Glossary

12.1 Maxwell relations

4 min read•july 30, 2024

Maxwell relations are powerful tools in thermodynamics, connecting different properties in unexpected ways. They stem from the fundamental equations and show how changing one property affects others. This interconnectedness helps predict hard-to-measure changes and understand complex systems.

These relations are super useful for solving real-world problems. By picking the right Maxwell relation, you can figure out tricky property changes during processes like heating or compression. They're key for designing efficient engines, refrigerators, and other thermodynamic systems.

Maxwell Relations

Derivation from Fundamental Thermodynamic Equations

Top images from around the web for Derivation from Fundamental Thermodynamic Equations

  • Maxwell equations (thermodynamics) - Knowino View original

    Is this image relevant?

  • Maxwell equations (thermodynamics) - Knowino View original

    Is this image relevant?

  • Maxwell equations (thermodynamics) - Knowino View original

    Is this image relevant?

1 of 3

Top images from around the web for Derivation from Fundamental Thermodynamic Equations

  • Maxwell equations (thermodynamics) - Knowino View original

    Is this image relevant?

  • Maxwell equations (thermodynamics) - Knowino View original

    Is this image relevant?

  • Maxwell equations (thermodynamics) - Knowino View original

    Is this image relevant?

1 of 3
  • The four fundamental thermodynamic equations are based on the first and second laws of thermodynamics
    • Relate the thermodynamic potentials (internal energy U, H, Helmholtz free energy F, and Gibbs free energy G) to their natural variables ( S, T, P, and V)
  • Maxwell relations are obtained by taking the of the fundamental thermodynamic equations with respect to different natural variables
    • Equate the mixed second partial derivatives
  • The four Maxwell relations are:
    • (∂T/∂V)S=−(∂P/∂S)V(\partial T/\partial V)_S = -(\partial P/\partial S)_V
    • (∂T/∂P)S=(∂V/∂S)P(\partial T/\partial P)_S = (\partial V/\partial S)_P
    • (∂S/∂V)T=(∂P/∂T)V(\partial S/\partial V)_T = (\partial P/\partial T)_V
    • (∂S/∂P)T=−(∂V/∂T)P(\partial S/\partial P)_T = -(\partial V/\partial T)_P
  • Each Maxwell relation connects four different thermodynamic properties
    • Enables the calculation of one property in terms of the others

Physical Interpretation

  • Maxwell relations demonstrate the interconnectedness of thermodynamic properties
    • Show how a change in one property affects the others
  • The slope of a thermodynamic property with respect to one variable, while holding another variable constant, is equal to the slope of a different thermodynamic property with respect to the second variable, while holding the first variable constant
  • Essential for understanding the behavior of thermodynamic systems
    • Predict changes in properties that are difficult to measure directly
  • Particularly useful in the study of phase transitions
    • Determine properties such as heat capacities, compressibilities, and thermal expansion coefficients

Applications of Maxwell Relations

Determining Changes in Thermodynamic Properties

  • Maxwell relations can be used to determine the change in a thermodynamic property that is difficult to measure directly
    • Express it in terms of properties that are easier to measure
  • To apply a Maxwell relation, integrate both sides of the equation with respect to the appropriate variable while holding the other variable constant
  • The choice of the integration variable depends on the property to be determined and the available experimental data
  • When integrating Maxwell relations, consider the path of integration and the initial and final states of the system

Examples of Applications

  • Determining the change in entropy during an isothermal process
    • Use the Maxwell relation (∂S/∂V)T=(∂P/∂T)V(\partial S/\partial V)_T = (\partial P/\partial T)_V and integrate with respect to volume
  • Calculating the change in enthalpy during an isobaric process
    • Use the Maxwell relation (∂T/∂P)S=(∂V/∂S)P(\partial T/\partial P)_S = (\partial V/\partial S)_P and integrate with respect to pressure
  • Estimating the change in Gibbs free energy during an isothermal-isobaric process
    • Combine the Maxwell relations (∂S/∂P)T=−(∂V/∂T)P(\partial S/\partial P)_T = -(\partial V/\partial T)_P and (∂S/∂V)T=(∂P/∂T)V(\partial S/\partial V)_T = (\partial P/\partial T)_V to express the change in Gibbs free energy in terms of measurable properties

