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Chapter 13: Problem 41

At \(20^{\circ} \mathrm{C}\), the surface tension of water is \(0.0728 \mathrm{~N} / \mathrm{m}\) and that of carbon tetrachloride \(\left(\mathrm{CCl}_{4}\right)\) is \(0.0268 \mathrm{~N} / \mathrm{m}\). If the gauge pressure is the same in two drops of these liquids, what is the ratio of the volume of the water drop to that of the \(\mathrm{CCl}_{4}\) drop?

Short Answer

Expert verified
The ratio of the volume of the water drop to that of the \( \text{CCl}_4 \) drop is approximately 20.05.

Step by step solution

01

Understand Surface Tension and Pressure Relation

The pressure difference between the inside and outside of a liquid drop due to surface tension can be given by the formula \( \Delta P = \frac{2T}{R} \), where \( T \) is the surface tension and \( R \) is the radius of the drop. Since the gauge pressure for both drops is the same, this implies \( \frac{2T_{\text{water}}}{R_{\text{water}}} = \frac{2T_{\text{CCl}_4}}{R_{\text{CCl}_4}} \).
02

Derive Relationship Between Radii

From the equation \( \frac{T_{\text{water}}}{R_{\text{water}}} = \frac{T_{\text{CCl}_4}}{R_{\text{CCl}_4}} \), we find that \( R_{\text{water}} = \frac{T_{\text{water}}}{T_{\text{CCl}_4}} R_{\text{CCl}_4} \). Substituting the given values, \( R_{\text{water}} = \frac{0.0728}{0.0268} R_{\text{CCl}_4} \).
03

Calculate Ratio of Radii

Calculate the ratio \( \frac{R_{\text{water}}}{R_{\text{CCl}_4}} = \frac{0.0728}{0.0268} = 2.716 \). So, the ratio of the radii of water to \( \text{CCl}_4 \) is 2.716.
04

Relate Volume to Radius

The volume of a sphere is given by \( V = \frac{4}{3} \pi R^3 \). For the ratio of the volumes of the two drops, we use \( \frac{V_{\text{water}}}{V_{\text{CCl}_4}} = \left(\frac{R_{\text{water}}}{R_{\text{CCl}_4}}\right)^3 \).
05

Calculate Ratio of Volumes

Substitute the previously calculated ratio of radii into the volume ratio formula: \( \frac{V_{\text{water}}}{V_{\text{CCl}_4}} = (2.716)^3 \approx 20.05 \). Thus, the volume ratio is approximately 20.05.

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Key Concepts

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

Gauge Pressure
Gauge pressure refers to the pressure inside a drop of liquid compared to the pressure outside it, without including atmospheric pressure. This type of pressure indicates the additional pressure exerted due to surface tension, which is especially important in small liquid drops.
When you have two drops with the same gauge pressure, it means the internal pressure (caused by surface tension) is equal for both drops. Understanding gauge pressure is essential because it helps explain how the shape and surface tension of droplets can lead to differences in pressure. This balance of pressure is why raindrops, for example, maintain their form while falling.
Liquid Drops
Liquid drops are fascinating due to their tendency to form spheres. This happens because surface tension, a cohesive force at the surface of a liquid, pulls the molecules into the tightest possible shape. A sphere offers the minimum surface area for a given volume, hence why drops naturally adopt this shape.
Understanding the dynamics of liquid drops is vital for explaining how surface tension influences the pressure inside a drop and impacts their form. When two different liquids, like water and carbon tetrachloride, form drops, their differing surface tensions lead to different radii if the same gauge pressure exists within them. This variance is a direct outcome of their differing molecular cohesion.
Volume Ratio
The volume ratio of two spheres (drops) is connected to their radius. Since both drops maintain the same gauge pressure, meaning the internal pressures are equal, the relationship between their surface tensions directly affects their radii and consequently, their volumes.
To calculate the volume ratio, you need to understand how volume scales with radius. Given the formula for the volume of a sphere, \(V = \frac{4}{3} \pi R^3\), the volume increases with the cube of the radius. If the ratio of the radii of the two drops is known, the volume ratio can be derived by cubing this radius ratio. For example, if the water drop's radius is 2.716 times that of the carbon tetrachloride, the volume ratio becomes approximately \(2.716^3\), giving a volume ratio of approximately 20.05.
Radius of Spheres
The radius of a sphere in the context of liquid drops is crucial. The radius is defined as the distance from the center of the drop to its surface. Because of surface tension, liquid drops attempt to minimize surface area, most often resulting in a spherical shape.
Surface tension's effect on radius is key to understanding the behavior of drops under the same gauge pressure. If surface tension is higher, as in water compared to carbon tetrachloride, the radius must also be larger for the pressure inside both drops to equilibrate. Thus, in problems involving liquid drops, relating radius to surface tension helps predict how different liquids conform to similar conditions.

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Most popular questions from this chapter

Blood pressure. Systemic blood pressure is defined as the ratio of two pressures-systolic and diastolic-both expressed in millimeters of mercury. Normal blood pressure is about \(\frac{120 \mathrm{~mm}}{80 \mathrm{~mm}}\) which is usually stated as \(\frac{120}{80}\). What would normal systemic blood pressure be if, instead of millimeters of mercury, we expressed pressure in each of the following units but continued to use the same ratio format? (a) atmospheres, (b) torr, (c) \(\mathrm{Pa},\) (d) \(\mathrm{N} / \mathrm{m}^{2},\) (e) psi.

A U-shaped tube open to the air at both ends contains some mercury. A quantity of water is carefully poured into the left arm of the U-shaped tube until the vertical height of the water column is \(15.0 \mathrm{~cm}\) (Figure 13.42 ). (a) What is the gauge pressure at the water-mercury interface? (b) Calculate the vertical distance \(h\) from the top of the mercury in the right-hand arm of the tube to the top of the water in the left-hand arm.

A \(975-\mathrm{kg}\) car has its tires each inflated to "32.0 pounds." (a) What are the absolute and gauge pressures in these tires in \(1 \mathrm{~b} /\) in. \(^{2}, \mathrm{~Pa},\) and atm? (b) If the tires were perfectly round, could the tire pressure exert any force on the pavement? (Assume that the tire walls are flexible so that the pressure exerted by the tire on the pavement equals the air pressure inside the tire.) (c) If you examine a car's tires, it is obvious that there is some flattening at the bottom. What is the total contact area for all four tires of the flattened part of the tires at the pavement?

A slab of ice floats on a freshwater lake. What minimum volume must the slab have for a \(45.0 \mathrm{~kg}\) woman to be able to stand on it without getting her feet wet?

Exploring Europa's oceans. Europa, a satellite of Jupiter, appears to have an ocean beneath its icy surface. Proposals have been made to send a robotic submarine to Europa to see if there might be life there. There is no atmosphere on Europa, and we shall assume that the surface ice is thin enough that we can ignore its weight and that the oceans are freshwater having the same density as on the earth. The mass and diameter of Europa have been measured to be \(4.78 \times 10^{22} \mathrm{~kg}\) and \(3130 \mathrm{~km},\) respectively. (a) If the submarine in tends to submerge to a depth of \(100 \mathrm{~m}\), what pressure must it be designed to withstand? (b) If you wanted to test this submarine before sending it to Europa, how deep would it have to go in our oceans to experience the same pressure as the pressure at a depth of \(100 \mathrm{~m}\) on Europa?
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