Gas mixtures are all around us, from the air we breathe to the fuel we burn. Understanding their properties is key to many engineering applications. This topic dives into how we can analyze and predict the behavior of these mixtures.

We'll explore Dalton's law, which helps us understand pressure in gas mixtures. We'll also learn how to calculate important properties like density and specific heat capacity. These concepts are crucial for designing and optimizing systems that use gas mixtures.

Dalton's Law of Partial Pressures

Fundamentals of Dalton's Law

• Dalton's law of partial pressures states that the total pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the individual gases in the mixture
• The partial pressure of a gas in a mixture is the pressure that the gas would exert if it occupied the same volume alone at the same temperature
• Dalton's law assumes that the gases in the mixture do not react chemically with each other and that the volume of the individual gas molecules is negligible compared to the total volume of the mixture

Mole Fractions and Partial Pressures

• The mole fraction of a component in an ideal gas mixture is the ratio of the number of moles of that component to the total number of moles in the mixture
  • For example, if a mixture contains 2 moles of nitrogen and 3 moles of oxygen, the mole fraction of nitrogen is 2/5 and the mole fraction of oxygen is 3/5
• The partial pressure of a component in an ideal gas mixture is equal to the product of its mole fraction and the total pressure of the mixture
  • For instance, if the total pressure of the mixture is 1 atm and the mole fraction of nitrogen is 0.4, the partial pressure of nitrogen is 0.4 atm

Ideal Gas Mixture Calculations

Partial Pressure, Total Pressure, and Volume Calculations

• The partial pressure of a component in an ideal gas mixture can be calculated using the equation: $P_i = y_i P_total$, where $P_i$ is the partial pressure of component $i$, $y_i$ is the mole fraction of component $i$, and $P_total$ is the total pressure of the mixture
• The total pressure of an ideal gas mixture is the sum of the partial pressures of all components in the mixture: $P_total = P_1 + P_2 + ... + P_n$, where $P_1, P_2, ..., P_n$ are the partial pressures of components $1, 2, ..., n$
  • For example, if a mixture contains three gases with partial pressures of 0.2 atm, 0.5 atm, and 0.8 atm, the total pressure of the mixture is 1.5 atm
• The volume of a component in an ideal gas mixture can be calculated using the ideal gas equation: $V_i = (n_i * R * T) / P_i$, where $V_i$ is the volume of component $i$, $n_i$ is the number of moles of component $i$, $R$ is the universal gas constant, $T$ is the absolute temperature, and $P_i$ is the partial pressure of component $i$
• The total volume of an ideal gas mixture is equal to the sum of the volumes of the individual components: $V_total = V_1 + V_2 + ... + V_n$, where $V_1, V_2, ..., V_n$ are the volumes of components $1, 2, ..., n$

Molar Mass and Density Calculations

• The molar mass of an ideal gas mixture can be calculated as the weighted average of the molar masses of the individual components, based on their mole fractions: $M_mix = Σ(y_i M_i)$, where $y_i$ is the mole fraction of component $i$ and $M_i$ is the molar mass of component $i$
  • For instance, if a mixture contains 60% nitrogen (molar mass 28 g/mol) and 40% oxygen (molar mass 32 g/mol), the molar mass of the mixture is 0.6 * 28 + 0.4 * 32 = 29.6 g/mol
• The density of an ideal gas mixture can be calculated using the ideal gas equation: $ρ = (P * M_mix) / (R * T)$, where $ρ$ is the density of the mixture, $P$ is the total pressure, $M_mix$ is the molar mass of the mixture, $R$ is the universal gas constant, and $T$ is the absolute temperature

Properties of Ideal Gas Mixtures

Specific Heat Capacities

• The specific heat capacity of an ideal gas mixture is the weighted average of the specific heat capacities of the individual components, based on their mass fractions or mole fractions
• The molar specific heat capacity at constant pressure ($C_p,m$) of an ideal gas mixture can be calculated using the equation: $C_p,m = Σ(y_i C_p,i)$, where $y_i$ is the mole fraction of component $i$ and $C_p,i$ is the molar specific heat capacity at constant pressure of component $i$
• The molar specific heat capacity at constant volume ($C_v,m$) of an ideal gas mixture can be calculated using the equation: $C_v,m = Σ(y_i C_v,i)$, where $y_i$ is the mole fraction of component $i$ and $C_v,i$ is the molar specific heat capacity at constant volume of component $i$
  • For example, if a mixture contains 50% helium (Cp = 20.8 J/mol·K, Cv = 12.5 J/mol·K) and 50% neon (Cp = 20.8 J/mol·K, Cv = 12.5 J/mol·K), the molar specific heat capacities of the mixture are Cp,m = 20.8 J/mol·K and Cv,m = 12.5 J/mol·K

Enthalpy and Internal Energy

• The molar enthalpy of an ideal gas mixture is the sum of the molar enthalpies of the individual components: $H_m = Σ(y_i H_i)$, where $y_i$ is the mole fraction of component $i$ and $H_i$ is the molar enthalpy of component $i$
  • For instance, if a mixture contains 70% methane (molar enthalpy 890 kJ/mol) and 30% ethane (molar enthalpy 1560 kJ/mol), the molar enthalpy of the mixture is 0.7 * 890 + 0.3 * 1560 = 1091 kJ/mol
• The molar internal energy of an ideal gas mixture is the sum of the molar internal energies of the individual components: $U_m = Σ(y_i U_i)$, where $y_i$ is the mole fraction of component $i$ and $U_i$ is the molar internal energy of component $i$

Ideal Gas Mixture Behavior

Ideal Gas Equation of State

• The ideal gas equation of state, $PV = nRT$, can be applied to ideal gas mixtures by considering the total number of moles ($n$) in the mixture and the total pressure ($P$) and volume ($V$) of the mixture
• For an ideal gas mixture, the compressibility factor ($Z$) is equal to 1, indicating that the mixture behaves as an ideal gas under all conditions of temperature and pressure
• The individual gas constant ($R_i$) for each component in an ideal gas mixture is equal to the universal gas constant ($R$), as the ideal gas equation applies to each component independently

Mixing and Separation of Ideal Gases

• When two or more ideal gases are mixed at constant temperature and pressure, the total volume of the mixture is equal to the sum of the volumes of the individual gases before mixing (Amagat's law)
  • For example, if 2 L of nitrogen and 3 L of oxygen are mixed at constant temperature and pressure, the total volume of the mixture is 5 L
• The separation of an ideal gas mixture into its components can be achieved through processes such as fractional distillation or membrane separation, which rely on differences in the physical properties (e.g., boiling points or permeabilities) of the individual components