Choosing Maxwell Relations

Identifying the Appropriate Maxwell Relation

  • The selection of the appropriate Maxwell relation depends on the thermodynamic properties involved in the problem and the available experimental data
  • If the problem involves changes in temperature, volume, pressure, and entropy, use the Maxwell relation (∂T/∂V)S=−(∂P/∂S)V(\partial T/\partial V)_S = -(\partial P/\partial S)_V
  • If the problem involves changes in temperature, pressure, volume, and entropy, use the Maxwell relation (∂T/∂P)S=(∂V/∂S)P(\partial T/\partial P)_S = (\partial V/\partial S)_P
  • If the problem involves changes in entropy, volume, pressure, and temperature, use the Maxwell relation (∂S/∂V)T=(∂P/∂T)V(\partial S/\partial V)_T = (\partial P/\partial T)_V
  • If the problem involves changes in entropy, pressure, volume, and temperature, use the Maxwell relation (∂S/∂P)T=−(∂V/∂T)P(\partial S/\partial P)_T = -(\partial V/\partial T)_P

Strategy for Selecting Maxwell Relations

  • Identify the thermodynamic properties given in the problem and the property to be determined
  • Choose the Maxwell relation that connects the given properties and the property of interest
  • Ensure that the chosen Maxwell relation is consistent with the available experimental data and the constraints of the problem (isothermal, isobaric, isochoric, or adiabatic processes)
  • If necessary, combine multiple Maxwell relations to express the desired property in terms of measurable quantities

Physical Significance of Maxwell Relations

Interconnectedness of Thermodynamic Properties

  • Maxwell relations demonstrate the interconnectedness of thermodynamic properties
    • Show how a change in one property affects the others
  • They reveal the symmetry and consistency of thermodynamic relationships
    • Provide a framework for understanding the behavior of thermodynamic systems

Prediction of Thermodynamic Behavior

  • Maxwell relations enable the prediction of changes in properties that are difficult to measure directly
    • Express these properties in terms of easily measurable quantities
  • They are particularly useful in the study of phase transitions
    • Determine properties such as heat capacities, compressibilities, and thermal expansion coefficients
  • Maxwell relations help in the design and optimization of thermodynamic processes and systems
    • Provide insights into the efficiency and feasibility of various processes (heat engines, refrigerators, and chemical reactors)

Experimental Verification

  • Maxwell relations can be experimentally verified by measuring the appropriate thermodynamic properties under controlled conditions
    • Measure the change in volume with respect to temperature at constant pressure to verify (∂V/∂T)P=−(∂S/∂P)T(\partial V/\partial T)_P = -(\partial S/\partial P)_T
    • Measure the change in entropy with respect to volume at constant temperature to verify (∂S/∂V)T=(∂P/∂T)V(\partial S/\partial V)_T = (\partial P/\partial T)_V
  • The agreement between the experimental results and the predictions based on Maxwell relations confirms the validity and universality of thermodynamic principles

Key Terms to Review (20)

