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Chapter 11: Problem 84

A weather balloon has a volume of \(750 \mathrm{~L}\) when filled with helium at \(8^{\circ} \mathrm{C}\) at a pressure of 380 torr. What is the final volume, in liters, of the balloon when the pressure is \(0.20 \mathrm{~atm}\), the temperature is \(-45^{\circ} \mathrm{C}\), and \(n\) remains constant? \((11.6)\)

Short Answer

Expert verified
The final volume of the balloon is approximately 1521.36 liters.

Step by step solution

01

- Convert Temperatures to Kelvin

First, convert the given temperatures from Celsius to Kelvin using the formula: \[ K = 掳C + 273.15 \] Initial temperature: \[ 8{^\text{C}} = 8 + 273.15 = 281.15 \text{ K} \] Final temperature: \[ -45{^\text{C}} = -45 + 273.15 = 228.15 \text{ K} \]
02

- Convert Pressures to the Same Unit

Convert the initial and final pressures to the same unit. We can convert the initial pressure from torr to atm using the conversion factor: \[ 1 \text{ atm} = 760 \text{ torr} \] Initial pressure: \[ 380 \text{ torr} \times \frac{1 \text{ atm}}{760 \text{ torr}} = 0.5 \text{ atm} \] Final pressure is already given in atm: \[ 0.2 \text{ atm} \]
03

- Apply the Combined Gas Law

Use the combined gas law formula, since the amount of gas (n) is constant: \[ \frac{P_1 \times V_1}{T_1} = \frac{P_2 \times V_2}{T_2} \] Where: \[ P_1 = 0.5 \text{ atm}, V_1 = 750 \text{ L}, T_1 = 281.15 \text{ K} \] \[ P_2 = 0.2 \text{ atm}, T_2 = 228.15 \text{ K} \]
04

- Solve for Final Volume

Rearrange the combined gas law to solve for the final volume, \[ V_2 \]: \[ V_2 = \frac{P_1 \times V_1 \times T_2}{T_1 \times P_2} \] Substitute the known values: \[ V_2 = \frac{0.5 \text{ atm} \times 750 \text{ L} \times 228.15 \text{ K}}{281.15 \text{ K} \times 0.2 \text{ atm}} \] Calculate the final volume: \[ V_2 鈮� \frac{0.5 \times 750 \times 228.15}{281.15 \times 0.2} \] \[ 鈮� \frac{85556.25}{56.23 } \] \[ 鈮� 1521.36 \text{ L} \]

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Key Concepts

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

Gas Laws Explained
The gas laws describe how gases behave under different conditions of temperature, pressure, and volume. They are crucial in understanding the behavior of gases in various scientific and practical applications. The main gas laws include:
  • Boyle's Law - Relates pressure and volume at constant temperature.
  • Charles's Law - Relates volume and temperature at constant pressure.
  • Avogadro's Law - Relates volume and amount of gas at constant temperature and pressure.
The combined gas law merges these individual laws and allows us to explore scenarios where temperature, pressure, and volume all change. It's particularly helpful in real-world applications where multiple variables simultaneously affect gas behavior.
Temperature Conversion
Temperature conversion is essential when working with gas laws, as these laws typically require temperature in Kelvin. The Kelvin scale is vital because it starts at absolute zero, the theoretically lowest temperature possible. The conversion from Celsius to Kelvin is straightforward: add 273.15 to the Celsius temperature for the Kelvin equivalent.

For instance:
  • Initial temperature: 8掳C + 273.15 = 281.15 K
  • Final temperature: -45掳C + 273.15 = 228.15 K
By using Kelvin, we ensure that our calculations remain consistent and accurate, adhering to the absolute scale needed for gas practices.
Pressure Conversion
When working with pressures, it's often necessary to convert between different units to maintain consistency in calculations. Common units of pressure include torr, atm (atmospheres), and pascals (Pa). For this exercise, we convert torr to atm, using the conversion factor: 1 atm = 760 torr.

Example:
  • Initial pressure: 380 torr 脳 (1 atm / 760 torr) = 0.5 atm
  • Final pressure: Already given as 0.20 atm
By converting both pressures to the same unit, we can accurately apply the combined gas law without discrepancies.
Boyle's Law
Boyle's Law explains the relationship between pressure and volume for a gas at constant temperature. Stated mathematically as:
\[ P_1 \times V_1 = P_2 \times V_2 \]
Where:
  • P represents pressure
  • V represents volume
  • The subscripts 1 and 2 refer to the initial and final states of the gas
According to Boyle鈥檚 Law, if the volume of a gas increases, the pressure decreases proportionally, provided the temperature remains unchanged, and vice versa. This principle helps us understand how gas volumes change with pressure variations and is essential for various real-life applications such as breathing and air pressure effects in tires.
Charles's Law
Charles's Law describes the direct relationship between the volume and temperature of a gas at constant pressure. Mathematically, it is expressed as:
\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
Where:
  • V represents volume
  • T represents temperature (in Kelvin)
  • The subscripts 1 and 2 refer to initial and final states
Charles's Law shows that if the temperature of a gas increases, its volume also increases proportionally, and vice versa, provided the pressure remains constant. This law helps us understand temperature-volume relationships, essential for real-life applications like hot air balloons and expanding gases under different thermal conditions.

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Most popular questions from this chapter

A gas at a pressure of \(2.0 \mathrm{~atm}\) is in a closed container. Indicate the changes (increases, decreases, does not change) in its volume that must have occurred if the pressure undergoes the following changes at constant temperature and amount of gas: a. The pressure increases to \(6.0\) atm. b. The pressure remains at \(2.0 \mathrm{~atm}\). c. The pressure drops to \(0.40 \mathrm{~atm}\).

An air bubble has a volume of \(0.500 \mathrm{~L}\) at \(18{ }^{\circ} \mathrm{C}\). What is the final volume, in liters, of the gas when the temperature changes to each of the following, if \(n\) and \(P\) do not change? a. \(0^{\circ} \mathrm{C}\) b. \(425 \mathrm{~K}\) c. \(-12^{\circ} \mathrm{C}\) d. \(575 \mathrm{~K}\)

Why does a sealed bag of chips expand when you take it to a higher altitude?

A sample of gas with a mass of \(1.62 \mathrm{~g}\) occupies a volume of \(941 \mathrm{~mL}\) at a pressure of 748 torr and a temperature of \(20.0{ }^{\circ} \mathrm{C}\). What is the molar mass of the gas? (11.8)

A gas with a volume of \(4.0 \mathrm{~L}\) is in a closed container. Indicate the changes (increases, decreases, does not change) in its pressure that must have occurred if the volume undergoes the following changes at constant temperature and amount of gas: a. The volume is compressed to \(2.0 \mathrm{~L}\). b. The volume expands to \(12 \mathrm{~L}\). c. The volume is compressed to \(0.40 \mathrm{~L}\).
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