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Chapter 8: Problem 36

A high altitude balloon is filled with \(1.41 \times 10^{4} \mathrm{L}\) of hydrogen at a temperature of \(21^{\circ} \mathrm{C}\) and a pressure of 745 torr. What is the volume of the balloon at a height of \(20 \mathrm{km}\), where the temperature is \(-48^{\circ} \mathrm{C}\) and the pressure is 63.1 torr?

Short Answer

Expert verified
V2 = (0.98026 atm * 1.41 * 10^4 L * 225.15 K) / (0.08303 atm * 294.15 K) = 1.20 * 10^5 L

Step by step solution

01

Convert Temperature to Kelvin

Convert the initial and final temperatures to Kelvin since absolute temperatures are required when applying the Gas Law. To convert Celsius to Kelvin, add 273.15 to the Celsius temperature. Initial temperature (T1) in Kelvin: T1 = 21°C + 273.15 = 294.15 K. Final temperature (T2) in Kelvin: T2 = -48°C + 273.15 = 225.15 K.
02

Convert Pressure to Atmospheres

Convert the initial and final pressures from torr to atmospheres, because the standard unit for pressure in gas laws is typically atmospheres. To convert from torr to atmospheres, divide by 760. Initial pressure (P1) in atmospheres: P1 = 745 torr / 760 = 0.98026 atm. Final pressure (P2) in atmospheres: P2 = 63.1 torr / 760 = 0.08303 atm.
03

Apply the Combined Gas Law

Use the Combined Gas Law which combines Charles's Law, Boyle's Law, and Gay-Lussac's Law. The formula is: (P1 * V1) / T1 = (P2 * V2) / T2. Solve for V2, the final volume: V2 = (P1 * V1 * T2) / (P2 * T1).
04

Insert Known Values

Insert the known values into the Combined Gas Law equation to solve for V2. Substitute P1 = 0.98026 atm, V1 = 1.41 * 10^4 L, T1 = 294.15 K, P2 = 0.08303 atm, and T2 = 225.15 K into the equation.
05

Calculate the Final Volume

Perform the calculations to find the final volume V2: V2 = (0.98026 atm * 1.41 * 10^4 L * 225.15 K) / (0.08303 atm * 294.15 K).
06

Simplify and Solve for V2

After placing all values into the equation and calculating, the final volume V2 is found.

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Key Concepts

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

Gas Laws
Gas laws are a fundamental part of chemistry and physics that describe the behavior of gases under various conditions of pressure, volume, and temperature. These empirical laws relate to the properties of gases, providing a way to predict how gases will respond when those properties change. Understanding gas laws is crucial for a variety of practical applications, from the design of car engines to predicting the behavior of the atmosphere.
Absolute Temperature in Kelvin
The Kelvin scale is the absolute temperature scale used in science and is one of the core aspects of gas laws. It starts at absolute zero, the theoretical point at which particles have minimum thermal motion. To convert Celsius to Kelvin, one must add 273.15. This is because the Kelvin scale and Celsius scale are aligned so that a difference of one degree Celsius is equivalent to a difference of one degree Kelvin, but 0 degrees Celsius is equivalent to 273.15 Kelvin. For gas laws, the Kelvin scale is essential as it allows the direct comparison of temperatures in the mathematical formulas.
Pressure Conversions
In gas law calculations, pressure often needs to be converted into a standard unit. The two common units are atmospheres (atm) and torr. One atmosphere is defined as the pressure exerted by a 760 mm column of mercury at sea level and at a temperature of 0 degrees Celsius. Conversely, 1 torr is the pressure exerted by a 1 mm column of mercury. Thus, to convert torr to atmospheres, you divide by 760. It's important to convert to a consistent unit to accurately apply the gas laws in calculations.
Charles's Law
Charles's Law is a principle that describes how the volume of a gas expands as the temperature increases, if the pressure is held constant. It states that the volume of a given mass of a gas is directly proportional to its temperature on the Kelvin scale. This can be represented mathematically as \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\) when the pressure is constant. In other words, if you double the temperature of a gas, its volume will also double, assuming the pressure remains unchanged.
Boyle's Law
Boyle's Law is another fundamental concept within gas laws which explains the inverse relationship between pressure and volume for a fixed amount of gas at a constant temperature. Mathematically, Boyle's Law states that \(\frac{P_1V_1}{P_2V_2}\) is a constant when the temperature is held steady. This tells us that when the volume of a gas decreases, the pressure increases, as long as the temperature doesn't change.
Gay-Lussac's Law
Gay-Lussac's Law relates the pressure of a gas to its absolute temperature, provided the volume of the gas remains constant. It states that the pressure of a fixed amount of gas held at a constant volume is directly proportional to the temperature. Represented as \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\), this law helps us understand that, if you heat a gas, increasing its temperature while keeping its volume constant, the pressure will also increase.

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Most popular questions from this chapter

Calculate the relative rate of diffusion of \(^{1} \mathrm{H}_{2}\) (molar mass \(2.0 \mathrm{g} / \mathrm{mol}\) ) compared with \(^{2} \mathrm{H}_{2}\) (molar mass 4.0 g/ mol) and the relative rate of diffusion of \(\mathrm{O}_{2}\) (molar mass \(32 \mathrm{g} / \mathrm{mol}\) ) compared with \(\mathrm{O}_{3}\) (molar mass \(48 \mathrm{g} / \mathrm{mol}\) ).

What is the molar mass of a gas if \(0.0494 \mathrm{g}\) of the gas occupies a volume of \(0.100 \mathrm{L}\) at a temperature \(26^{\circ} \mathrm{C}\) and a pressure of 307 torr?

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Why is it necessary to use a nonvolatile liquid in a barometer or manometer?

An alternate way to state Avogadro's law is "All other things being equal, the number of molecules in a gas is directly proportional to the volume of the gas." (a) What is the meaning of the term "directly proportional?" (b) What are the "other things" that must be equal?
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