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

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

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

문제 46

17장, 문제 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?

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.