Glossary

8.3 Thermodynamic equations of state

3 min read•july 23, 2024

Equations of state are the mathematical backbone of thermodynamics. They link , , and , allowing us to predict system behavior. From the simple ideal gas law to more complex models like van der Waals, these equations are essential tools.

These equations aren't just abstract math. They help us calculate real-world properties like and . By comparing different models, we can choose the best one for a given situation, improving our ability to analyze and predict thermodynamic systems.

Equations of State

Concept of equation of state

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Top images from around the web for Concept of equation of state

  • Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law | General Chemistry View original

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  • Mathematical relationship between state variables (pressure, volume, temperature) describes thermodynamic behavior of a system
  • Allows calculation of thermodynamic properties and prediction of system behavior based on limited set of known variables
  • Different equations of state used depending on specific system and range of conditions (ideal gas equation, , )

Derivation from fundamental equation

  • Fundamental equation in thermodynamics relates (UU) to entropy (SS), volume (VV), and number of particles (NiN_i) of each species dU=TdS−PdV+∑iμidNidU = TdS - PdV + \sum_i \mu_i dN_i TT temperature, PP pressure, μi\mu_i chemical potential of species ii
  • For simple system with single species and constant particle number dU=TdS−PdVdU = TdS - PdV
  • Apply reciprocity relation (∂T/∂V)S=−(∂P/∂S)V(\partial T/\partial V)_S = -(\partial P/\partial S)_V to obtain dU=TdS−TdV(∂P/∂S)VdU = TdS - TdV(\partial P/\partial S)_V
  • Integrate equation to obtain thermodynamic equation of state U=TS−∫TdV(∂P/∂S)VU = TS - \int TdV(\partial P/\partial S)_V

Calculation of thermodynamic properties

  • Thermodynamic equation of state used to calculate various properties (pressure, volume, temperature) when some properties known
  • Using ideal gas equation (PV=nRTPV = nRT) Temperature calculated if pressure and volume known T=PV/nRT = PV/nR Pressure calculated if temperature and volume known P=nRT/VP = nRT/V
  • Other properties (enthalpy, entropy, ) calculated using appropriate equations of state and thermodynamic relations

Analysis of thermodynamic systems

  • Equations of state predict and analyze behavior of thermodynamic systems under different conditions
  • Using van der Waals equation (P+aVm2)(Vm−b)=RT\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT aa and bb constants specific to gas, VmV_m molar volume Accounts for intermolecular attractions and finite volume of gas molecules, more accurate description of compared to ideal gas equation
  • Analyzing system behavior using equations of state determines (gas to liquid) Deviations from ideal behavior

Comparing Equations of State

Ideal gas vs van der Waals equations

  • Ideal gas equation PV=nRTPV = nRT Assumes negligible size of gas molecules and no intermolecular interactions Accurately describes gas behavior at low pressures and high temperatures Fails to account for real gas behavior (condensation, critical phenomena)
  • Van der Waals equation (P+aVm2)(Vm−b)=RT\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT Accounts for intermolecular attractions (a/Vm2a/V_m^2 term) and finite volume of gas molecules (bb term) More accurate description of real gas behavior, particularly at higher pressures and lower temperatures Predicts existence of critical point and formation of liquid phase
  • Other equations (Redlich-Kwong, Peng-Robinson) build upon van der Waals equation Improve accuracy and applicability to wider range of systems Introduce additional parameters and modifications to better represent behavior of specific substances or mixtures

Key Terms to Review (18)

