13.3 The Ideal Gas Law

The ideal gas law is a powerful tool for understanding gas behavior. It relates pressure, volume, temperature, and the number of gas particles, allowing us to predict how gases respond to changing conditions. This fundamental equation forms the basis for many gas laws and applications in thermodynamics.

Mastering the ideal gas law opens doors to solving real-world problems involving gases. From weather balloons to scuba diving, this equation helps explain phenomena and make critical calculations. Understanding its various forms and related laws equips you to tackle diverse gas-related scenarios.

Ideal Gas Law

Ideal gas law representations

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• Molecular representation expresses ideal gas law using number of molecules ($PV = NkT$)
  • $P$ represents pressure of the gas measured in pascals (Pa)
  • $V$ represents volume of the gas measured in cubic meters (m³)
  • $N$ represents total number of gas molecules
  • $k$ is Boltzmann constant with value $1.38 \times 10^{-23}$ J/K
  • $T$ represents absolute temperature measured in Kelvin (K)
• Molar representation expresses ideal gas law using number of moles ($PV = nRT$)
  • $n$ represents number of moles of the gas
  • $R$ is universal gas constant with value $8.314$ J/(mol·K)
  • Other variables ($P$, $V$, $T$) have same meaning as molecular representation

Applications of ideal gas law

• Calculate changes in gas properties when one variable changes while others remain constant
  • Pressure halves when volume doubles at constant temperature and moles (Boyle's law)
  • Volume doubles when temperature doubles at constant pressure and moles (Charles's law)
• Determine how variables must change simultaneously to maintain ideal gas law equality
  • Both pressure and volume doubling requires doubling temperature or moles
  • Tripling pressure and halving volume requires temperature to decrease by factor of 6
• Apply combined gas law ($\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$) when temperature changes
  • Subscripts 1 and 2 denote initial and final states of the gas
• Use Boyle's law ($P_1V_1 = P_2V_2$) for processes at constant temperature
• Employ Charles's law ($\frac{V_1}{T_1} = \frac{V_2}{T_2}$) for processes at constant pressure
• Utilize Gay-Lussac's law ($\frac{P_1}{T_1} = \frac{P_2}{T_2}$) for processes at constant volume
• Calculate partial pressure in gas mixtures using Dalton's law

Molecule-mole conversions in gas laws

• Avogadro's number ($N_A = 6.022 \times 10^{23}$ molecules/mol) relates number of molecules to moles
  • One mole contains $N_A$ number of molecules of a substance
• Convert from molecules to moles by dividing number of molecules by $N_A$
  • $n = \frac{N}{N_A}$
  • $2.5 \times 10^{24}$ molecules of helium equals $\frac{2.5 \times 10^{24}}{6.022 \times 10^{23}} = 4.15$ moles
• Convert from moles to molecules by multiplying number of moles by $N_A$
  • $N = n \times N_A$
  • 2.7 moles of nitrogen contains $2.7 \times 6.022 \times 10^{23} = 1.63 \times 10^{24}$ molecules
• Substitute conversions into ideal gas law when given molecules or moles
  • $PV = \frac{N}{N_A}RT$ for molecular to molar conversion
  • $PV = (n \times N_A)kT$ for molar to molecular conversion

Theoretical foundations and limitations

• Kinetic theory of gases provides microscopic explanation for ideal gas behavior
• Ideal gas law is a fundamental equation in thermodynamics
• Real gases deviate from ideal behavior under certain conditions
  • Van der Waals equation accounts for molecular interactions and volume in real gases