Understanding Excess Pressure Inside a Liquid Drop

Excess pressure inside a liquid drop is a fundamental concept in the study of surface tension. It arises due to the molecular forces acting at the liquid surface, resulting in a difference between the internal and external pressures. Understanding this pressure difference is essential for topics related to the mechanical properties of fluids and is frequently tested in competitive examinations such as JEE Main.

Surface Tension and Its Role in Liquid Drops

Surface tension is the property of a liquid’s surface that enables it to resist an external force, due to the cohesive nature of its molecules. This property is responsible for the spherical shape of small liquid drops, as the system tends to minimize its surface area for a given volume, thereby reducing surface energy. More details on this phenomenon can be found at Surface Tension and Contact Angle.

Origin of Excess Pressure Inside a Liquid Drop

The curved surface of a drop creates a situation where molecules on the surface experience a net inward force due to unbalanced molecular interactions. This causes the pressure inside the drop to exceed the external atmospheric pressure. This pressure difference, known as excess pressure, is necessary to maintain equilibrium and keep the surface stable.

Derivation of Excess Pressure Inside a Liquid Drop

Consider a spherical liquid drop of radius $r$. Let $P_{\text{in}}$ be the pressure inside the drop and $P_{\text{out}}$ be the pressure outside. The surface tension of the liquid is denoted by $T$. The excess pressure is defined as $\Delta P = P_{\text{in}} - P_{\text{out}}$.

Suppose the radius of the drop increases by a small amount $\delta r$. The increase in surface area is given by:

$\Delta A = 4\pi (r + \delta r)^2 - 4\pi r^2 \approx 8\pi r\, \delta r$

The work done in increasing the surface area is used in overcoming surface tension and is:

$\delta W = T \cdot \Delta A = T \cdot 8\pi r\, \delta r$

The forces acting are due to the difference in pressure across the surface of the drop. The work done by the excess pressure for the small increase in volume is:

$\delta W = \Delta P \cdot 4\pi r^2 \delta r$

Equate the two expressions for work:

$\Delta P \cdot 4\pi r^2 \delta r = T \cdot 8\pi r\, \delta r$

Simplifying gives:

$\Delta P = \dfrac{2T}{r}$

Therefore, the excess pressure inside a liquid drop is inversely proportional to its radius and directly proportional to the surface tension of the liquid.

Physical Significance and Applications

Excess pressure explains why smaller droplets exhibit higher internal pressure compared to larger ones. This relationship impacts fluid behaviors in biological systems, technology, and meteorology. The tendency of drops to become spherical is a direct result of surface tension minimizing the pressure difference.

Comparison: Excess Pressure in Drops, Bubbles, and Soap Bubbles

In a liquid drop, only the outer surface contributes to surface tension. For a soap bubble, both the inner and outer surfaces create tension. The table below summarizes the key differences:

Key Points on Excess Pressure Inside a Liquid Drop

• Surface tension leads to excess internal pressure
• Excess pressure is inversely proportional to radius
• Liquid drops have one surface, soap bubbles have two
• Relevant to the stability of droplets and bubbles

Related Fluid Properties

Excess pressure in liquid drops is closely linked to other fluid properties such as viscosity and general fluid pressure. Additional information on these topics can be reviewed at Fluid Pressure and Viscosity and Viscous Force.

Summary

Excess pressure inside a liquid drop is quantitatively described by $\Delta P = \dfrac{2T}{r}$, where $T$ is the surface tension and $r$ is the radius of the drop. This concept underpins the behavior of small droplets, soap bubbles, and similar structures, and is an important aspect of fluid mechanics covered in the JEE Main curriculum.

To further strengthen the understanding of this topic, students may also study topics like Statics and Dynamics and Latent Heat for a comprehensive view of mechanical properties of fluids.