13.1 Composition of gas mixtures
7 min read•july 30, 2024
Gas mixtures are a crucial concept in thermodynamics. They're all around us, from the air we breathe to industrial processes. Understanding their composition is key to analyzing their behavior and properties.
Mass, mole, and fractions are essential tools for describing gas mixtures. These fractions help us quantify the relative amounts of each component in a mixture, allowing us to calculate important properties like density and .
Mass, Mole, and Volume Fractions in Gas Mixtures
Defining and Differentiating Fractions
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- represents the ratio of the mass of a component to the total mass of the mixture
- Dimensionless quantity
- Sum of mass fractions for all components in a mixture equals 1
- expresses the ratio of the number of moles of a component to the total number of moles in the mixture
- Also a dimensionless quantity
- Sum of mole fractions for all components in a mixture equals 1
- denotes the ratio of the volume occupied by a component to the total volume of the mixture
- Dimensionless quantity
- Sum of volume fractions for all components in a mixture equals 1
- Mass, mole, and volume fractions are interrelated quantities
- Can be converted from one to another using the molecular weights and densities of the components in the mixture
- Conversion allows for flexibility in analyzing and characterizing gas mixtures
Interrelationships and Conversions
- Mass, mole, and volume fractions are interconnected quantities in gas mixtures
- Conversion between fractions requires molecular weights and densities of components
- Molecular weight used to convert between mass and moles
- Density used to convert between mass and volume
- Example: Converting mass fraction to mole fraction
- Divide mass fraction by molecular weight to obtain moles of component
- Sum moles of all components to get total moles in mixture
- Divide moles of component by total moles to calculate mole fraction
- Example: Converting mole fraction to volume fraction
- Use ideal gas law () to calculate volume of each component from moles
- Sum volumes of all components to obtain total volume
- Divide volume of component by total volume to determine volume fraction
Calculating Component Fractions in Gas Mixtures
Calculating Mass Fraction
- Mass fraction calculated by dividing the mass of each component by the total mass of the mixture
- Example: Gas mixture with 3 kg of nitrogen and 2 kg of oxygen
- Total mass of mixture = 3 kg + 2 kg = 5 kg
- Mass fraction of nitrogen = 3 kg / 5 kg = 0.6
- Mass fraction of oxygen = 2 kg / 5 kg = 0.4
- Ensure consistent units of mass for all components and total mixture
Calculating Mole Fraction
- Mole fraction calculated by dividing the number of moles of each component by the total number of moles in the mixture
- Number of moles calculated from mass and molecular weight of each component
- Moles = Mass / Molecular Weight
- Example: Gas mixture with 50 g of CO2 (MW = 44 g/mol) and 25 g of CH4 (MW = 16 g/mol)
- Moles of CO2 = 50 g / 44 g/mol = 1.14 mol
- Moles of CH4 = 25 g / 16 g/mol = 1.56 mol
- Total moles in mixture = 1.14 mol + 1.56 mol = 2.70 mol
- Mole fraction of CO2 = 1.14 mol / 2.70 mol = 0.42
- Mole fraction of CH4 = 1.56 mol / 2.70 mol = 0.58
Calculating Volume Fraction
- Volume fraction calculated by dividing the volume of each component by the total volume of the mixture
- Volume of each component calculated from its mass, molecular weight, and the ideal gas law ()
- Volume = (nRT) / P, where n = moles, R = gas constant, T = , P =
- Example: Gas mixture with 10 g of He (MW = 4 g/mol) and 20 g of Ar (MW = 40 g/mol) at 300 K and 1 atm
- Moles of He = 10 g / 4 g/mol = 2.5 mol
- Moles of Ar = 20 g / 40 g/mol = 0.5 mol
- Volume of He = (2.5 mol × 0.08206 L·atm/(mol·K) × 300 K) / 1 atm = 61.5 L
- Volume of Ar = (0.5 mol × 0.08206 L·atm/(mol·K) × 300 K) / 1 atm = 12.3 L
- Total volume of mixture = 61.5 L + 12.3 L = 73.8 L
- Volume fraction of He = 61.5 L / 73.8 L = 0.83
- Volume fraction of Ar = 12.3 L / 73.8 L = 0.17
Ensuring Consistent Units
- When solving problems involving mass, mole, and volume fractions, ensure consistent units for all components and the total mixture
- Convert masses to a common unit (e.g., grams or kilograms)
- Use appropriate units for moles (mol) and volume (e.g., liters or cubic meters)
- Maintain consistent temperature and pressure units when applying the ideal gas law
- Consistent units prevent errors in calculations and ensure accurate results
Apparent Molecular Weight of Gas Mixtures
Definition and Significance
- Apparent molecular weight of a gas mixture is the weighted average of the molecular weights of its components
- Weighting based on mass or mole fractions of components
- Represents the molecular weight of a hypothetical pure gas that would have the same density as the mixture at the same temperature and pressure
- Used to determine the density and specific volume of the gas mixture using the ideal gas law
Calculating Apparent Molecular Weight using Mass Fractions
- Multiply the mass fraction of each component by its molecular weight
- Sum the products to obtain the apparent molecular weight
- Example: Gas mixture with 60% N2 (MW = 28 g/mol) and 40% O2 (MW = 32 g/mol) by mass
