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Differential Forms of Fundamental Equations

    The fundamental thermodynamic equations follow from five primary thermodynamic definitions and describe internal energy, enthalpy, Helmholtz energy, and Gibbs energy in terms of their natural variables. Here they will be presented in their differential forms.

    1. 1. Introduction
    2. 2. Five Principle Equations
        1. 2.1.1. First Law of Thermodynamics
        2. 2.1.2. Principle of Clausius
        3. 2.1.3. Definition of Enthalpy
        4. 2.1.4. Definition of Gibbs Energy
        5. 2.1.5. Definition of Helmholtz Energy
        6. 2.1.6. Assumptions
    3. 3. Differential Fundamental Equation for Internal Energy
    4. 4. Differential Fundamental Equation for Enthalpy
    5. 5. Differential Fundamental Equation for Gibbs Energy
    6. 6. Differential Fundamental Equation for Helmholtz Energy
    7. 7. Fundamental Equations Including n-Dependence
    8. 8. Importance/Relevance of Fundamental Equations
        1. 8.1.1. Understanding Natural Variables
        2. 8.1.2. Maxwell Relations
    9. 9. References
    10. 10. Outside Links
    11. 11. Problems
    12. 12. Answers
    13. 13. Contributors

    Introduction

    The fundamental thermodynamic equations describe the thermodynamic quantities U, H, G, and A in terms of their natural variables. The term "natural variable" simply denotes a variable that is one of the convenient variables to describe U, H, G, or A. When considered as a whole, the four fundamental equations demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentially equations of state, and using the fundamental equations, experimental data can be used to determine sought-after quantities like G or H.

    Five Principle Equations

    First Law of Thermodynamics

    The first law of thermodynamics is represented below in its differential form

    \[ dU = \delta \; q+\delta \; w \]

    where U is the internal energy of the system, q is heat flow of the system, and w is the work of the system. The "đ" notation indicates that both q and w are path functions. Recall that U is a state function. The first law states that internal energy changes occur only as a result of heat flow and work done.

    It is assumed that w refers only to PV work, where

    \[ w = -\int{pdV}\]

    Principle of Clausius

    The principle of Clausius states that the entropy change of a system is equal to the ratio of heat flow in a reversible process to the temperature at which the process occurs. Mathematically this is written as

    \[ dS = \frac{\delta q_{rev}}{T}\]

    where S is the entropy of the system, qrev is the heat flow of a reversible process, and T is the temperature in Kelvin.

    Definition of Enthalpy

    Mathematically, enthalpy is defined as

    \[ H = U + pV\]

    where H is enthalpy of the system, p is pressure, and V is volume.

    Definition of Gibbs Energy

    The mathematical description of Gibbs energy is as follows

     

    \[ G = H - TS\]

    where G is the Gibbs energy of the system.

    Definition of Helmholtz Energy

    Mathematically, Helmholtz energy is defined as

    \[ A = U - TS\]

    where A is the Helmholtz energy of the system. The Helmholtz energy is often written as the symbol F.

    Assumptions

    In order for the definitions to hold, it is assumed that only P-V work is done and that only reversible processes are used. These assumptions are required for the first law and the principle of Clausius to remain valid.

    Also, these equations do not account include n, the number of moles, as a variable. When n is included, the equations appear different, but the essence of their meaning is captured without including the n-dependence.

    Differential Fundamental Equation for Internal Energy

    The fundamental thermodynamic equation for internal energy follows directly from the first law and the principle of Clausius:

     

    \[ dU = \delta \; q + \delta \; w\] \[ dS = \frac{\delta q_rev}{T} \]

    we have

    \[ dU = TdS + \delta w\]

    Since only P-V work is performed,

    \[ dU = TdS - pdV\]
    The above equation is the fundamental equation for U. The natural variables of internal energy are S and V, entropy and volume.

     

    Differential Fundamental Equation for Enthalpy

    The fundamental thermodynamic equation for enthalpy follows directly from the definition of enthalpy and the fundamental equation for internal energy:

    \[ dH = dU + d(pV)\] \[ = dU + pdV + VdP\] \[ dU = TdS - pdV\] \[ dH = TdS - pdV + pdV + Vdp\] \[ dH = TdS + Vdp\]

    The above equation is the fundamental equation for H. The natural variables of enthalpy are S and p, entropy and pressure.

