6.1 Gas laws

Gas laws are the backbone of understanding how gases behave under different conditions. They explain the relationships between pressure, volume, temperature, and amount of gas. These laws are crucial for predicting gas behavior in various real-world scenarios.

The ideal gas law, Boyle's law, Charles's law, and Avogadro's law form the foundation of gas behavior. These laws help us calculate and predict changes in gas properties. Understanding their limitations and real gas behavior is essential for accurate applications in chemistry and engineering.

Apply the ideal gas law to calculate properties of gases under various conditions

Ideal gas law equation and variables

• The ideal gas law is $PV = nRT$, where:
  • $P$ is pressure
  • $V$ is volume
  • $n$ is the number of moles of gas
  • $R$ is the ideal gas constant
  • $T$ is the absolute temperature in Kelvin
• The ideal gas law can be used to calculate any of the five properties ($P$, $V$, $n$, $R$, or $T$) if the other four are known

Ideal gas constant values

• The value of the ideal gas constant $R$ depends on the units used for pressure, volume, and temperature
• Common values include:
  • $R = 0.08206 \text{ L} \cdot \text{atm} / \text{mol} \cdot \text{K}$
  • $R = 8.314 \text{ J} / \text{mol} \cdot \text{K}$

Assumptions of the ideal gas law

• The ideal gas law assumes that:
  • Gas particles have negligible volume
  • No intermolecular forces exist between gas particles
  • Gas particles undergo perfectly elastic collisions
• The ideal gas law is most accurate at low pressures and high temperatures, where the gas behavior is closest to ideal

Describe the relationships between pressure, volume, temperature, and amount of gas using Boyle's law, Charles's law, and Avogadro's law

Boyle's law

• Boyle's law states that the pressure and volume of a gas are inversely proportional at constant temperature and amount of gas
• Mathematical expression: $P_1V_1 = P_2V_2$
• Example: If the volume of a gas is halved while temperature and amount remain constant, the pressure will double

Charles's law

• Charles's law states that the volume and absolute temperature of a gas are directly proportional at constant pressure and amount of gas
• Mathematical expression: $V_1/T_1 = V_2/T_2$
• Example: If the temperature of a gas is doubled while pressure and amount remain constant, the volume will also double

Avogadro's law

• Avogadro's law states that the volume and amount (in moles) of a gas are directly proportional at constant pressure and temperature
• Mathematical expression: $V_1/n_1 = V_2/n_2$
• Example: If the amount of a gas is tripled while pressure and temperature remain constant, the volume will also triple

Combined gas law

• The combined gas law is formed by combining Boyle's law, Charles's law, and Avogadro's law
• Mathematical expression: $(P_1V_1)/(n_1T_1) = (P_2V_2)/(n_2T_2)$
• The combined gas law can be used to calculate the change in any of the four variables ($P$, $V$, $n$, or $T$) when the other three are known for two different states of a gas

Explain the concept of partial pressure and use Dalton's law to calculate partial pressures in a gas mixture

Partial pressure

• In a mixture of gases, each gas exerts its own pressure, called the partial pressure, as if it were the only gas present in the container
• The partial pressure of each gas can be calculated using the mole fraction of that gas ($x_i$) and the total pressure: $P_i = x_i \times P_\text{total}$
• The mole fraction of a gas is the ratio of the number of moles of that gas to the total number of moles in the mixture: $x_i = n_i / n_\text{total}$

Dalton's law

• Dalton's law states that the total pressure of a gas mixture is equal to the sum of the partial pressures of the individual gases
• Mathematical expression: $P_\text{total} = P_1 + P_2 + ... + P_n$
• Example: In a mixture of nitrogen ($P_\text{N2} = 0.8 \text{ atm}$) and oxygen ($P_\text{O2} = 0.2 \text{ atm}$), the total pressure is $P_\text{total} = 0.8 \text{ atm} + 0.2 \text{ atm} = 1.0 \text{ atm}$
• Dalton's law assumes that the gases in the mixture do not react with each other and that they behave ideally

Understand the limitations and assumptions of the ideal gas law and when it breaks down for real gases

Assumptions of the ideal gas law

• The ideal gas law assumes that:
  • Gas particles have negligible volume
  • No intermolecular forces exist between gas particles
  • Gas particles undergo perfectly elastic collisions
• These assumptions are not entirely accurate for real gases, especially at high pressures and low temperatures

Deviations from ideal behavior

• At high pressures, the volume of the gas particles becomes significant compared to the volume of the container, leading to deviations from ideal behavior
• Intermolecular forces, such as van der Waals forces, become more significant at low temperatures and high pressures, causing the gas to deviate from ideal behavior

Compressibility factor

• The compressibility factor ($Z$) is used to quantify the deviation of a real gas from ideal behavior
• Mathematical expression: $Z = PV / nRT$
  • For an ideal gas, $Z = 1$
  • For a real gas, $Z \neq 1$

Van der Waals equation

• The van der Waals equation is a modification of the ideal gas law that accounts for the volume of gas particles and intermolecular forces
• It is more accurate for describing the behavior of real gases, particularly at high pressures and low temperatures
• Mathematical expression: $(P + an^2/V^2)(V - nb) = nRT$, where $a$ and $b$ are constants specific to each gas

Critical point

• The critical temperature and critical pressure are the conditions above which a gas cannot be liquefied by increasing pressure alone
• Near the critical point, the behavior of gases deviates significantly from the ideal gas law
• Example: Carbon dioxide has a critical temperature of 304.13 K and a critical pressure of 73.8 atm