Glossary

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

4 min read•july 30, 2024

Thermodynamic property relations are the backbone of understanding energy changes in systems. These equations link , , , 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, , and 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)
    • TT represents temperature
    • ss represents specific entropy
    • PP represents pressure
    • vv 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)
    • TT represents temperature
    • ss represents specific entropy
    • vv represents specific volume
    • PP 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)
    • uu represents specific internal energy
    • PP represents pressure
    • vv represents specific volume
    • TT 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
    • states that energy cannot be created or destroyed, only converted from one form to another
    • 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)vcv = (∂u/∂T)v
    • cvcv 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)Pcp = (∂h/∂T)P
    • cpcp 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=Rcp - cv = R for ideal gases
    • RR represents the specific gas constant
    • Ideal gases are hypothetical gases that follow the ideal gas law PV=nRTPV = 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−Tsg = h - Ts
    • Helmholtz free energy is defined as f=u−Tsf = 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=cvΔTΔu = cv ΔT
    • cvcv represents the specific heat capacity at constant volume
    • ΔTΔT represents the change in temperature
  • For an ideal gas, the change in specific enthalpy (Δh) is calculated as Δh=cpΔTΔh = cp ΔT
    • cpcp represents the specific heat capacity at constant pressure
    • ΔTΔT represents the change in temperature

Calculating entropy changes

  • For an (constant temperature), the change in specific entropy (Δs) is calculated as Δs=Rln(v2/v1)Δs = R ln(v2/v1) for an ideal gas
    • RR represents the specific gas constant
    • v1v1 and v2v2 represent the initial and final specific volumes, respectively
  • For a reversible (no heat transfer), the general relations simplify to ds=0ds = 0
    • This leads to the equation Pvγ=constantPvγ = constant for an ideal gas
    • γγ represents the specific heat ratio (cp/cv)(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 , derived from the general relations, provide additional relationships between partial derivatives of thermodynamic properties
    • For example, (∂T/∂v)s=−(∂P/∂s)v(∂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)z (∂y/∂z)x (∂z/∂x)y = -1
    • xx, yy, and zz 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)

Key Terms to Review (20)

Adiabatic process: An adiabatic process is a thermodynamic process in which no heat is transferred into or out of the system. During this type of process, any change in the internal energy of the system is solely due to work done on or by the system, making it essential in understanding how systems behave under different conditions.
Carnot Cycle: The Carnot cycle is an idealized thermodynamic cycle that represents the most efficient possible heat engine operating between two temperature reservoirs. It provides a standard for measuring the performance of real engines and illustrates the principles of energy transfer, work, and heat efficiency in thermodynamic processes.
Closed System: A closed system is a physical system that does not exchange matter with its surroundings but can exchange energy in the form of heat and work. This concept is vital in understanding how energy flows and transforms within a defined environment without any mass transfer, influencing various thermodynamic processes and principles.
Cp: The specific heat at constant pressure, denoted as $c_p$, measures the amount of heat required to raise the temperature of a substance by one degree Celsius while maintaining constant pressure. It plays a crucial role in understanding how substances respond to heat transfer and is essential for calculations involving enthalpy, phase changes, and thermal processes in thermodynamics.
Cv: The term cv, or specific heat at constant volume, refers to the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius while keeping the volume constant. This concept is crucial when analyzing processes that occur at constant volume, and it helps distinguish between different heat capacities, specifically in understanding the energy changes in thermodynamic systems.
Dh = t ds + v dp: The equation 'dh = t ds + v dp' represents a fundamental relationship in thermodynamics that connects changes in enthalpy (dh) to changes in entropy (ds) and pressure (dp) through temperature (t) and specific volume (v). This expression illustrates how enthalpy changes can be influenced by variations in both entropy and pressure, highlighting the interdependence of these thermodynamic properties. Understanding this relationship is crucial for analyzing energy transfer processes, particularly during phase changes and chemical reactions.
Ds = (du + p dv) / t: This equation represents the change in entropy (ds) in terms of the internal energy change (du), the pressure-volume work term (p dv), and the temperature (t). It highlights how energy transfers, volume changes, and temperature interrelate in thermodynamic systems, emphasizing that entropy is a measure of disorder or randomness within these systems.
Du = t ds - p dv: The equation 'du = t ds - p dv' represents the differential change in internal energy (du) of a system in thermodynamics, connecting it to changes in entropy (ds) and volume (dv) under constant temperature and pressure conditions. It illustrates how internal energy is affected by heat transfer (t ds) and work done on or by the system (-p dv), providing a foundational relationship in understanding energy conservation and transformations.
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.
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.
Gibbs Free Energy Relation: The Gibbs free energy relation is a thermodynamic equation that describes the change in Gibbs free energy (G) of a system in terms of its enthalpy (H), entropy (S), and temperature (T). This relation is crucial for understanding the spontaneity of processes, as it connects energy changes with the direction of chemical reactions and phase transitions.
Ideal Gas Equation: The ideal gas equation is a mathematical relationship that describes the behavior of an ideal gas, expressed as $$PV=nRT$$, where $$P$$ is pressure, $$V$$ is volume, $$n$$ is the number of moles, $$R$$ is the universal gas constant, and $$T$$ is temperature in Kelvin. This equation connects various properties of gases and serves as a foundational concept in thermodynamics, enabling the exploration of how gases behave under different conditions, as well as the calculation of other thermodynamic properties.
Internal energy: Internal energy is the total energy contained within a system, resulting from the kinetic and potential energies of its molecules. It plays a crucial role in determining the thermodynamic state of the system, affecting properties like temperature and pressure, and is essential for understanding energy transfer processes.
Isothermal process: An isothermal process is a thermodynamic process in which the temperature of a system remains constant while the system undergoes a change in volume or pressure. This type of process is crucial for understanding how systems interact with their surroundings and how energy is exchanged in various thermodynamic cycles.
Maxwell Relations: Maxwell relations are a set of equations derived from the equality of mixed partial derivatives of thermodynamic potentials. These relations connect different thermodynamic quantities, allowing us to express one property in terms of others, providing insight into the relationships among entropy, temperature, pressure, and volume, among other variables.
Open System: An open system is a physical system that can exchange both matter and energy with its surroundings. This type of system is crucial for understanding how various processes occur in real-life applications, where energy and mass flow in and out, influencing the system's behavior over time.
Refrigeration cycle: The refrigeration cycle is a thermodynamic process that removes heat from a designated area to lower its temperature, typically using a refrigerant. This cycle involves a series of phase changes and energy transfers that allow heat to be absorbed from the surroundings and expelled elsewhere, making it essential for various applications such as cooling systems and heat pumps.
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.
Van der Waals equation: The van der Waals equation is a modified version of the ideal gas law that accounts for the volume occupied by gas molecules and the intermolecular forces present in real gases. It introduces two constants, 'a' and 'b', which correct for the attractive forces between particles and the finite size of the particles themselves, thus providing a more accurate representation of gas behavior under various conditions.
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