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# Mixtures of Gases

Mixtures of Gases

Contents

Recall the concept of Real Gases 1

Recall the following Laws: i. Dalton’s Law of Partial Pressure ii. Amagat’s Law iii. Boyle’s Law iv. Charles Law 2

Recall the following Thermodynamic Relations: i. Gibbs function ii. Helmholtz function iii. Van Derwaals equation iv. Maxwell equation v. Joule Thomson coefficient vi. Clausius Clapeyron equation 3

# Recall the concept of Real Gases

Real gases are gases that deviate from ideal gas behavior due to the effects of intermolecular forces and finite molecular size. Ideal gases are theoretical gases that are assumed to follow the ideal gas law, which states that the pressure, volume, and temperature of a gas are directly proportional to one another.

In contrast, real gases have intermolecular forces that affect the behavior of the gas. These intermolecular forces cause real gases to have properties that are different from those of ideal gases, such as a pressure-volume curve that deviates from the ideal gas law, and a lower critical temperature and pressure.

One of the most important deviations from ideal gas behavior is the effect of intermolecular forces on the compressibility of a gas. Real gases have a lower compressibility than ideal gases, meaning that they are less able to be compressed than an ideal gas of the same temperature and pressure. This is due to the fact that the intermolecular forces in a real gas resist compression, causing the gas to occupy a larger volume than an ideal gas of the same temperature and pressure.

Another important deviation from ideal gas behavior is the effect of intermolecular forces on the energy of a gas. Real gases have a higher energy than ideal gases due to the intermolecular forces between the gas molecules. This results in a higher internal energy for a real gas, which affects its behavior and properties, such as its temperature and pressure.

In conclusion, real gases are gases that deviate from ideal gas behavior due to the effects of intermolecular forces and finite molecular size. Real gases have intermolecular forces that cause them to have properties that are different from those of ideal gases, such as a pressure-volume curve that deviates from the ideal gas law and a lower critical temperature and pressure. The study of real gases is important for understanding the behavior of gases in real-world conditions, and for predicting and understanding the behavior of gases in various applications.

# Recall the following Laws: i. Dalton’s Law of Partial Pressure ii. Amagat’s Law iii. Boyle’s Law iv. Charles Law

i. Dalton’s Law of Partial Pressure: This law states that the total pressure exerted by a mixture of gases is equal to the sum of the pressures that each gas would exert if it were present alone in the same volume at the same temperature. This law is expressed mathematically as:

Ptotal = P1 + P2 + … + Pn

Where Ptotal is the total pressure of the gas mixture, and P1, P2, …, Pn are the partial pressures of each individual gas in the mixture.

ii. Amagat’s Law: This law states that the volume of a mixture of gases is equal to the sum of the volumes that each gas would occupy if it were present alone at the same temperature and pressure. This law is expressed mathematically as:

Vtotal = V1 + V2 + … + Vn

Where Vtotal is the total volume of the gas mixture, and V1, V2, …, Vn are the volumes occupied by each individual gas in the mixture.

iii. Boyle’s Law: This law states that the pressure and volume of a gas are inversely proportional to each other, provided the temperature and number of moles of the gas remain constant. This law is expressed mathematically as:

P * V = k

Where P is the pressure of the gas, V is the volume of the gas, and k is a constant.

iv. Charles’ Law: This law states that the volume of a gas is directly proportional to its absolute temperature, provided the pressure and number of moles of the gas remain constant. This law is expressed mathematically as:

V / T = k

Where V is the volume of the gas, T is the absolute temperature of the gas in kelvin, and k is a constant.

In conclusion, these laws provide fundamental relationships between the pressure, volume, temperature, and number of moles of gases. They are important tools for understanding and predicting the behavior of gases in different conditions and for understanding the behavior of gases in various applications.