Adiabatic process: An adiabatic process is a thermodynamic process in which no heat is exchanged between the system and its surroundings. This means that any change in the internal energy of the system is entirely due to work done on or by the system, making it a critical concept in understanding various thermodynamic cycles and processes.
Charles's Law: Charles's Law states that the volume of a gas is directly proportional to its temperature when the pressure is held constant. This relationship is fundamental in understanding how gases behave under varying temperature conditions and connects to other important concepts such as state variables, thermodynamic equations, and the behavior of ideal gases and solids.
Compressibility Factors: Compressibility factors are dimensionless numbers that indicate how much a real gas deviates from ideal gas behavior. These factors help in adjusting the ideal gas law for real gases under various conditions, showing how factors like pressure and temperature influence the volume of gases compared to what would be expected if they were ideal.
Critical Points: Critical points are specific conditions at which a phase of a substance transitions into another, particularly at the boundary between liquid and gas phases. At this point, the properties of the liquid and gas become indistinguishable, leading to the concept of a critical temperature and critical pressure that define this unique state. These points are essential in understanding thermodynamic equations of state, as they mark the limits where traditional phase behavior changes.
Enthalpy: Enthalpy is a thermodynamic property that represents the total heat content of a system, defined as the sum of its internal energy and the product of its pressure and volume. This concept is crucial in understanding how energy is exchanged in processes, especially in the context of thermodynamic systems and the transformations they undergo.
Entropy: Entropy is a measure of the degree of disorder or randomness in a system, reflecting the number of microscopic configurations that correspond to a thermodynamic system's macroscopic state. It connects to various principles of thermodynamics, indicating how energy disperses and the direction of spontaneous processes.
Gibbs Free Energy: Gibbs free energy is a thermodynamic potential that measures the maximum reversible work obtainable from a closed system at constant temperature and pressure. It is a crucial concept because it helps predict the direction of chemical reactions and phase transitions, determining whether a process will occur spontaneously based on changes in enthalpy and entropy.
Gibbs Phase Rule: The Gibbs Phase Rule is a principle in thermodynamics that relates the number of phases present in a system to the number of independent variables that can be altered without changing the number of phases. It is expressed mathematically as $$F = C - P + 2$$, where $$F$$ is the degrees of freedom, $$C$$ is the number of components, and $$P$$ is the number of phases. This rule is critical for understanding phase equilibria, particularly when analyzing how changes in temperature and pressure can affect the state of a system.
Internal Energy: Internal energy is the total energy contained within a thermodynamic system, encompassing kinetic and potential energies of all the molecules in the system. This energy is crucial in determining the state of the system and plays a key role in thermodynamic processes, including heat transfer and work done on or by the system.
Isothermal process: An isothermal process is a thermodynamic process in which the temperature of the system remains constant while heat is exchanged with the surroundings. This constant temperature implies that any internal energy changes in the system are fully compensated by heat transfer, making it an essential concept in understanding how systems behave under thermal equilibrium and the laws governing energy conservation.
Maxwell Relations: Maxwell relations are a set of equations in thermodynamics derived from the equality of mixed partial derivatives of state functions, linking different thermodynamic variables. They allow for the conversion between various thermodynamic properties, such as temperature, pressure, volume, and entropy, providing deep insights into the relationships and behaviors of these properties under different conditions.
Phase Transitions: Phase transitions are processes where a substance changes from one state of matter to another, such as solid to liquid or liquid to gas, often due to changes in temperature or pressure. These transitions involve significant changes in the energy, structure, and organization of the particles within the substance, which directly connects to various thermodynamic principles and properties.
Pressure: Pressure is defined as the force exerted per unit area on a surface, which plays a crucial role in various thermodynamic processes and systems. It is a key state variable that influences how substances behave under different conditions, affecting phase changes, equilibria, and the behavior of gases. Understanding pressure is essential for analyzing the relationships between temperature, volume, and the state of matter in both microscopic and macroscopic contexts.
Real gas behavior: Real gas behavior refers to how gases deviate from ideal gas laws due to interactions between gas molecules and the volume occupied by them. Unlike ideal gases, which are hypothetical and follow simple equations, real gases experience effects such as intermolecular forces and finite molecular size that become significant under certain conditions, like high pressure and low temperature. Understanding real gas behavior is crucial for accurately predicting the properties of gases in various thermodynamic processes.
Redlich-Kwong Equation: The Redlich-Kwong equation is a thermodynamic model used to describe the behavior of real gases, formulated to improve upon the ideal gas law by incorporating factors like pressure and temperature. It is particularly useful for predicting the properties of substances at high pressures and moderate temperatures, making it a vital tool in various engineering applications. This equation relates state variables to account for non-ideal gas behavior, enabling better understanding and calculations regarding phase transitions and other thermodynamic processes.
Temperature: Temperature is a measure of the average kinetic energy of the particles in a substance, reflecting how hot or cold that substance is. It plays a crucial role in various physical processes and influences the behavior of materials and systems in both macroscopic and microscopic contexts.
Van der Waals equation: The van der Waals equation is an adjustment of the ideal gas law that accounts for the volume occupied by gas molecules and the attractive forces between them, providing a more accurate description of real gas behavior. This equation is crucial for understanding the state variables of a system and how they relate to thermodynamic properties, as it offers insights into the deviations from ideal behavior observed in real gases under various conditions.
Volume: Volume is a measure of the three-dimensional space occupied by a substance or object, often expressed in units such as liters or cubic meters. It plays a crucial role in understanding how substances interact, especially under varying temperature and pressure conditions, linking it to essential properties like state variables and equations of state.
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