gases and gas law

Gases and Gas Laws

Overview

  • Focus Areas:

    • Structural study of atoms and molecules through chemical bonding (Focus 1 & 2).

    • Examination of the properties of bulk matter in the form of gases, liquids, and solids (Focus 3).

  • Source: Atkins Chemical Principles: The Quest for Insight – Macmillan Learning ©2023

Importance of Studying Gases

  • Gases in Atmosphere:

    • Essential for understanding Earth's atmosphere.

  • Industrial Gas Processes:

    • Include combustion, distillation, explosion, etc.

  • Relevance in Industry:

    • Used in chemical and medical fields (e.g., oxygen, Nitrous oxide (N2O), inhalers).

    • Example: Air Products Inc. generates $12 billion in annual revenue.

  • Historical Significance:

    • Early gas observations were foundational for chemistry.

Properties of Gases

  1. Expansibility: Gases expand or can be compressed to fill the volume of any container.

  2. Density: Gases have much lower densities than solids or liquids.

  3. Variable Densities: Gases display highly variable densities with changes in volume due to compression/expansion.

  4. Diffusion: Gases mix readily and thoroughly with one another.

State of a Gas

  • Defined by four properties:

    • V = Volume (L)

    • T = Temperature (K)

    • n = Amount (moles)

    • P = Pressure (atmospheres, atm)

Pressure: What Causes It?

  • Unit of Pressure:

    • SI unit: Pascal (Pa)

    • 1 ext{ Pa} = 1 ext{ kg·m}^{-1} ext{s}^{-2}

  • Basic Equations:

    • F = ma

    • a = rac{dv}{dt}

    • Pressure also considered in different units:

    • Force (cgs unit: dyne)

    • MKS unit: Newton

    • Pressure unit 1 ext{ Bar} = 10,000 ext{ Pa}

  • Molecular Collisions:

    • Pressure results from molecular collisions between gas molecules and container walls.

    • For a balloon, at pressure P_{ ext{atm}}, with n moles in volume V at temperature T, pressure equals atmospheric pressure.

Measurement of Pressure

  • Barometer:

    • Developed by Torricelli in 1643 to measure air pressure.

  • Equation:

    • P = rac{F}{A} (pressure as function of force and area).

  • Balancing Forces:

    • At the surface, the mercury (Hg) density
      ho and height h relation:

    • P = rac{F}{A} = rac{mG}{A} = rac{V
      ho G}{A} = hA
      ho G/A = h
      ho G

  • Gas above Mercury:

    • For gas in a container: P{atm} = P{g} + h
      ho G

Example: Atmospheric Pressure Calculation

  • Given:

    • Height of Hg column: 760 mm at 15 °C, density of Hg: 13.595 g·cm−3

    • Standard gravity: g = 9.80665 ext{ m·s}^{-2}

  • Pressure Calculation:
    P =
    ho imes h imes g
    P = (13,595 ext{ kg·m}^{-3}) imes (0.760 ext{ m}) imes (9.80665 ext{ m·s}^{-2})
    P ext{ in Pascals: } 1.01 imes 10^{5} ext{ Pa}

  • Indicated forces balance at height of the mercury column.

Standard Atmospheric Pressure

  • Standard Measurement:

  • 1 ext{ atm} = 760 ext{ mm Hg} = 760 ext{ Torr} = 1.013 imes 10^{5} ext{ Pa} = 101.325 ext{ kPa} = 14.7 ext{ psi}

  • Unit Conversion:

    • 1 Newton = 10^{5} dyne

Self-test Example

  • Water Barometer Query:

    • Calculate column equivalent of water at 760 mm of mercury pressure (refer to density of water: 0.998 g·cm−3; results in a significantly tall column).

Measuring Pressure in the Laboratory

  • Open-Tube Manometer (a):

    • Height difference in mercury levels indicates pressure difference with atmosphere.

  • Closed-Tube Manometer (b):

    • Height difference corresponds to pressure in the system.

Example Problem: Open-tube Manometer

  • Calculation Problem:

    • Given: pressure on system-side is 10 mm higher than atmospheric, 756 mmHg at 15 °C.

  • Results:

    • P_{ ext{internal}} = 746 ext{ mmHg}.

    • P = dhg = (13,595 ext{ kg·m}^{-3}) imes (0.746 ext{ m}) imes (9.80665 ext{ m·s}^{-2}) = 9.95 imes 10^{4} ext{ Pa}

The Gas Laws

  • Overview:

    • Properties of gases include pressure, volume, temperature, and amount (moles).

  • Historically: Robert Boyle's (1662) studies led to foundational gas laws. Patented experiments included effects of pressure on gas volume with later contributions from Jacques Charles and Joseph-Louis Gay-Lussac.

  • Contributions by Amedeo Avogadro:

    • Introduced relationship between volume and amount of molecules, enhancing atomic theory understanding.

Boyle’s Law

  • Definition:

    • At constant n and T, PV = k; if V decreases, then P increases.

  • Experiment:

    • Boyle utilized a J-shaped tube with mercury to compress air, showing the direct relationship of P and V.

  • Graphical Analysis:

    • Provides linear data for plotting P against 1/V.