# Recall the following Thermodynamic Relations: i. Gibbs function ii. Helmholtz function iii. Van Derwaals equation iv. Maxwell equation v. Joule Thomson coefficient vi. Clausius Clapeyron equation

i. Gibbs Function (also known as Gibbs Free Energy): This function is a measure of the maximum amount of work that can be performed by a system at constant temperature and pressure. The Gibbs function is defined as:

G = H – T x S

Where G is the Gibbs function, H is the enthalpy of the system, T is the temperature, and S is the entropy of the system. The Gibbs function is important because it allows us to predict the direction of a thermodynamic process, whether it is spontaneous or not, based on the change in Gibbs function. If the change in Gibbs function is negative, the process is spontaneous and will occur naturally. If the change in Gibbs function is positive, the process is not spontaneous and requires external energy input.

ii. Helmholtz Function (also known as Helmholtz Free Energy): This function is a measure of the maximum amount of non-expansion work that can be performed by a system at constant temperature and volume. The Helmholtz function is defined as:

A = U – T x S

Where A is the Helmholtz function, U is the internal energy of the system, T is the temperature, and S is the entropy of the system. The Helmholtz function is useful for predicting the stability of a thermodynamic system and for understanding the behavior of a system under conditions of constant temperature and volume.

iii. Van der Waals Equation: This equation is an equation of state for real gases that takes into account the effects of intermolecular forces and finite molecular size on the behavior of a gas. The Van der Waals equation is expressed mathematically as:

(P + a / V2) * (V – b) = RT

Where P is the pressure of the gas, V is the volume of the gas, T is the temperature, R is the gas constant, a is the Van der Waals constant related to the intermolecular forces, and b is the Van der Waals constant related to the finite molecular size. The Van der Waals equation provides a more accurate prediction of the behavior of real gases compared to the ideal gas law.

In conclusion, these thermodynamic relations provide important tools for understanding the behavior of thermodynamic systems, predicting the direction of thermodynamic processes, and understanding the behavior of gases under different conditions. They are fundamental concepts in thermodynamics and are essential for understanding and solving real-world problems in thermodynamics.

iv. Maxwell Equations: These equations are a set of four differential equations that describe the behavior of electric and magnetic fields and their interactions with each other and with matter. The Maxwell equations are expressed mathematically as:

1. Gauss’s Law: ∇ ∙ E = ρ/ε₀
2. Gauss’s Law for Magnetic Fields: ∇ ∙ B = 0
3. Faraday’s Law: ∇ × E = – ∂B/∂t
4. Ampere’s Law: ∇ × B = μ₀J + μ₀ε₀ ∂E/∂t

Where E is the electric field, B is the magnetic field, J is the electric current density, ρ is the electric charge density, t is time, ε₀ is the vacuum permittivity, μ₀ is the vacuum permeability, and ∇ is the del operator. The Maxwell equations provide a complete description of the behavior of electromagnetic fields and their interactions with matter and are fundamental concepts in electromagnetism.

v. Joule Thomson Coefficient: This coefficient is a measure of the change in temperature that occurs when a gas is expanded or compressed isothermally, meaning at constant temperature. The Joule Thomson coefficient is defined as:

ΔT / ΔP = μ / Cp

Where ΔT is the change in temperature, ΔP is the change in pressure, μ is the coefficient of thermal expansion, and Cp is the specific heat capacity at constant pressure. The Joule Thomson coefficient is important because it determines the behavior of gases during isothermal expansion or compression and is used in many practical applications, such as in refrigeration and cooling systems.

vi. Clausius-Clapeyron Equation: This equation is an equation that relates the vapour pressure of a substance to its temperature. The Clausius-Clapeyron equation is expressed mathematically as:

dP / dT = L / T2 * dHvap / dP

Where P is the vapour pressure, T is the temperature, L is the heat of evaporation, and dHvap / dP is the slope of the evaporation line on a phase diagram. The Clausius-Clapeyron equation provides a way to determine the relationship between the vapour pressure of a substance and its temperature and is important for understanding the behavior of liquids and vapours.

In conclusion, these thermodynamic relations provide important tools for understanding the behavior of thermodynamic systems and the relationships between different thermodynamic properties. They are fundamental concepts in thermodynamics and are essential for understanding and solving real-world problems in thermodynamics.