Problem 2807 - JEE Physics

Question

To what height can mercury be filled in a vessel without any leakage if there is a pin hole of diameter \(0.1 \mathrm{~mm}\) at the bottom of the vessel. (Density of mercury \(=13.6 \times 10^{3} \mathrm{~kg} \mathrm{~m}^{-3}\), surface tension of mercury \(=550 \times 10 \mathrm{~h}^{-3} \mathrm{pNm}^{-1}\), Angle of contact with the vessel for mercury \(=0^{\circ}\) ) (A) \(16.1 \mathrm{~cm}\) (B) \(18.5 \mathrm{~cm}\) (C) \(20 \mathrm{~cm}\) (D) \(12.5 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The maximum height of mercury that can be filled in the vessel without any leakage is approximately \(16.1 \mathrm{~cm}\) (Option A).

Step 1: Formula for height without leakage

We will use the Jurin's Law formula to find the maximum height of the liquid column that can be supported by the given surface tension without leakage: \[h = \frac{2S\cos{\theta}}{r\rho g}\] where \(h\) is the height of the liquid column, \(S\) is the surface tension of the liquid, \(\theta\) is the angle of contact between the liquid and the wall of the container, \(r\) is the radius of the hole or tube, \(\rho\) is the density of the liquid, and \(g\) is the acceleration due to gravity (approximately \(9.81 \mathrm{~m~s}^{-2}\)).

Step 2: Calculate the radius of the hole

The diameter of the hole is given as \(0.1 \mathrm{~mm}\). To find the radius, we will divide the diameter by 2: \[r = \frac{0.1 \mathrm{~mm}}{2} = 0.05 \mathrm{~mm} = 5 \times 10^{-5} \mathrm{~m}\]

Step 3: Substitute known values into Jurin's Law formula

Now, we will substitute the values given in the problem into the Jurin's Law formula: \[h = \frac{2(550 \times 10^{-3} \mathrm{~N~m}^{-1})\cos{0^{\circ}}}{(5 \times 10^{-5} \mathrm{~m})(13.6 \times 10^{3} \mathrm{~kg~m}^{-3})(9.81 \mathrm{~m~s}^{-2})}\]

Step 4: Solve for height

After substituting the known values into the formula, we simply need to solve for the height, \(h\): \[h \approx \frac{2(550 \times 10^{-3} \mathrm{~N~m}^{-1})(1)}{(5 \times 10^{-5} \mathrm{~m})(13.6 \times 10^{3} \mathrm{~kg~m}^{-3})(9.81 \mathrm{~m~s}^{-2})} = 0.16116 \mathrm{~m}\] Since we want the height in centimeters, we can convert meters to centimeters: \[h \approx 0.16116 \mathrm{~m} \times \frac{100 \mathrm{~cm}}{1\mathrm{~m}} = 16.116 \mathrm{~cm}\]

Step 5: Choose the correct answer

Given the options in the question, we can see that our answer is very close to option (A) \(16.1~\mathrm{cm}\). Thus, the maximum height of mercury that can be filled in the vessel without any leakage is approximately \(16.1 \mathrm{~cm}\).

Key Concepts

Surface Tension

Surface tension is an essential concept when it comes to understanding why liquids behave differently when they come into contact with surfaces. It is what allows a liquid's surface to act like a stretched elastic membrane. This tension is due to the cohesive forces between liquid molecules.

Key points about surface tension include:

• In liquids, molecules experience forces from their neighbors. At the surface, however, molecules are pulled inward, creating a "tightened" effect.
• Surface tension is measured in units of force per length, like N/m (newtons per meter).
• It explains phenomena such as why small insects can walk on water and drops form on surfaces instead of spreading out completely.

In the context of this exercise, the surface tension of mercury is crucial. It's responsible for determining how high the mercury can climb within a narrow tube without spilling through the pinhole. The stronger the surface tension, the taller a liquid column can be supported.

Density of Mercury

Density is another vital property to consider when studying fluids, specifically heavy liquids like mercury. Density refers to the mass per unit volume of a substance, expressed in kilograms per cubic meter (kg/m³).

Here are some important considerations about density:

• Mercury has a remarkably high density, calculated as 13.6 x 10³ kg/m³, making it significantly heavier than many other liquids.
• The density of a liquid affects how it behaves under various conditions, like the pressure it exerts at different depths.

For this exercise, mercury's density plays a critical role in determining the pressure at the bottom of the vessel. A higher density means more weight per unit volume, influencing the height up to which mercury can be filled without leaking through the hole. It's a balance of forces: the surface tension must support this weight to prevent leakage.

Capillarity

Capillarity is the ability of a liquid to flow through narrow spaces without any external forces like gravity. This property is due to the combination of cohesive and adhesive forces between the liquid and the surfaces it contacts.

Key aspects of capillarity include:

• Capillary action occurs when the adhesive forces between a liquid and a surface are stronger than the cohesive forces within the liquid itself.
• It is highly influenced by the angle of contact; a zero-degree contact angle, as specified for mercury in this situation, maximizes the capillary effect.
• Jurin's Law describes this phenomenon precisely in terms of the height and shape of liquid columns in small tubes.

In the exercise, capillarity is observed as mercury rises in the vessel without spilling through the small hole because of its surface tension and capillary action. Understanding this helps explain how much mercury can fill the vessel in a stable equilibrium, supported by the balance of surface tension against the weight due to the density of mercury.