Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Problem 48

Chapter 17, Problem 48

A helium balloon rises because of a buoyant force. By what percentage does the buoyant force on a helium balloon change if the temperature of the helium is increased from 15°C to 25°C while the temperature of the surrounding air is unchanged? Assume that the pressure of the helium remains constant.

Verified step by step guidance

Start by recalling the formula for buoyant force: \( F_b = \rho_{air} \cdot V_{balloon} \cdot g \), where \( \rho_{air} \) is the density of the surrounding air, \( V_{balloon} \) is the volume of the balloon, and \( g \) is the acceleration due to gravity. Note that \( \rho_{air} \) and \( g \) remain constant in this problem.

Recognize that the volume of the helium balloon, \( V_{balloon} \), is related to the temperature of the helium gas through the ideal gas law: \( PV = nRT \). Since the pressure \( P \) and the number of moles \( n \) of helium are constant, the volume \( V \) is directly proportional to the temperature \( T \) in kelvins: \( V \propto T \).

Convert the given temperatures from Celsius to kelvins using the formula \( T(K) = T(°C) + 273.15 \). For 15°C, \( T_1 = 15 + 273.15 = 288.15 \ \text{K} \), and for 25°C, \( T_2 = 25 + 273.15 = 298.15 \ \text{K} \).

Determine the percentage change in the volume of the balloon, and hence the buoyant force, using the relationship \( \text{Percentage change} = \left( \frac{T_2 - T_1}{T_1} \right) \cdot 100 \). Substitute \( T_1 = 288.15 \ \text{K} \) and \( T_2 = 298.15 \ \text{K} \) into this formula.

Conclude that the percentage change in the buoyant force is equal to the percentage change in the volume of the balloon, as the buoyant force is directly proportional to the volume. This completes the solution.

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Key Concepts

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

Buoyant Force

Buoyant force is the upward force exerted by a fluid on an object submerged in it, which is equal to the weight of the fluid displaced by the object. According to Archimedes' principle, this force depends on the density of the fluid and the volume of the object. In the case of a helium balloon, the buoyant force allows it to rise in the denser surrounding air.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. For a constant pressure scenario, an increase in temperature will lead to an increase in volume for a given amount of gas, which is crucial for understanding how the helium inside the balloon behaves as its temperature changes.

Density and Temperature Relationship

The density of a gas is affected by its temperature; as temperature increases, the density of the gas typically decreases if pressure is held constant. This relationship is important for understanding how the buoyant force changes when the temperature of the helium in the balloon increases, as a decrease in density will affect the amount of air displaced and thus the buoyant force experienced by the balloon.

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