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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
All textbooksGiancoli Douglas 5th editionCh. 17 - Temperature, Thermal Expansion, and the Ideal Gas LawProblem 46
Chapter 17, Problem 46

A helium-filled balloon escapes a child’s hand at sea level and 20.0°C. When it reaches an altitude of 3600 m, where the temperature is 5.0°C and the pressure only 0.64 atm, how will its volume compare to that at sea level?

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Step 1: Recognize that this problem involves the Ideal Gas Law, which is expressed as PV = nRT. Since the number of moles of gas (n) and the gas constant (R) remain constant, we can use the combined gas law: \(\frac{P_1 V_1}{T_1}\) = \(\frac{P_2 V_2}{T_2}\), where subscripts 1 and 2 refer to the initial and final states, respectively.
Step 2: Convert the given temperatures from Celsius to Kelvin, as the Ideal Gas Law requires absolute temperature. Use the formula T(K) = T(°C) + 273.15. For the initial temperature, T_1 = 20.0 + 273.15. For the final temperature, T_2 = 5.0 + 273.15.
Step 3: Identify the initial and final pressures. The initial pressure at sea level is P_1 = 1.00 \; \(\text{atm}\), and the final pressure at 3600 m is P_2 = 0.64 \; \(\text{atm}\).
Step 4: Rearrange the combined gas law to solve for the final volume V_2: V_2 = V_1 \(\cdot\) \(\frac{P_1 T_2}{P_2 T_1}\). Substitute the known values for P_1, P_2, T_1, and T_2 into this equation.
Step 5: Simplify the expression to find the ratio \(\frac{V_2}{V_1}\), which represents how the final volume compares to the initial volume. This ratio will indicate whether the balloon's volume increases or decreases as it rises to the higher altitude.

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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. This law is fundamental in understanding how gases behave under varying conditions of temperature and pressure, which is crucial for analyzing the balloon's volume change as it ascends.
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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 helps explain how the volume of the helium balloon will change as it rises to a lower pressure at higher altitudes.
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Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its absolute temperature when pressure is held constant (V1/T1 = V2/T2). This concept is important for understanding how the temperature drop from 20.0°C to 5.0°C affects the volume of the helium balloon as it ascends.
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