  • Equation Form:

    • PV = ext{ constant} ; also, P2V2 = P1V1 (indicating final and initial conditions).

Self-Test Example: Boyle's Law Application

  • Context: Petroleum refinery scenario where ethylene gas at 750 L (1.00 bar) was compressed to 5.00 bar; apply Boyle's Law for final volume calculation.

Charles’s Law

  • Principle: Volume of gas directly correlates with temperature increase when n and P are held constant:

    • V = k'T

  • Experimentations: Conducted by balloonists Charles and Gay-Lussac, verifying volume increases with temperature.

  • Graphing:

    • Exhibits linear relationship of V against T,

    • V/T = ext{constant} ; extrapolation indicates gases reach zero volume at −273.15 °C.

Self-test Example: Charles’s Law Application

  • Scenario Task: Calculate the volume of a gas heated from 50 °C to 100 °C. Convert to Kelvin from degrees Celsius approval: K = 273.15 + °C.

Gay-Lussac’s Law

  • Context: For gas under constant volume, pressure increases with temperature (

  • Formula Formulation: Expressed as P ext{ proportional to } T.

Avogadro’s Law

  • Definition: At the same T and P, equal volumes of gases contain equal numbers of molecules.

  • Equation: Presented as V = k''n under constant conditions.

    • Molar volume for gases under standard conditions: approximately 22 L·mol−1 at 0 °C and 1 atm.

Ideal Gas Law

  • Full Expression: Combines earlier laws into a singular equation relating P, V, T, and n:

    • PV = nRT and includesAvogadro’s, Boyle’s, Charles’s Law, Gay-Lussac’s relationships.

  • Ideal Gas Behavior Contexts: Gases maintain ideal behavior under STP conditions (0 °C/273.15 K, 1 atm).

Utilizing PV = nRT for Calculations

  • Example Problem: Determining gaseous molecules in a room:

    • Volume: 180,000 ft³ (converted to L), calculating pressure at P = 745 mmHg and temperature at 25 °C.

Mixtures of Gases & Dalton’s Law of Partial Pressures

  • Understanding Gas Behavior:

    • Air as a mixture of nitrogen, oxygen, among other gases.

  • Pressure Relationships: Total pressure as a sum of partial pressures; P{tot} = P1 + P_2 + ext{…} .

  • Mathematical Expression: Derived from relationships relating to mole fractions and total pressure experience.

Example Problem: Calculating Partial Pressures

  • Hydrogen and Oxygen Case Study: Using an equation to assign mol ratios for H₂ and O₂ under total pressure conditions.

Applications of Gas Stoichiometry

  • Practical Usage: Understanding volumetric needs in combustion or aerobic respiration scenarios.

Relevant Gas Reactions and Applications

  • Airbags in Automobile Safety:

    • Nitrogen gas generated by sodium azide decomposition for crash safety systems.

    • Reaction: 2 ext{ NaN}3 ightarrow 2 ext{ Na} + 3 ext{ N}2

    • The decomposition creates the volume of nitrogen required to fill the airbag rapidly upon impact.

Economic Reaction Calculations

  • Evaluating reactions, work done by gas expansion, and temperature calculations to anticipate exigencies when gases reach high temperatures.

Reactions with Carbon Dioxide

  • Removing CO₂ from Submarines: Utilizing potassium superoxide for purification, releasing O₂.

Example Problem: CO₂ Reactivity Calculations

  • Assessing moles necessary to remove CO₂ and applying molar volume to convert volume to required gas units.

Gases and Gas Laws Equations

Pressure Equations
  • SI unit of Pressure: 1 \text{ Pa} = 1 \text{ kg·m}^{-1} \text{s}^{-2}
  • Force: F = ma
  • Acceleration: a = \frac{dv}{dt}
  • Bar Unit: 1 \text{ Bar} = 10,000 \text{ Pa}
  • Pressure definition: P = \frac{F}{A}
  • Pressure with density, height, and gravity: P = h\rho G
  • Atmospheric pressure for gas in container (manometer): P_{atm} = P_{g} + h\rho G
  • Standard Atmospheric Pressure Conversions:
    • 1 \text{ atm} = 760 \text{ mm Hg} = 760 \text{ Torr} = 1.013 \times 10^{5} \text{ Pa} = 101.325 \text{ kPa} = 14.7 \text{ psi}
    • Unit Conversion: 1 \text{ Newton} = 10^{5} \text{ dyne}
Gas Laws
  • Boyle’s Law (Constant n, T):
    • PV = k
    • P_1V_1 = P_2V_2
  • Charles’s Law (Constant n, P):
    • V = k'T
    • V/T = \text{constant}
    • Temperature Conversion: K = 273.15 + \text{°C}
  • Gay-Lussac’s Law (Constant V, n):
    • P \text{ proportional to } T
  • Avogadro’s Law (Constant T, P):
    • V = k''n
  • Ideal Gas Law (Combines all):
    • PV = nRT
Mixtures of Gases
  • Dalton’s Law of Partial Pressures:
    • P_{tot} = P_1 + P_2 + \text{…}
Relevant Gas Reactions
  • Airbag Decomposition Reaction:
    • 2 \text{ NaN}_3 \rightarrow 2 \text{ Na} + 3 \text{ N}_2