2.3 Real gases and equations of state

Real gases don't always play by the rules of the ideal gas law. They've got their own quirks, like taking up space and having forces between molecules. This can mess with how they behave, especially when things get intense.

To deal with this, we've got special equations that account for these real-world factors. The van der Waals equation is a popular one. It helps us predict how real gases will act in different situations, which is super useful in many fields.

Limitations of Ideal Gas Law

Assumptions and Deviations

Top images from around the web for Assumptions and Deviations

• 

  Non-Ideal Gas Behavior | Chemistry: Atoms First View original

  Is this image relevant?

• 

  Deviation of Gas from Ideal Behavior | Boundless Chemistry View original

  Is this image relevant?

• 

  real gases View original

  Is this image relevant?

• 

  Non-Ideal Gas Behavior | Chemistry: Atoms First View original

  Is this image relevant?

• 

  Deviation of Gas from Ideal Behavior | Boundless Chemistry View original

  Is this image relevant?

1 of 3

Top images from around the web for Assumptions and Deviations

• 

  Non-Ideal Gas Behavior | Chemistry: Atoms First View original

  Is this image relevant?

• 

  Deviation of Gas from Ideal Behavior | Boundless Chemistry View original

  Is this image relevant?

• 

  real gases View original

  Is this image relevant?

• 

  Non-Ideal Gas Behavior | Chemistry: Atoms First View original

  Is this image relevant?

• 

  Deviation of Gas from Ideal Behavior | Boundless Chemistry View original

  Is this image relevant?

1 of 3

• The ideal gas law assumes that gas particles have negligible volume and no intermolecular forces, which is not true for real gases, especially at high pressures and low temperatures
• Real gases deviate from ideal gas behavior due to the presence of intermolecular forces (attractive and repulsive) and the finite volume occupied by the gas molecules
• The ideal gas law fails to accurately predict the behavior of real gases under conditions where intermolecular forces and molecular size become significant, such as near the critical point or at high pressures

Need for Accurate Equations of State

• Equations of state for real gases, such as the van der Waals equation, aim to account for the effects of intermolecular forces and molecular size to provide more accurate predictions of gas behavior
• These equations introduce additional parameters that consider the attractive forces between molecules (van der Waals forces) and the finite volume occupied by the molecules
• Accurate equations of state are crucial for understanding and predicting the behavior of real gases in various applications, such as in the design of industrial processes, refrigeration systems, and high-pressure equipment

Van der Waals Equation of State

Modifying the Ideal Gas Law

• The van der Waals equation modifies the ideal gas law by introducing two parameters: "a" accounts for the attractive intermolecular forces, and "b" represents the volume occupied by the gas molecules
• The van der Waals equation is $(P + a/V^2)(V - b) = nRT$, where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is temperature
• The parameter "a" is a measure of the strength of attractive intermolecular forces, and its value depends on the specific gas (larger "a" value indicates stronger attractive forces between molecules)
• The parameter "b" represents the volume occupied by the gas molecules and is a measure of the size of the molecules (specific to the gas and related to the excluded volume due to the finite size of the molecules)

Calculating Pressure, Volume, or Temperature

• To calculate the pressure, volume, or temperature of a real gas using the van der Waals equation, substitute the known values into the equation and solve for the unknown variable
• For example, to find the pressure of a real gas, rearrange the equation to solve for $P$: $P = \frac{nRT}{V-b} - \frac{a}{V^2}$
• Similarly, to find the volume or temperature, rearrange the equation and solve for the desired variable using the known values of the other variables and the van der Waals parameters "a" and "b"

Real Gases vs Ideal Gases

Behavior at Different Conditions

• At low pressures and high temperatures, real gases behave more like ideal gases because the intermolecular forces are relatively weak, and the molecular volume is negligible compared to the total volume
• As pressure increases or temperature decreases, real gases deviate more significantly from ideal gas behavior due to the increasing importance of intermolecular forces and molecular volume
• Real gases have lower compressibility than ideal gases at high pressures because the molecules are already close together, and the repulsive forces resist further compression

Condensation and Liquefaction

• Real gases may exhibit condensation and liquefaction at high pressures and low temperatures, while ideal gases are assumed to remain in the gaseous state under all conditions
• Condensation occurs when the attractive forces between molecules overcome the kinetic energy of the molecules, causing them to form a liquid phase
• Liquefaction is the process of converting a gas into a liquid by increasing pressure and/or decreasing temperature

Volume Deviations

• The volume of a real gas is larger than that predicted by the ideal gas law at high pressures due to the finite volume occupied by the molecules, which is not accounted for in the ideal gas law
• The van der Waals equation introduces the parameter "b" to account for the volume occupied by the molecules, resulting in a more accurate prediction of the volume of real gases at high pressures

Compressibility Factor for Real Gases

Definition and Deviation from Unity

• The compressibility factor (Z) is defined as the ratio of the actual volume of a gas to the volume predicted by the ideal gas law at the same pressure and temperature: $Z = V_{actual} / V_{ideal}$
• For an ideal gas, Z is always equal to 1, indicating that the gas follows the ideal gas law perfectly
• For real gases, Z can deviate from 1, depending on the pressure and temperature conditions (Z > 1 indicates that the gas is less compressible than an ideal gas, while Z < 1 indicates that the gas is more compressible)

Measuring Deviation from Ideal Behavior

• The deviation of Z from unity is a measure of the extent to which a real gas deviates from ideal gas behavior
• A larger deviation suggests a more significant influence of intermolecular forces and molecular size on the gas behavior
• The compressibility factor can be calculated using experimental data for pressure, volume, and temperature, or it can be estimated using equations of state like the van der Waals equation

Compressibility Factor Diagrams

• The compressibility factor is often plotted against pressure at constant temperature (Z-P diagrams) or against temperature at constant pressure (Z-T diagrams) to visualize the behavior of real gases and their deviation from ideality
• Z-P diagrams show how the compressibility factor changes with pressure at a given temperature, revealing the extent of deviation from ideal gas behavior at different pressures
• Z-T diagrams show how the compressibility factor changes with temperature at a given pressure, indicating the influence of temperature on the deviation from ideality

Critical Properties of Real Gases

Definition of Critical Point

• The critical point of a substance is the temperature and pressure at which the liquid and vapor phases become indistinguishable, and the properties of the two phases converge
• At the critical point, the distinctions between liquid and gas disappear, and the substance exists as a single, homogeneous phase
• The critical point is characterized by the critical temperature (Tc), critical pressure (Pc), and critical volume (Vc)

Critical Temperature, Pressure, and Volume

• The critical temperature (Tc) is the highest temperature at which a gas can be liquefied by increasing pressure (above Tc, the gas cannot be liquefied, regardless of the applied pressure)
• The critical pressure (Pc) is the minimum pressure required to liquefy a gas at its critical temperature (at pressures above Pc and temperatures below Tc, the gas will condense into a liquid)
• The critical volume (Vc) is the volume occupied by one mole of a substance at its critical point

Calculating Critical Properties using Equations of State

• The van der Waals equation can be used to calculate the critical properties of a real gas by setting the first and second derivatives of pressure with respect to volume equal to zero at the critical point
• For the van der Waals equation, the critical properties are given by: $T_c = 8a/27Rb$, $P_c = a/27b^2$, and $V_c = 3b$, where "a" and "b" are the van der Waals parameters specific to the gas
• Other equations of state, such as the Redlich-Kwong equation or the Peng-Robinson equation, can also be used to calculate critical properties by applying similar mathematical techniques