- Apparent MW = 0.60 × 28 g/mol + 0.40 × 32 g/mol
- Apparent MW = 16.8 g/mol + 12.8 g/mol = 29.6 g/mol
Calculating Apparent Molecular Weight using Mole Fractions
- Multiply the mole fraction of each component by its molecular weight
- Sum the products to obtain the apparent molecular weight
- Example: Gas mixture with 75% CH4 (MW = 16 g/mol) and 25% C2H6 (MW = 30 g/mol) by moles
- Apparent MW = 0.75 × 16 g/mol + 0.25 × 30 g/mol
- Apparent MW = 12 g/mol + 7.5 g/mol = 19.5 g/mol
Application in Density and Specific Volume Calculations
- Apparent molecular weight is used to calculate the density and specific volume of gas mixtures
- Density = (Pressure × Apparent MW) / (Gas Constant × Temperature)
- Specific Volume = (Gas Constant × Temperature) / (Pressure × Apparent MW)
- Example: Gas mixture with apparent MW of 29.6 g/mol at 300 K and 1 atm
- Density = (1 atm × 29.6 g/mol) / (0.08206 L·atm/(mol·K) × 300 K) = 1.20 g/L
- Specific Volume = (0.08206 L·atm/(mol·K) × 300 K) / (1 atm × 29.6 g/mol) = 0.83 L/g
Analyzing Gas Mixture Composition with Psychrometrics
Psychrometric Charts and Tables
- Psychrometric charts and tables used to determine properties and composition of moist air (mixture of dry air and water vapor)
- Composition of moist air typically expressed using:
- Specific humidity: mass of water vapor per unit mass of dry air
- Humidity ratio: mass of water vapor per unit mass of dry air (same as specific humidity)
- Psychrometric charts plot humidity ratio against dry-bulb temperature
- Include lines of constant relative humidity, wet-bulb temperature, and specific volume
- Allow visual determination of moist air properties and composition
- Psychrometric tables provide numerical values for moist air properties at various dry-bulb temperatures, wet-bulb temperatures, and pressures
Analyzing Moist Air Composition
- To analyze the composition of moist air using psychrometric charts or tables:
- Locate the point corresponding to the given dry-bulb temperature and humidity ratio (or relative humidity)
- Read the values of other properties such as wet-bulb temperature, specific volume, and enthalpy at this point
- Example: Moist air at 25°C dry-bulb temperature and 50% relative humidity
- On psychrometric chart, locate the point at the intersection of 25°C dry-bulb temperature and 50% relative humidity line
- Read the values of humidity ratio (0.0098 kg water/kg dry air), wet-bulb temperature (18.5°C), and specific volume (0.855 m³/kg dry air) at this point
Applications of Psychrometric Analysis
- Psychrometric analysis is essential for designing and operating air-conditioning and refrigeration systems
- Determines the cooling and dehumidification requirements for maintaining desired indoor air quality and comfort
- Helps in selecting appropriate equipment and processes for air treatment
- Psychrometric data is also used in drying processes, where moist air is used as a drying medium
- Allows optimization of drying conditions (temperature, humidity) for efficient moisture removal
- Helps in estimating the energy requirements and drying time for a given product and process
Key Terms to Review (20)
Apparent molecular weight: Apparent molecular weight is a concept used to describe the average molecular weight of a gas mixture, taking into account the individual contributions of each component in the mixture. It is essential for understanding the behavior of gas mixtures in various thermodynamic processes, as it helps to predict how mixtures will respond to changes in conditions such as temperature and pressure.
Behavior of gases under different conditions: The behavior of gases under different conditions refers to how gas properties, such as pressure, volume, and temperature, change in response to varying external factors. Understanding this behavior is crucial for predicting how gases will react in various situations, such as during compression or heating, and it is fundamentally tied to the principles of gas laws and the composition of gas mixtures.
Collision frequency: Collision frequency refers to the number of collisions that occur per unit time between gas molecules in a given volume. This concept is important in understanding the behavior of gas mixtures, as it helps explain how different gases interact with each other, affecting their overall properties like pressure and temperature. The collision frequency can be influenced by factors such as the size of the molecules, their speed, and the density of the gas mixture.
Dalton's Law of Partial Pressures: Dalton's Law of Partial Pressures states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of each individual gas. This law helps us understand how different gases contribute to the overall pressure in a system, which is crucial when analyzing gas mixtures and atmospheric air composition.
Equilibrium: Equilibrium is the state in which a system experiences no net change over time, meaning that the macroscopic properties of the system remain constant. This concept is crucial as it underpins various thermodynamic principles, allowing for the analysis of systems at rest or in balanced conditions. Understanding equilibrium helps in exploring relationships among variables, such as temperature, pressure, and volume, leading to deeper insights into system behaviors in contexts like thermodynamics and gas mixtures.