Calculating thermodynamic properties: Calculating thermodynamic properties involves determining measurable characteristics of a system, such as pressure, volume, temperature, and internal energy. These calculations are essential for understanding how systems behave under different conditions and are interconnected through various relationships in thermodynamics, such as Maxwell relations, which link different thermodynamic potentials and variables.
Critical Point: The critical point is the temperature and pressure at which the phase of a substance becomes indistinguishable between liquid and gas, resulting in a unique state known as the supercritical fluid. At this point, distinct liquid and gas phases cease to exist, leading to significant changes in properties and behavior. Understanding the critical point is crucial for analyzing phase changes, interpreting phase diagrams, utilizing property tables, and applying Maxwell relations in thermodynamics.
Cross partial derivatives: Cross partial derivatives are the second-order derivatives of a multivariable function, taken with respect to different variables. This concept is essential in thermodynamics, as it helps analyze how different state functions change with respect to one another, particularly in the context of Maxwell relations, which connect various thermodynamic properties through these derivatives.
Enthalpy: Enthalpy is a thermodynamic property defined as the sum of a system's internal energy and the product of its pressure and volume, represented by the equation $$H = U + PV$$. This concept is crucial for understanding energy transfer in processes involving heat and work, especially in closed systems, where enthalpy changes can indicate how much energy is absorbed or released during physical and chemical transformations.
Entropy: Entropy is a measure of the disorder or randomness in a system, reflecting the degree of energy dispersal at a specific temperature. It connects to fundamental concepts like the direction of processes, equilibrium states, and the efficiency of energy transformations in various thermodynamic cycles.
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.
First Law of Thermodynamics: The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed from one form to another, which means the total energy of an isolated system remains constant. This principle underlies various processes, cycles, and energy interactions that involve heat, work, and mass transfer in different systems.
James Clerk Maxwell: James Clerk Maxwell was a Scottish physicist best known for his formulation of the classical theory of electromagnetic radiation, which unified electricity, magnetism, and light as manifestations of the same phenomenon. His work laid the groundwork for modern physics, influencing thermodynamics and statistical mechanics, particularly through the Maxwell relations.
Maxwell's relations: Maxwell's relations are a set of equations in thermodynamics that relate different thermodynamic quantities to one another, derived from the equality of mixed partial derivatives. These equations are powerful tools because they allow us to derive properties of systems without needing direct measurements. They connect the changes in entropy, volume, pressure, and temperature, enabling the calculation of one property based on others.
Molar heat capacity: Molar heat capacity is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). It is a key concept in understanding how different materials respond to heat, which varies between solids, liquids, and gases, and plays a significant role in thermodynamic calculations.
Partial Derivatives: Partial derivatives are a fundamental concept in multivariable calculus, representing the rate of change of a function with respect to one variable while keeping the other variables constant. This concept is crucial for understanding how thermodynamic properties depend on different variables, allowing us to analyze systems with multiple interacting components.
Path Function: A path function is a property that depends on the specific way in which a system transitions from one state to another, rather than just the initial and final states. This means that the value of a path function varies based on the process taken, making it different from state functions, which are determined solely by the state of the system regardless of the path taken. Understanding path functions is crucial when analyzing systems, energy interactions, work done, and thermodynamic relations in various scenarios.
Phase Transition: A phase transition is the process where a substance changes from one state of matter to another, such as from solid to liquid or liquid to gas. This transformation occurs at specific temperatures and pressures, which are critical for understanding the conditions under which materials exist in various states. Recognizing phase transitions helps to describe how substances behave under different environmental conditions and is essential for predicting their physical properties during these changes.
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.
Relating different state variables: Relating different state variables refers to the mathematical relationships that connect various thermodynamic properties of a system, such as temperature, pressure, volume, and internal energy. These relationships are crucial for understanding how changes in one state variable can influence others, enabling predictions about system behavior during processes. Key equations and principles, such as the Maxwell relations, arise from these interconnections and help us analyze thermodynamic systems more effectively.
Rudolf Clausius: Rudolf Clausius was a German physicist and mathematician known for formulating the second law of thermodynamics and introducing the concept of entropy. His work laid the groundwork for understanding energy transformations and the irreversible nature of real processes, influencing key concepts such as entropy changes in pure substances, isentropic processes, and the relationships between temperature and entropy.
Second Law of Thermodynamics: The Second Law of Thermodynamics states that the total entropy of an isolated system can never decrease over time, and it tends to increase, leading to the concept that energy transformations are not 100% efficient. This law establishes the directionality of processes, implying that certain processes are irreversible and energy has a quality that degrades over time, connecting tightly to concepts of heat transfer, work, and system analysis.
State Function: A state function is a property of a system that depends only on the current state of the system and not on the path taken to reach that state. This concept is crucial in understanding how different thermodynamic properties interact, allowing us to assess energy changes and equilibrium states without needing to track the entire process history.
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.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide