Gaseous Mixtures: Dalton's and Amagat's Laws, Applications & Calculations

Module 2: Gases & Vapors

Mixture of Gases

Gaseous mixtures are common in chemical engineering and general chemistry. Understanding how individual gases behave within a mixture is essential for predicting properties such as pressure, volume, and composition.

• Mixture of Gases: When two or more gases are combined, each gas behaves independently as if it alone occupied the entire volume at the same temperature.

• Example: Mixing steam (120 psi) and air (30 psi) results in a mixture with a total pressure (e.g., 150 psi), illustrating the additive nature of partial pressures.

Dalton's Law of Partial Pressure

Definition and Principle

Dalton's Law of Partial Pressure states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of the individual gases.

• Partial Pressure: The pressure exerted by a single component gas if it were alone in the container at the same volume and temperature as the mixture.

• Key Points:

  • Each gas in a mixture exerts its own pressure independently.

  • The total pressure is the sum of these individual pressures.

Mathematical Formulation

• General Equation:

$P = P_A + P_B + P_C + \cdots + P_N$

• Derivation:

$n_T = n_A + n_B + n_C + \cdots + n_N$ $\frac{PV}{RT} = \frac{P_A V_A}{RT_A} + \frac{P_B V_B}{RT_B} + \frac{P_C V_C}{RT_C} + \cdots + \frac{P_N V_N}{RT_N}$ $V = V_A = V_B = V_C = \cdots = V_N$ $T = T_A = T_B = T_C = \cdots = T_N$ $P = P_A + P_B + P_C + \cdots + P_N$

• Partial Pressure in Terms of Mole Fraction:

$P_A = y_A P$ $y_A = \frac{n_A}{n}$ $P = y_A P + y_B P + y_C P + \cdots + y_n P$ $1 = y_A + y_B + y_C + \cdots + y_n$

• Example: Calculating the partial pressure of a component in a natural gas mixture using its mole fraction and total pressure.

Amagat's Law of Pure Component Volume

Definition and Principle

Amagat's Law states that the total volume occupied by a gaseous mixture is equal to the sum of the volumes that each component would occupy if it were alone at the mixture's temperature and pressure.

• Pure Component Volume: The volume a single gas would occupy alone at the same temperature and pressure as the mixture.

• Key Points:

  • Each gas's volume is calculated as if it were the only gas present.

  • The total volume is the sum of these individual volumes.

Mathematical Formulation

• General Equation:

$V = V_A + V_B + V_C + \cdots + V_N$

• Derivation:

$n_T = n_A + n_B + n_C + \cdots + n_N$ $\frac{PV}{RT} = \frac{P_A V_A}{RT_A} + \frac{P_B V_B}{RT_B} + \frac{P_C V_C}{RT_C} + \cdots + \frac{P_N V_N}{RT_N}$ $P = P_A = P_B = P_C = \cdots = P_N$ $T = T_A = T_B = T_C = \cdots = T_N$ $V = V_A + V_B + V_C + \cdots + V_N$

• Pure Component Volume in Terms of Mole Fraction:

$V_A = y_A V$ $y_A = \frac{n_A}{n}$ $V = y_A V + y_B V + y_C V + \cdots + y_n V$ $1 = y_A + y_B + y_C + \cdots + y_n$

• Example: Calculating the pure component volume of ethane in a natural gas mixture using its mole fraction and total volume.

Sample Problems

Sample Problem 5

A natural gas mixture contains 94.1% CH4, 3% C2H6, and 2.9% N2 by volume. The gas is at 27°C and 345 kPa. Assume ideal gas behavior.

• A. Partial pressure of C2H6: Use $P_{C_2H_6} = y_{C_2H_6} P_{total}$

• B. Pure component volume of C2H6 per 100 m3: Use $V_{C_2H_6} = y_{C_2H_6} V_{total}$

• C. Density of the mixture: Use $\rho = \frac{m}{V}$, where $m$ is the total mass calculated from the molar masses and mole fractions.

Sample Problem 6

Producer gas (from coal, steam, and air) has a given composition and flow conditions. Calculate:

• A. Volumetric flow rate

• B. Complete analysis of the mixture

• C. Partial pressure of each component (including water)

• D. Mass flow rate

Apply Dalton's and Amagat's laws, ideal gas law, and mole fraction relationships to solve.

Summary Table: Dalton's vs. Amagat's Law

Law	Property Summed	Equation	Key Assumption
Dalton's Law	Pressure	$P = P_A + P_B + \cdots + P_N$	Each gas exerts pressure independently
Amagat's Law	Volume	$V = V_A + V_B + \cdots + V_N$	Each gas occupies volume independently

Additional info: These laws are foundational for understanding gas mixtures in both general chemistry and chemical engineering. They are widely used in laboratory calculations, industrial processes, and environmental science.