    Differential Fundamental Equation for Gibbs Energy

    The fundamental thermodynamic equation for Gibbs Energy follows directly from the definition of Gibbs energy and the fundamental equation for enthalpy:

    \[ dG = dH - d(TS)\] \[ = dH - TdS - SdT\]

    Since

    \[ dH = TdS + Vdp\] \[ dG = TdS + Vdp - TdS - SdT\] \[ dG = Vdp - SdT\] The above equation is the fundamental equation for G. The natural variables of Gibbs energy are p and T, pressure and temperature.

     

    Differential Fundamental Equation for Helmholtz Energy

    The fundamental thermodynamic equation for Helmholtz energy follows directly from the definition of Helmholtz energy and the fundamental equation for internal energy:

    \[ dA = dU - d(TS)\] \[ = dU - TdS - SdT\]

    Since

    \[ dU = TdS - pdV\]

    \[ dA = TdS - pdV -TdS - SdT\]

    \[ dA = -pdV - SdT\]

    The above equation is the fundamental equation for A. The natural variables of enthalpy are V and T, volume and temperature.

    Fundamental Equations Including n-Dependence

    The fundamental equations derived above were not dependent on changes in the amounts of species in the system. Below the n-dependent forms are presented1,4.

    \[ dU = TdS - PdV + \sum_{i=1}^{N}\mu_idn_i \] \[ dH = TdS + VdP + \sum_{i=1}^{N}\mu_idn_i \] \[ dG = -SdT + Vdp + \sum_{i=1}^{N}\mu_idn_i \] \[ dA = -SdT - PdV + \sum_{i=1}^{N}\mu_idn_i\]

    where μi is the chemical potential of species i and dni is the change in number of moles of substance i.

    Importance/Relevance of Fundamental Equations

    Understanding Natural Variables

    The differential fundamental equations describe U, H, G, and A in terms of their natural variables. The natural variables become useful in understanding not only how thermodynamic quantities are related to each other, but also in analyzing relationships between measurable quantities (i.e. P, V, T) in order to learn about the thermodynamics of a system.

    Below is a table summarizing the natural variables for U, H, G, and A:

    Thermodynamic Quantity Natural Variables
    U (internal energy) S, V
    H (enthalpy) S, P
    G (Gibbs energy) T, P
    A (Helmholtz energy) T, V
    Maxwell Relations

    The fundamental thermodynamic equations are the means by which the Maxwell relations are derived1,4. The Maxwell Relations can, in turn, be used to group thermodynamic functions and relations into more general "families"2,3. See the sample problems and the Maxwell Relation section for details.

    References

    1. DOI: 10.1063/1.1749582
    2. DOI: 10.1063/1.1749549
    3. DOI:10.1103/PhysRev.3.273
    4. A Treatise on Physical Chemistry, 3rd ed.; Taylor, H. S. and Glasstone, S., Eds.; D. Van Nostrand Company: New York, 1942; Vol. 1; p 454-485.

    Problems

    1. If the assumptions made in the derivations above were not made, what would effect would that have? Try to think of examples were these assumptions would be violated. Could the definitions, principles, and laws used to derive the fundamental equations still be used? Why or why not?
    2. For what kind of system does the number of moles not change? This said, do the fundamental equations without n-dependence apply to a wide range of processes and systems?
    3. Derive the Maxwell Relations.
    4.  Derive the expression

    \[ \left (\frac{\partial H}{\partial P} \right)_{T,n} = -T \left(\frac{\partial V}{\partial T} \right)_{P,n} +V \]

    Then apply this equation to an ideal gas. Does the result seem reasonable?
    5. Using the definition of Gibbs energy and the conditions observed at phase equilibria, derive the Clapeyron equation.

    Answers

    1. If it was not assumed that PV-work was the only work done, then the work term in the second law of thermodynamics equation would include other terms (e.g. for electrical work, mechanical work). If reversible processes were not assumed, the Principle of Clausius could not be used. One example of such situations could the movement of charged particles towards a region of like charge (electrical work) or an irreversible process like combustion of hydrocarbons or friction.
    2. In general, a closed system of non-reacting components would fit this description. For example, the number of moles would not change for a closed system in which a gas is sealed (to prevent leaks) in a container and allowed to expand/is contracted.
    3. See the Maxwell Relations section.
    4. \((\frac{\partial H}{\partial P})_{T,n} = 0 \)  for an ideal gas. Since there are no interactions between ideal gas molecules, changing the pressure will not involve the formation or breaking of any intermolecular interactions or bonds.
    5. See the third outside link.

    Contributors

    • Andreana Rosnik, Hope College

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