91Ó°ÊÓ

The role of surface tension in bubble formation can be demonstrated by considering a spherical bubble of pure saturated vapor in mechanical and thermal equilibrium with its superheated liquid. (a) Beginning with an appropriate free-body diagram of the bubble, perform a force balance to obtain an expression of the bubble radius, $$ r_{b}=\frac{2 \sigma}{p_{\mathrm{sta}}-p_{l}} $$ where \(p_{\text {sat }}\) is the pressure of the saturated vapor and \(p_{l}\) is the pressure of the superheated liquid outside the bubble. (b) On a \(p-v\) diagram, represent the bubble and liquid states. Discuss what changes in these conditions will cause the bubble to grow or collapse. (c) Calculate the bubble size under equilibrium conditions for which the vapor is saturated at \(101^{\circ} \mathrm{C}\) and the liquid pressure corresponds to a saturation temperature of \(100^{\circ} \mathrm{C}\).

Step by step solution

01

Draw a free-body diagram of the bubble

Draw a spherical bubble with pressure inside the bubble, \(p_{sta}\), and pressure outside the bubble, \(p_l\).

02

Force balance

To obtain an expression for the bubble radius, perform a force balance on the bubble considering the pressure forces and surface tension force. The force due to pressure inside the bubble: \(F_{sta} = p_{sta} \cdot A\) where \(A = 4\pi{r_b^2}\) is the surface area of the bubble The force due to pressure outside the bubble: \(F_l = p_l \cdot A\) The force due to surface tension: \(F_{\sigma} = \sigma \cdot C\) where \(C = 8\pi{r_b}\) is the circumference of the bubble

03

Equate forces and find the expression for bubble radius

Equate the forces on the bubble and solve for the bubble radius. \(F_{sta} + F_{\sigma} = F_l\) \(p_{sta} \cdot 4\pi{r_b^2} + \sigma \cdot 8\pi{r_b} = p_l \cdot 4\pi{r_b^2}\) Solve for \(r_b\): \(r_b = \frac{2 \sigma}{p_{sta} - p_l}\) For part (b):

04

Sketch a p-v diagram

Draw a pressure-volume (p-v) diagram. Mark the bubble state on the diagram with pressure \(p_{sta}\) and label it 'Bubble'. Mark the liquid state on the diagram with pressure \(p_l\) and label it 'Liquid'.

05

Discuss factors affecting bubble growth or collapse

The factors that cause the bubble to grow or collapse are: 1. If the pressure outside the bubble, \(p_l\), decreases, then the bubble will grow as the force balance between the internal and external pressure changes. 2. If the surface tension, \(\sigma\), decreases, the bubble will also grow as the force exerted by the surface tension becomes smaller. 3. If the pressure of the saturated vapor inside the bubble, \(p_{sta}\), increases, the bubble will grow. 4. If the pressure of the liquid, \(p_l\), increases or the surface tension, \(\sigma\), increases, the bubble will collapse as the force exerted by the surface tension becomes larger, or the pressure force outside the bubble increases. For part (c):

06

Find vapor and liquid pressures

To calculate the bubble size, we will use the given vapor and liquid pressures. For a saturated vapor at \(101^{\circ} \mathrm{C}\), we find its pressure, \(p_{sta}\), using a steam table or other reference. Similarly, we find the liquid pressure, \(p_l\), corresponding to a saturation temperature of \(100^{\circ} \mathrm{C}\).

07

Calculate bubble size

Now, use the expression for bubble radius obtained in part (a) to calculate the equilibrium bubble size. Substitute values for \(\sigma\), \(p_{sta}\), and \(p_l\) in the expression and compute the value for the bubble radius, \(r_b\). \(r_{b} = \frac{2 \sigma}{p_{sta} - p_l}\) By calculating the bubble radius, \(r_b\), we find the equilibrium bubble size for the given conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Tension

Surface tension plays a fundamental role in the formation of bubbles. Imagine a bubble as a thin elastic surface that is slightly stretched. Surface tension is this elasticity that wants to minimize the bubble's surface area. It acts along the surface of a bubble, providing an inward pull that balances the pressures inside and outside the bubble.

But how does surface tension come into play? It's due to molecules at the liquid's surface experiencing a net inward force, as they are not surrounded by as many similar molecules compared to those inside the liquid.

• This creates a 'film' or layer at the surface, acting almost like a stretched rubber band around the bubble.
• The strength of this pull or tension largely depends on the type of liquid being used.

Surface tension is vital because it determines the shape and stability of bubbles. In our exercise, it was critical for calculating the bubble radius using the formula: \[ r_{b} = \frac{2 \sigma}{p_{\text{sta}} - p_{l}} \]

Force Balance

In bubble formation, force balance is an essential concept. It's what maintains the bubble in a stable or equilibrium state. There are several forces acting on a bubble:

• Pressure forces from the inside and outside of the bubble.
• The forces caused by surface tension acting along its circumference.

Understanding how these forces interact helps us determine why a bubble keeps its shape or might change in size.

By performing a force balance, you find how these different forces equate to give stability. For a bubble, it means finding when the force due to internal pressure plus the surface tension force equals the force due to external pressure.

Expressed as an equation: \( F_{\text{sta}} + F_{\sigma} = F_{l} \). From this, we can derive the expression for the bubble radius as shown in the exercise.

Thermodynamic Equilibrium

Thermodynamic equilibrium is a state where there are no net macroscopic flows of matter or energy. In bubble formation, maintaining thermodynamic equilibrium is crucial for stability. The bubble is in equilibrium when:

• The temperature across it doesn't vary significantly inside or outside.
• The pressure is balanced as per the force balance.

Additionally, for a bubble in thermal equilibrium:

• The energy exchanges among molecules are even, preventing shifts that could destabilize the bubble structure.

This concept is key to bubble stability and ensures that changes, such as temperature variations, don't lead to further bubble growth or collapse. Continued equilibrium means the bubble will neither expand nor shrink on its own without external influence.

Pressure-Volume Diagram

A pressure-volume (p-v) diagram is an excellent tool for visualizing the state of a bubble and the surrounding liquid. This diagram helps highlight how changes in pressure and volume can affect bubble growth or collapse.

In our exercise, visualizing bubble and liquid states on a p-v diagram can illustrate:

• The condition inside the bubble, marked by pressure \( p_{\text{sta}} \), on one point of the diagram.
• The external liquid condition, marked by pressure \( p_{l} \), on another point.

By examining this graph:

• An increase in vapor pressure or a decrease in liquid pressure would generally allow the bubble to expand.
• Conversely, higher external pressure or increased surface tension would lead the bubble to shrink or collapse.

This representation simplifies understanding how different pressure conditions impact the bubble, making complex concepts easier to grasp. Thus, the p-v diagram is a valuable tool in analyzing the physical changes a bubble undergoes.