful logarithmic differentiation, recognizing that the decreasing surface tension (numerator) and increasing radius (denominator) both contribute to a rapid decay in capillary height, while the decreasing density partially counteracts this effect.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Thermal Expansion of Area and Volume
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free