12.1 Maxwell relations

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?

• 

  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?

• 

  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, enthalpy H, Helmholtz free energy F, and Gibbs free energy G) to their natural variables (entropy S, temperature T, pressure P, and volume V)
• Maxwell relations are obtained by taking the partial derivatives of the fundamental thermodynamic equations with respect to different natural variables
  • Equate the mixed second partial derivatives
• The four Maxwell relations are:
  • $(\partial T/\partial V)_S = -(\partial P/\partial S)_V$
  • $(\partial T/\partial P)_S = (\partial V/\partial S)_P$
  • $(\partial S/\partial V)_T = (\partial P/\partial T)_V$
  • $(\partial S/\partial P)_T = -(\partial V/\partial T)_P$
• 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 $(\partial S/\partial V)_T = (\partial P/\partial T)_V$ and integrate with respect to volume
• Calculating the change in enthalpy during an isobaric process
  • Use the Maxwell relation $(\partial T/\partial P)_S = (\partial V/\partial S)_P$ and integrate with respect to pressure
• Estimating the change in Gibbs free energy during an isothermal-isobaric process
  • Combine the Maxwell relations $(\partial S/\partial P)_T = -(\partial V/\partial T)_P$ and $(\partial S/\partial V)_T = (\partial P/\partial T)_V$ 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 $(\partial T/\partial V)_S = -(\partial P/\partial S)_V$
• If the problem involves changes in temperature, pressure, volume, and entropy, use the Maxwell relation $(\partial T/\partial P)_S = (\partial V/\partial S)_P$
• If the problem involves changes in entropy, volume, pressure, and temperature, use the Maxwell relation $(\partial S/\partial V)_T = (\partial P/\partial T)_V$
• If the problem involves changes in entropy, pressure, volume, and temperature, use the Maxwell relation $(\partial S/\partial P)_T = -(\partial V/\partial T)_P$

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 $(\partial V/\partial T)_P = -(\partial S/\partial P)_T$
  • Measure the change in entropy with respect to volume at constant temperature to verify $(\partial S/\partial V)_T = (\partial P/\partial T)_V$
• The agreement between the experimental results and the predictions based on Maxwell relations confirms the validity and universality of thermodynamic principles