Thermodynamic potentials are key to understanding how energy flows in molecular systems. They help us predict how systems change under different conditions, like temperature and pressure changes. These potentials are crucial for figuring out when a system is in equilibrium.

Knowing how to use thermodynamic potentials lets us solve real-world problems in chemistry and physics. We can predict chemical reactions, phase changes, and even how materials behave under extreme conditions. It's like having a roadmap for molecular behavior.

Thermodynamic potentials and natural variables

Relationships between thermodynamic potentials

• The four main thermodynamic potentials are internal energy (U), enthalpy (H), Helmholtz free energy (F), and Gibbs free energy (G)
  • Each potential is a function of specific natural variables
• The natural variables for the thermodynamic potentials
  • Internal energy (U) - entropy (S) and volume (V)
  • Enthalpy (H) - entropy (S) and pressure (P)
  • Helmholtz free energy (F) - temperature (T) and volume (V)
  • Gibbs free energy (G) - temperature (T) and pressure (P)
• Relationships between thermodynamic potentials can be derived using Legendre transformations
  • Example: Gibbs free energy is related to the enthalpy and entropy by the equation $G = H - TS$

Expressing thermodynamic potentials using natural variables

• The total differential of each thermodynamic potential can be expressed in terms of its natural variables
  • Example: $dG = -SdT + VdP$, where S and V are the partial derivatives of G with respect to T and P, respectively
• The partial derivatives of the thermodynamic potentials with respect to their natural variables have physical interpretations
  • Example: $(\partial G/\partial P)_T = V$, which means that the partial derivative of Gibbs free energy with respect to pressure at constant temperature is equal to the volume
  • Other physical interpretations include $(\partial U/\partial S)_V = T$ and $(\partial H/\partial S)_P = T$

Conditions for thermodynamic equilibrium

Equilibrium criteria for different systems

• Thermodynamic equilibrium is achieved when a system reaches a state of maximum entropy or minimum Gibbs free energy
  • At equilibrium, the system's macroscopic properties remain constant over time
• For a closed system at constant temperature and pressure, the condition for thermodynamic equilibrium is that the Gibbs free energy of the system is at a minimum ($dG = 0$)
• In a system with multiple phases, thermodynamic equilibrium is achieved when the chemical potentials of each component are equal across all phases
  • This condition is known as the equality of chemical potentials

Chemical reaction equilibrium

• For chemical reactions, the condition for equilibrium is that the Gibbs free energy of reaction ($\Delta G_r$) is zero
  • This occurs when the chemical potentials of the reactants and products are balanced
• The equilibrium constant (K) of a chemical reaction is related to the standard Gibbs free energy of reaction ($\Delta G^\circ_r$) by the equation $\Delta G^\circ_r = -RT \ln K$
  • R is the gas constant and T is the absolute temperature
• Le Chatelier's principle states that when a system at equilibrium is subjected to a disturbance, the system will shift its equilibrium position to counteract the disturbance and establish a new equilibrium state
  • Examples of disturbances include changes in concentration, pressure, or temperature

Calculating changes in thermodynamic potentials

Maxwell relations

• Maxwell relations are a set of equations that relate the partial derivatives of thermodynamic potentials with respect to their natural variables
  • These relations are derived from the equality of mixed second partial derivatives of the thermodynamic potentials
• The four Maxwell relations are:
  • $(\partial S/\partial V)_T = (\partial P/\partial T)_V$
  • $(\partial S/\partial P)_T = -(\partial V/\partial T)_P$
  • $(\partial T/\partial V)_S = (\partial P/\partial S)_V$
  • $(\partial T/\partial P)_S = -(\partial V/\partial S)_P$

Applying Maxwell relations to molecular processes

• Maxwell relations can be used to calculate changes in thermodynamic potentials for molecular processes when direct measurements of the desired quantities are not possible or convenient
• Example: To calculate the change in entropy ($\Delta S$) for an isothermal process, one can use the Maxwell relation $(\partial S/\partial P)_T = -(\partial V/\partial T)_P$ and integrate: $\Delta S = -\int (\partial V/\partial T)_P dP$
• Example: To calculate the change in enthalpy ($\Delta H$) for an isobaric process, one can use the Maxwell relation $(\partial S/\partial P)_T = -(\partial V/\partial T)_P$ and the definition of enthalpy ($H = U + PV$) to obtain: $\Delta H = \Delta U + P\Delta V = \Delta U + P\Delta V + \int V dP = Q_P$, where $Q_P$ is the heat absorbed at constant pressure
• Maxwell relations can also be used to derive other thermodynamic relationships, such as the Clapeyron equation, which relates the slope of a phase boundary in a P-T diagram to the entropy and volume changes during a phase transition