Dalton's Law of Partial Pressure, Molar Volume, and Ideal Gas Law

Dalton's Law of Partial Pressure

• Dalton's law of partial pressure states that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of each individual gas.

• This is analogous to calculating your grade, where each component contributes to the total.

Partial Pressure

• Partial pressure is the individual pressure exerted by a gas in a mixture of gases.

• Example: Air is composed of oxygen, nitrogen, argon, water vapor, and carbon dioxide, each exerting its own pressure.

Calculating Partial Pressure

• The partial pressure of a gas can be calculated as:

  $Partial\ Pressure\ of\ O<em>2 = \frac{Pressure\ exerted\ by\ O</em>2}{Total\ Pressure}$

• The total pressure in a system is the sum of the partial pressures of all gases:

  $P<em>{total} = P</em>{O<em>2} + P</em>{N<em>2} + P</em>{Ar} + P<em>{H</em>2O} + P<em>{CO</em>2}$

• Example values: Oxygen (20.9 kPa), Nitrogen, Argon, Water vapor, leading to a total pressure of 101.3 kPa.

Example Calculation

• To find the percentage of total pressure exerted by oxygen:

  $\frac{20.9\ kPa}{101.3\ kPa} \approx 0.2063$

• Converting to percentage:

  $0.2063 \times 100\% = 20.63\%$

• This means 20.63% of the total pressure is due to oxygen gas.

Mole Fraction

• Instead of using actual pressures, mole fraction can be used if the mole quantities are known.

• Mole fraction is the ratio of the moles of a particular gas to the total moles of all gases in the mixture.

Application Example

• Given: A container with hydrogen, nitrogen, and oxygen gas has a total pressure of 646 Torr.

• Total pressure is the sum of individual partial pressures:

  $P<em>{total} = P</em>{H<em>2} + P</em>{N<em>2} + P</em>{O_2} = 646\ Torr$

• Hydrogen makes up 23%, oxygen 67%, and nitrogen 10% of the total pressure.

Calculating Partial Pressures from Percentages

• If hydrogen is 23% of the total pressure, its partial pressure is:

  $P<em>{H</em>2} = 0.23 \times 646 \ Torr = 148.58 \ Torr$

• Similarly, for oxygen:

  $P<em>{O</em>2} = 0.67 \times 646 \ Torr = 432.82 \ Torr$

• To find the partial pressure of nitrogen, subtract the partial pressures of hydrogen and oxygen from the total pressure:

  $P<em>{N</em>2} = 646 \ Torr - (148.58 \ Torr + 432.82 \ Torr) = 64.6 \ Torr$

Key Idea

• Dalton's law states that the total pressure of a gas mixture is the sum of the partial pressures of each gas.

• This is similar to calculating a grade, where knowing the percentages allows you to determine the value contributed by each component.

Collecting Gas Over Water

• A common method to collect gas produced in a chemical reaction involves water displacement.

• However, the collected gas is mixed with water vapor, affecting pressure readings.

Accounting for Water Vapor

• The total pressure recorded when collecting gas over water is the sum of the gas's pressure and water vapor pressure:

  $P<em>{total} = P</em>{gas} + P<em>{H</em>2O}$

• To find the true pressure of the gas, subtract the partial pressure of water vapor.

• Use a chart to find the partial pressure of water at a specific temperature.

Example: Nitrogen Gas Collection

• Nitrogen gas collected over water at 25°C shows a total pressure of 98.9 kPa.

• The chart indicates water's partial pressure is 23.756 Torr at 25°C.

• Convert Torr to kPa:

  $23.756\ Torr \times \frac{101.325\ kPa}{760\ Torr} \approx 3.167\ kPa$

• Subtract water's partial pressure to find nitrogen's partial pressure:

  $P<em>{N</em>2} = 98.9\ kPa - 3.167\ kPa \approx 95.733\ kPa$

Avogadro's Law

• Avogadro's Law: As the number of moles of gas increases, the volume also increases, given constant pressure and temperature.

• This relationship leads to the concept of molar volume.

Molar Volume

• STP (Standard Temperature and Pressure): 273 K and 1 atm.

• SATP (Standard Ambient Temperature and Pressure): 298 K (25°C) and 1 atm.

• At STP, one mole of any gas occupies 22.4 liters.

• At SATP, one mole of any gas occupies 24.8 liters.

Molar Volume Application

• Problem: Find the volume of four moles of nitrogen monoxide gas at STP.

• At STP, one mole occupies 22.4 liters, so:

  $4\ moles \times 22.4\ L/mol = 89.6\ liters$

Example Calculation 2

• Problem: How many moles are in 25.4 liters of carbon dioxide at STP?

• Using molar volume:

  $\frac{25.4\ L}{22.4\ L/mol} \approx 1.1339\ moles$

• This result can be extended to find mass (using molar mass) or the number of molecules (using Avogadro's number).

Ideal Gas Law

• Ideal gases are theoretical gases with negligible particle volume and no intermolecular interactions.

• Collisions between ideal gas particles are perfectly elastic.

Ideal Gas Law Equation

• The ideal gas law relates pressure, volume, number of moles, and temperature:

  $PV = nRT$

  Where:

  • P = Pressure

  • V = Volume (in liters)

  • n = Number of moles

  • R = Ideal gas constant

  • T = Temperature (in Kelvin)

Gas Constant Values

• Two common values for R:

  • $R = 8.314\ \frac{L \cdot kPa}{mol \cdot K}$

  • $R = 0.08206\ \frac{L \cdot atm}{mol \cdot K}$

Ideal Gas Law Application

• Problem: What volume does 5 moles of $O_2$ occupy at 28°C (301.15 K) and 0.998 atm?

• Using R = 0.08206:

  $V = \frac{nRT}{P} = \frac{5\ mol \times 0.08206\ \frac{L \cdot atm}{mol \cdot K} \times 301.15\ K}{0.998\ atm} \approx 123.8\ liters$

Finding Mass with Ideal Gas Law

• Problem: What mass of neon gas is needed to produce a pressure of 90 kPa in a 0.88 liter tube at 30°C (303.15 K)?

• First, find the number of moles using the ideal gas law:

  $n = \frac{PV}{RT} = \frac{90\ kPa \times 0.88\ L}{8.314\ \frac{L \cdot kPa}{mol \cdot K} \times 303.15\ K} \approx 0.03142\ moles$

• Then, convert moles to mass using neon's molar mass (20.18 g/mol):

  $0.03142\ moles \times 20.18\ \frac{g}{mol} \approx 0.63\ g$

Choosing The Right Gas Law

• Use Charles, Boyle, Gay-Lussac, or the combined gas law when a property changes in the system, and you want to see that change impact another property in the system.

• Use the ideal gas law when trying to find a particular property in the system.