Problem 54 - Master Resource Book in JEE Main Physics

Question

While measuring surface tension of water using capillary rise method, height of the lower meniscus from free surface of water is \(3 \mathrm{~cm}\) while inner radius of capillary tube is found to be \(0.5 \mathrm{~cm}\). Then compute surface tension of water using this data. [Take contact angle between glass and water as \(0^{\circ}\) and \(\left.g=9.81 \mathrm{~ms}^{-2}\right]\) (a) \(0.72 \mathrm{Nm}^{-1}\) (b) \(0.77 \mathrm{Nm}^{-1}\) (c) \(1.67 \mathrm{Nm}^{-1}\) (d) None of the above

Step-by-Step Solution

Answer

The closest value is (a) 0.72 Nm^{-1}.

Step 1: Understand the Capillary Rise Method

The capillary rise method uses the phenomenon where a liquid climbs up a narrow tube due to surface tension. The height the liquid rises to is related to the surface tension, the density of the liquid, the acceleration due to gravity, and the radius of the tube.

Step 2: Formula for Surface Tension in Capillary Rise

The surface tension \( \gamma \) can be calculated using the formula: \[ \gamma = \frac{h r \rho g}{2 \cos \theta} \]where:- \( h \) is the height of the liquid column (3 cm or 0.03 m),- \( r \) is the radius of the capillary tube (0.5 cm or 0.005 m),- \( \rho \) is the density of water (approximately 1000 kg/m³),- \( g \) is the acceleration due to gravity (9.81 m/s²),- \( \theta \) is the contact angle (0° so \( \cos \theta = 1 \)).

Step 3: Calculate Surface Tension using Given Values

Substitute the given values into the formula:\[ \gamma = \frac{(0.03 \, m) \times (0.005 \, m) \times (1000 \, kg/m^3) \times (9.81 \, m/s^2)}{2 \times 1} \]Calculating further:\[ \gamma = \frac{(1.47)}{2} = 0.735 \mathrm{Nm}^{-1} \].

Step 4: Round Off the Result

Round the calculated value to two decimal places to match standard surface tension results. Thus, \( \gamma = 0.74 \mathrm{Nm}^{-1} \).

Step 5: Select the Closest Option

Examine the given options: (a) 0.72 Nm^{-1}, (b) 0.77 Nm^{-1}, (c) 1.67 Nm^{-1}, (d) None of the above. The closest option to the calculated surface tension (0.74 Nm^{-1}) is (a) 0.72 Nm^{-1).

Key Concepts

capillary rise method

The capillary rise method is a fascinating phenomenon that reveals the wonders of surface tension. This method works on the principle that a liquid will rise or fall in a narrow tube, such as a capillary, depending on the liquid's cohesive and adhesive forces. When the adhesive forces between the liquid and the tube's surface are stronger, the liquid climbs up, creating what we call a meniscus.

The height to which the liquid rises is directly linked to the liquid's surface tension. To calculate it, we consider several factors: the height of the liquid column, the radius of the tube, the density of the liquid, the angle of contact, and the gravity acting on the system. Understanding these elements helps us accurately measure the surface tension of a liquid using a capillary tube.

contact angle

The contact angle in the context of surface tension can influence the behavior of liquids in capillary tubes. The contact angle, denoted by \( \theta \), is the angle at which a liquid interface meets a solid surface.

In the exercise, the contact angle between glass and water is given as \(0^{\circ}\), which implies perfect wetting. In such a case, \( \cos \theta \) equals 1, simplifying the calculations of surface tension. When the liquid perfectly wets the surface, the liquid tends to rise higher inside the capillary tube, enhancing the measurement accuracy.

Different liquids and tube materials can have varying contact angles, affecting how much a liquid may rise or fall within the tube. Understanding contact angles is crucial for accurate surface tension measurements.

density of liquid

The density of a liquid, represented as \( \rho \), is an intrinsic property that indicates how much mass is contained in a given volume of the substance. In the context of surface tension measurements, knowing the density of the liquid is essential as it contributes to understanding how the liquid behaves under gravitational forces.

A denser liquid will exert more force due to gravity, affecting the overall capillary rise. In the provided exercise, the density of water is approximated at 1000 kg/m³, which is a standard value widely used in calculations.

By incorporating the density into the formula for surface tension, it ensures a more accurate determination of how surface tension counteracts the gravitational forces pulling the liquid down.

acceleration due to gravity

Acceleration due to gravity, denoted as \( g \), is the acceleration that Earth imparts to objects, pulling them towards its center. This universally constant force significantly influences how liquids rise in a capillary tube.

In the experiment presented, \( g \) is approximated to be \(9.81 \text{ m/s}^2\). This force directly affects the weight of the liquid column in the tube.

• Stronger gravitational pull counteracts the surface tension more vigorously, potentially lowering the height of the liquid rise.
• However, accurate accounting of \( g \) in surface tension calculations allows for a precise determination of how high a liquid can rise due to the surface tension forces counterbalancing gravity.

Understanding gravity's role helps simplify complex systems into models that accurately reflect real-world phenomena.