12.3 General relations for du, dh, ds, cv, and cp

Thermodynamic property relations are the backbone of understanding energy changes in systems. These equations link internal energy, enthalpy, entropy, and specific heat capacities, showing how they're all connected. They're essential for analyzing real-world processes and designing efficient machines.

The general relations for du, dh, ds, cv, and cp give us a roadmap for calculating energy changes in any thermodynamic process. By mastering these equations, you'll be able to tackle complex problems involving heat transfer, work, and efficiency in various systems.

Internal energy, enthalpy, and entropy changes

General relations for thermodynamic property changes

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• The general relation for change in internal energy (du) is expressed as [du = T ds - P dv](https://www.fiveableKeyTerm:du_=_t_ds_-_p_dv)
  • $T$ represents temperature
  • $s$ represents specific entropy
  • $P$ represents pressure
  • $v$ represents specific volume
• The general relation for change in enthalpy (dh) is expressed as [dh = T ds + v dP](https://www.fiveableKeyTerm:dh_=_t_ds_+_v_dp)
  • $T$ represents temperature
  • $s$ represents specific entropy
  • $v$ represents specific volume
  • $P$ represents pressure
• The general relation for change in entropy (ds) is expressed as [ds = (du + P dv) / T](https://www.fiveableKeyTerm:ds_=_(du_+_p_dv)_/_t)
  • $u$ represents specific internal energy
  • $P$ represents pressure
  • $v$ represents specific volume
  • $T$ represents temperature

Derivation and validity of general relations

• The general relations are derived from the first and second laws of thermodynamics and the definitions of the thermodynamic properties
  • First law of thermodynamics states that energy cannot be created or destroyed, only converted from one form to another
  • Second law of thermodynamics introduces the concept of entropy and states that the total entropy of an isolated system always increases over time
• The general relations are valid for any pure substance in any thermodynamic process, reversible or irreversible
  • Reversible processes are idealized processes where the system is always in equilibrium with its surroundings (frictionless pistons, perfect heat transfer)
  • Irreversible processes are real-world processes where the system is not in equilibrium with its surroundings (friction, heat loss)

Specific heat capacities and thermodynamic properties

Specific heat capacities as partial derivatives

• The specific heat capacity at constant volume (cv) is expressed as $cv = (∂u/∂T)v$
  • $cv$ represents the partial derivative of specific internal energy with respect to temperature at constant specific volume
  • Measures the amount of heat required to raise the temperature of a substance by one degree while keeping the volume constant
• The specific heat capacity at constant pressure (cp) is expressed as $cp = (∂h/∂T)P$
  • $cp$ represents the partial derivative of specific enthalpy with respect to temperature at constant pressure
  • Measures the amount of heat required to raise the temperature of a substance by one degree while keeping the pressure constant

Relationship between specific heat capacities

• The specific heat capacities are related by the equation $cp - cv = R$ for ideal gases
  • $R$ represents the specific gas constant
  • Ideal gases are hypothetical gases that follow the ideal gas law $PV = nRT$ (negligible intermolecular forces, point-like particles)
• The specific heat capacities can also be expressed in terms of second partial derivatives of Gibbs free energy (g) or Helmholtz free energy (f)
  • Gibbs free energy is defined as $g = h - Ts$
  • Helmholtz free energy is defined as $f = u - Ts$

Calculating thermodynamic property changes

Applying general relations to calculate changes

• The general relations can be used to calculate changes in internal energy, enthalpy, and entropy for any process by integrating the appropriate equation
  • Integration is a mathematical operation that finds the area under a curve or the accumulated value of a quantity over a range
• For an ideal gas, the change in specific internal energy (Δu) is calculated as $Δu = cv ΔT$
  • $cv$ represents the specific heat capacity at constant volume
  • $ΔT$ represents the change in temperature
• For an ideal gas, the change in specific enthalpy (Δh) is calculated as $Δh = cp ΔT$
  • $cp$ represents the specific heat capacity at constant pressure
  • $ΔT$ represents the change in temperature

Calculating entropy changes

• For an isothermal process (constant temperature), the change in specific entropy (Δs) is calculated as $Δs = R ln(v2/v1)$ for an ideal gas
  • $R$ represents the specific gas constant
  • $v1$ and $v2$ represent the initial and final specific volumes, respectively
• For a reversible adiabatic process (no heat transfer), the general relations simplify to $ds = 0$
  • This leads to the equation $Pvγ = constant$ for an ideal gas
  • $γ$ represents the specific heat ratio $(cp/cv)$

Relationships between thermodynamic properties

Interdependence of thermodynamic properties

• The general relations demonstrate the interdependence of thermodynamic properties and how changes in one property affect the others
• The Maxwell relations, derived from the general relations, provide additional relationships between partial derivatives of thermodynamic properties
  • For example, $(∂T/∂v)s = -(∂P/∂s)v$, which relates the partial derivatives of temperature with respect to volume at constant entropy and pressure with respect to entropy at constant volume
• The cyclic rule, also derived from the general relations, states that $(∂x/∂y)z (∂y/∂z)x (∂z/∂x)y = -1$
  • $x$, $y$, and $z$ represent any three thermodynamic properties

Analyzing thermodynamic processes

• The general relations can be used to analyze the behavior of substances during various thermodynamic processes
  • Isothermal processes occur at constant temperature (heat engines, chemical reactions)
  • Isobaric processes occur at constant pressure (open systems, phase changes)
  • Isochoric processes occur at constant volume (closed systems, explosions)
  • Adiabatic processes occur without heat transfer (rapid compression or expansion, insulated systems)
• The relationships between thermodynamic properties, as described by the general relations, are crucial for understanding and solving problems in thermodynamics
  • Engineers use these relations to design and optimize thermodynamic systems (engines, refrigerators, power plants)
  • Scientists use these relations to study the behavior of materials under different conditions (high pressure, low temperature)