Question

The relation between surface tension T. Surface area \(\mathrm{A}\) and surface energy \(\mathrm{E}\) is given by. (A) \(\mathrm{T}=(\mathrm{E} / \mathrm{A})\) (B) \(\mathrm{T}=\mathrm{EA}\) (C) \(\mathrm{E}=(\mathrm{T} / \mathrm{A})\) (D) \(\mathrm{T}=(\mathrm{A} / \mathrm{E})\)

Short answer

The correct relation between surface tension (T), surface area (A), and surface energy (E) is given by option (B): \(T=EA\).

Step-by-step solution

Recall the definition of surface tension

Surface tension is a property of liquid surfaces which makes them behave as if they are covered with a stretched elastic membrane. This tension arises due to an imbalance in the molecular cohesive forces at the surface. Surface energy is the extra energy that molecules at the surface possess due to the imbalance of force.

Relate surface tension to surface energy and surface area

The relation between surface tension, surface energy, and surface area can be established by considering a change in surface area. The work done to increase the surface area is equal to the increase in surface energy. Mathematically, this can be represented as: \[ dE = T dA \] This equation states that the differential work (\(dE\)) required to change the surface area by a small amount (\(dA\)) is equal to the product of the surface tension (T) and change in surface area (\(dA\)). Integrating both sides of the equation will give us the relation between surface energy, surface tension, and surface area: \[ E = T A \]

Identify the correct option

Now that we have derived the relationship between surface tension (T), surface area (A), and surface energy (E): \[ E = TA \] We can see that the correct relation corresponds to option (B): \(T=EA\).

Key concepts

Surface Energy

Surface energy involves the energy that is present at the surface of a material. This energy arises due to the uneven forces experienced by molecules at the surface as compared to those well inside the material. When molecules are on the surface, they are not completely surrounded by other molecules, which leads to extra energy. This is why liquid surfaces tend to minimize their area — to reduce this surface energy.

• Surface energy is crucial in phenomena such as wetting, spreading, and adhesion.
• It is denoted by the symbol \(E\).
• Measured in units of Joules per square meter (J/m²).

Understanding surface energy allows us to predict how surfaces will interact, which is important in various applications in engineering and science.

Surface Area

Surface area is the measure of the total area that the surface of an object occupies. In the context of liquid surfaces, increasing or decreasing the surface area involves work against the cohesive forces present at the surface.

• The surface area is a key factor in determining the surface energy.
• It is denoted by the symbol \(A\).
• A larger surface area means more energy is required to maintain that area due to increased surface energy.

The balance between surface energy and surface area is crucial for understanding phenomena in liquids, such as the formation of droplets and bubbles.

Molecular Cohesive Forces

Molecular cohesive forces refer to the force of attraction between molecules of the same substance. In the case of liquid surfaces, these forces are responsible for creating surface tension. Cohesive forces are stronger within the interior of a liquid where molecules are surrounded equally by neighbors than at the surface where the number of neighboring molecules is reduced.

• These forces lead to a minimized surface area as molecules are pulled together.
• When the surface area changes, energy is transferred due to these cohesive interactions.
• Cohesive forces are integral to maintaining the integrity of a surface layer in a liquid.

These cohesive forces are key players in phenomena such as capillarity, where liquids move in narrow spaces without external forces.

Work-Energy Principle

The Work-Energy Principle states that work done on an object is equal to the change in its energy. When applied to surface tension, it suggests that work is needed to increase the surface area of a liquid, and this work results in a change in surface energy. Mathematically, this can be expressed as: \[ dE = T dA \] This equation inside tells us that a change in surface energy \(dE\) is proportional to the surface tension \(T\) and the infinitesimal change in the surface area \(dA\).

• The fundamental understanding provided by this principle assists in calculating how much work is required to change a surface area.
• Allows prediction of how much energy is needed or released when altering a surface.
• It is critical in applications such as manufacturing, where controlling surface properties is essential.

Hence, the work-energy principle offers a foundational insight into the relationships between surface tension, surface area, and energy.