Problem 14 - Master Resource Book in JEE Main Physics

Question

In experiment for measuring surface tension by capillary rise method, readings for positions \(A, B, C\) and \(D\) for internal diameter of capillary tube are given as under. Mean internal radius of capillary is \(\begin{aligned} &A(\mathrm{~cm})=1,000 \\ &B(\mathrm{~cm})=1,000 \\ &C(\mathrm{~cm})=1,000 \\ &D(\mathrm{~cm})=1,000 \end{aligned}\) (a) \(0.002 \mathrm{~cm}\) (b) \(0.003 \mathrm{~cm}\) (c) \(0.004 \mathrm{~cm}\) (d) \(0.005 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The mean internal radius of the capillary tube is 0.002 cm.

Step 1: Understand the Variables

The experiment involves measuring surface tension using a capillary tube. The positions A, B, C, and D indicate the height of the liquid rise in the capillary tube, but they are identical. We need to focus on the mean internal radius for the calculations.

Step 2: Calculate Capillary Rise

In the capillary rise method, the capillary rise (h) is inversely proportional to the radius (r) of the capillary tube. Since all position readings are identical, the mean internal radius is the primary variable for calculating surface tension.

Step 3: Determine Mean Internal Radius

Given multiple choice answers (a) to (d) provide the radius in centimeters. We are to choose the value that best relates to the height of the liquid rise, keeping in mind that the smaller the radius, the higher the liquid will rise.

Step 4: Choose Correct Value

Based on capillary action principles, narrower tubes result in higher liquid rises due to greater surface tension forces. The correct choice is therefore the smallest provided mean radius, which is 0.002 cm.

Key Concepts

Capillary Action

Capillary action occurs when a liquid moves up or down a narrow space, such as a capillary tube, without the aid of external forces. This movement is due to two key factors: cohesion and adhesion. Cohesion is the force that holds the liquid molecules together, while adhesion is the force between the liquid molecules and the walls of the tube.
In the context of surface tension measurement, capillary action is critical because it helps demonstrate how liquids interact with solid surfaces. When the adhesive force between the liquid and the tube's surface is greater than the cohesive forces within the liquid, the liquid rises in the tube. This phenomenon is clearly visible in the capillary rise method used for measuring surface tension.
Key aspects of capillary action include:

• It only works for liquids that wet the solid surface.
• The narrower the tube, the higher the liquid can rise due to stronger cohesive forces.
• Gravity opposes this action, thus the liquid only rises to a certain height.

Capillary Rise Method

The capillary rise method is a simple experimental technique to measure a liquid's surface tension by observing how high the liquid rises in a capillary tube. It exploits the capillary action principle to quantify surface tension based on the rise height of the liquid column.
The height to which a liquid can rise is inversely proportional to the radius of the tube. If you have a narrow tube, the liquid will rise higher due to increased capillary action. This relationship can be used to calculate the surface tension using the formula:
\[ h = \frac{2\gamma}{\rho g r} \]
where:

• \( h \) = height of the liquid rise
• \( \gamma \) = surface tension
• \( \rho \) = density of the liquid
• \( g \) = acceleration due to gravity
• \( r \) = radius of the capillary tube

Understanding these parameters helps in accurately determining the surface tension. Ensuring the tube is clean and the liquid is pure is also essential for precise results.

Internal Radius of Capillary Tube

The internal radius of a capillary tube is a crucial parameter in determining how high a liquid can rise within it, which directly relates to measuring surface tension. A smaller radius causes the liquid to rise higher due to stronger capillary action than in a tube with a larger radius.
In practical terms, the measurement of the tube's internal radius needs to be accurate because even tiny deviations can affect the rise height of the liquid, leading to errors in surface tension calculations. As seen in the exercise provided, selecting the smallest radius (0.002 cm) ensures a higher rise, reflecting the correct application of the capillary rise method.
To summarize, keep in mind:

• Ensure precision in radius measurement for accurate results.
• A smaller internal radius often leads to higher accuracy in surface tension assessments due to significant capillary action effects.
• This concept underscores the importance of careful selection and measurement of capillary tubes in surface tension experiments.