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Ch.5 - Gases
Chapter 5, Problem 115

A scuba diver creates a spherical bubble with a radius of 2.5 cm at a depth of 30.0 m where the total pressure (including atmospheric pressure) is 4.00 atm. What is the radius of the bubble when it reaches the surface of the water? (Assume that the atmospheric pressure is 1.00 atm and the temperature is 298 K.)

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. In this scenario, the pressure and volume of the gas bubble change as it ascends, while the temperature remains constant. Understanding this law is crucial for calculating how the volume (and thus the radius) of the bubble changes with pressure.
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Ideal Gas Law Formula

Boyle's Law

Boyle's Law states that the pressure of a gas is inversely proportional to its volume when temperature is held constant (P1V1 = P2V2). This principle is essential for determining how the volume of the bubble will change as the diver ascends and the pressure decreases from 4.00 atm to 1.00 atm, allowing us to find the new radius of the bubble at the surface.
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Spherical Geometry

The volume of a sphere is calculated using the formula V = (4/3)πr³, where r is the radius. When the volume of the bubble changes due to pressure changes, the radius can be derived from the new volume. Understanding spherical geometry is necessary to relate the volume of the bubble to its radius, which is the final step in solving the problem.
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Related Practice
Open Question
Twenty-five milliliters of liquid nitrogen (density = 0.807 g/mL) is poured into a cylindrical container with a radius of 10.0 cm and a length of 20.0 cm. The container initially contains only air at a pressure of 760.0 mmHg (atmospheric pressure) and a temperature of 298 K. If the liquid nitrogen completely vaporizes, what is the total force (in lb) on the interior of the container at 298 K?
Open Question
A 160.0-L helium tank contains pure helium at a pressure of 1855 psi and a temperature of 298 K. How many 3.5-L helium balloons can be filled with the helium in the tank? (Assume an atmospheric pressure of 1.0 atm and a temperature of 298 K.)
Textbook Question

An 11.5-mL sample of liquid butane (density = 0.573 g/mL) is evaporated in an otherwise empty container at a temperature of 28.5 °C. The pressure in the container following evaporation is 892 torr. What is the volume of the container?

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Open Question
A particular balloon can be stretched to a maximum surface area of 1257 cm². The balloon is filled with 3.0 L of helium gas at a pressure of 755 torr and a temperature of 298 K. The balloon is then allowed to rise in the atmosphere. If the atmospheric temperature is 273 K, what pressure will the balloon burst at? (Assume the balloon is the shape of a sphere.)
Textbook Question

A catalytic converter in an automobile uses a palladium or platinum catalyst (a substance that increases the rate of a reaction without being consumed by the reaction) to convert carbon monoxide gas to carbon dioxide according to the reaction: 2 CO(g) + O2(g) → 2 CO2(g) A chemist researching the effectiveness of a new catalyst combines a 2.0:1.0 mole ratio mixture of carbon monoxide and oxygen gas (respectively) over the catalyst in a 2.45-L flask at a total pressure of 745 torr and a temperature of 552 °C. When the reaction is complete, the pressure in the flask has dropped to 552 torr. What percentage of the carbon monoxide was converted to carbon dioxide?

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Textbook Question

A quantity of N2 occupies a volume of 1.0 L at 300 K and 1.0 atm. The gas expands to a volume of 3.0 L as the result of a change in both temperature and pressure. Find the density of the gas at these new conditions.

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