Gas Stoichiometry: Gas stoichiometry is the application of stoichiometric principles to calculate the relationships between the quantities of reactants and products in chemical reactions involving gases. It involves using gas laws to understand how volume, temperature, and pressure relate to the amounts of gas molecules involved in reactions, often expressed in moles or liters. This understanding is crucial for predicting how gases behave during reactions, especially when dealing with mixtures of different gases.
Graham's Law of Effusion: Graham's Law of Effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. This means that lighter gases effuse faster than heavier gases, which is crucial for understanding the behavior of gas mixtures and their compositions.
Ideal gas mixture: An ideal gas mixture is a collection of multiple gases that behave independently of one another while occupying the same volume and maintaining uniform temperature and pressure conditions. Each component in the mixture follows the ideal gas law, and the overall behavior of the mixture can be analyzed using simple mathematical relationships, which make it easier to calculate properties like density, pressure, and temperature of the mixture.
Mass fraction: Mass fraction is the ratio of the mass of a particular component in a mixture to the total mass of the mixture. This concept is essential for understanding the composition of gas mixtures, as it allows for the quantification of each component's contribution to the overall properties of the mixture, such as density and specific heat capacity.
Mean Free Path: Mean free path is the average distance a molecule travels between collisions with other molecules in a gas. This concept is crucial for understanding the behavior of gases, especially when looking at how gas mixtures interact and how their composition influences properties like diffusion and viscosity.
Mixing entropy: Mixing entropy is a measure of the increase in disorder or randomness that occurs when two or more different substances are combined. It quantifies how the mixing process enhances the number of possible arrangements of particles, thus reflecting the inherent tendency of systems to move toward states of greater entropy. This concept is particularly significant when discussing gas mixtures and their properties, as it helps to explain the thermodynamic behavior of these mixtures.
Mole fraction: Mole fraction is the ratio of the number of moles of a specific component in a mixture to the total number of moles of all components in that mixture. This dimensionless quantity is important for understanding the composition of gas mixtures and how different gases behave collectively. By expressing concentrations in this way, mole fraction helps in calculations related to partial pressures and other properties of ideal gas mixtures.
P_total = p1 + p2 + ... + pn: The equation p_total = p1 + p2 + ... + pn represents the total pressure of a gas mixture as the sum of the partial pressures of each individual gas present in the mixture. This principle is grounded in Dalton's Law of Partial Pressures, which states that in a mixture of non-reacting gases, each gas exerts its own pressure independently of the others. Understanding this equation is crucial for analyzing the behavior of gas mixtures in various thermodynamic processes.
Partial pressure: Partial pressure is the pressure exerted by a single component of a gas mixture when it occupies the same volume as the entire mixture. This concept is essential for understanding how different gases behave when mixed, as each gas contributes to the total pressure based on its own properties and concentration. The idea of partial pressure helps in analyzing gas mixtures, calculating properties of ideal gases, and applying the first and second laws of thermodynamics to systems containing multiple gases.
Pressure: Pressure is defined as the force exerted per unit area on the surface of an object. It plays a crucial role in understanding the behavior of substances in various states, how systems reach equilibrium, and is a key parameter in equations that describe the relationships between different properties of gases and fluids.
Pv=nrt: The equation $$pv=nrt$$, known as the ideal gas law, relates the pressure (p), volume (v), number of moles (n), the ideal gas constant (R), and temperature (T) of a gas. This fundamental equation is essential for understanding the behavior of gases, particularly in how different gases interact in mixtures and the overall composition of those mixtures. The ideal gas law provides insights into the properties of gases under various conditions, helping to analyze and predict how changes in temperature or pressure can affect gas behavior.
Real gas mixture: A real gas mixture refers to a combination of different gases that do not behave ideally under all conditions, typically due to interactions between gas molecules and variations in pressure and temperature. Unlike ideal gas mixtures, real gas mixtures take into account factors such as non-ideal behavior and the physical properties of each component, which can significantly affect the overall thermodynamic characteristics of the mixture.
Temperature: Temperature is a measure of the average kinetic energy of the particles in a substance, providing an indication of how hot or cold that substance is. It plays a critical role in understanding properties, state changes, and equilibrium conditions of substances, influencing how they interact with one another and their environments.
Volume: Volume is the amount of space occupied by a substance, typically measured in cubic units. It plays a crucial role in understanding the physical properties of matter, the state of a system, and the equilibrium conditions. Knowing the volume helps in analyzing gas behavior, calculating densities, and applying equations of state that describe how substances behave under varying conditions.
Volume Fraction: Volume fraction is a dimensionless quantity that represents the ratio of the volume of a component to the total volume of a mixture. It is important in understanding the composition of gas mixtures, as it provides insight into how different gases contribute to the overall properties and behavior of the mixture. By knowing the volume fractions of each gas in a mixture, one can determine important characteristics such as density, pressure, and mole fractions, which are essential for various calculations in thermodynamics.