Thermodynamics: Maximum Work and Free Energy
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Questions and Answers

What does the fundamental equation apply to?

  • Closed systems with constant composition
  • All types of thermodynamic systems (correct)
  • Open systems with variable composition
  • Closed systems with variable temperature

    • Which statement correctly describes U in relation to state functions?

  • U is dependent on temperature and pressure.
  • U can only be calculated through reversible processes.
  • U is independent of the path taken between two states. (correct)
  • U is not a state function.

    • What expression represents the differential form of U?

  • $dU = T dV + p dS$
  • $dU = T dS - p dV$ (correct)
  • $dU = T dS + p dV$
  • $dU = -T dS + p dV$

    • What condition must be satisfied for df to be an exact differential?

    <p>$ rac{ iny{ rac{ ext{d}g}{ ext{d}x}}}{ ext{d}y} = rac{ iny{ rac{ ext{d}h}{ ext{d}y}}}{ ext{d}x}$ (B)</p> Signup and view all the answers

    How are Maxwell relations useful in thermodynamics?

    <p>They allow derivation of relationships between seemingly unrelated quantities. (A)</p> Signup and view all the answers

    What does the symbol $T$ represent in the expression $T = rac{ iny{ rac{ ext{d}U}{ ext{d}S}}}{ ext{d}V}$?

    <p>Absolute temperature (B)</p> Signup and view all the answers

    What does the derivative $ rac{ iny{ rac{ ext{d}U}{ ext{d}S}}}{ ext{d}V}$ imply about the relationship between U, S, and V?

    <p>Both S and V influence the value of U. (D)</p> Signup and view all the answers

    What does the negative sign in $- rac{ iny{ rac{ ext{d}p}{ ext{d}S}}}{ ext{d}V}$ indicate?

    <p>An increase in volume leads to a decrease in pressure. (B)</p> Signup and view all the answers

    What is the relationship used to determine if a spontaneous change is enthalpy driven?

    <p>dG = dH - TdS (B)</p> Signup and view all the answers

    How is Gibbs energy defined mathematically?

    <p>G = H - TS (C)</p> Signup and view all the answers

    What does a decrease in Gibbs energy indicate at constant temperature and pressure?

    <p>Spontaneous changes or reactions (A)</p> Signup and view all the answers

    What is the correct equation for the Helmholtz energy?

    <p>A = U - TS (C)</p> Signup and view all the answers

    What is true about Helmholtz energy (A) and its relation to work?

    <p>A is the maximum work function. (C)</p> Signup and view all the answers

    Which of the following statements about Gibbs energy is true?

    <p>∆G is the maximum work at constant temperature and pressure. (A)</p> Signup and view all the answers

    What does the relationship dU = TdS - pdV represent?

    <p>The Fundamental equation combining both laws. (D)</p> Signup and view all the answers

    What type of work does ∆G represent at constant temperature and pressure?

    <p>Non-expansion work (D)</p> Signup and view all the answers

    What does the Clausius inequality imply for a system in thermal and mechanical contact with its surroundings?

    <p>The change in entropy of the system is greater than or equal to the negative change in entropy of the surroundings. (D)</p> Signup and view all the answers

    What is the Gibbs energy a measure of in a system?

    <p>Maximum non-expansion work obtainable from a system. (D)</p> Signup and view all the answers

    In the context of the First and Second Laws of Thermodynamics, what does the Fundamental equation relate?

    <p>The internal energy of a system to its volume and pressure changes. (C)</p> Signup and view all the answers

    At constant volume, what can be said about the relationship between heat transfer and the internal energy change of a system?

    <p>Change in internal energy equals the heat transferred since no work is done. (B)</p> Signup and view all the answers

    When heat transfer occurs at constant pressure, what does the equation $dqp = dH$ imply?

    <p>Heat transferred corresponds to the change in enthalpy during a process. (C)</p> Signup and view all the answers

    What condition must be met for the equation $dS_{sys} ≥ -dq_{sys}/T$ to hold true?

    <p>The system must be in equilibrium with the surroundings. (A)</p> Signup and view all the answers

    What happens to the heat transfer ($dq$) if the system is isolated?

    <p>No heat transfer occurs. (D)</p> Signup and view all the answers

    What can be concluded about spontaneity if the change in entropy of the system ($dS_{sys}$) is positive?

    <p>The process will occur without external influence. (A)</p> Signup and view all the answers

    Flashcards

    Clausius Inequality

    For a system undergoing spontaneous change, the total change in entropy (system + surroundings) must be greater than or equal to zero. This inequality relates heat transfer and temperature to the change in entropy.

    System-only criteria for spontaneity

    A way to determine spontaneous change in a system without considering its surroundings, useful for practical applications and calculations.

    Gibbs Free Energy (G)

    A thermodynamic state function that measures the maximum non-expansion work obtainable from a closed system at constant temperature and pressure.

    Helmholtz Free Energy (H)

    A thermodynamic potential that measures the maximum reversible work obtainable from a closed system at constant temperature and volume.

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    Fundamental Equation

    Combining the First and Second Laws of Thermodynamics to derive a relationship that expresses the system's energy, entropy, and other variables.

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    Maxwell Relations

    Important mathematical relationships derived from the Fundamental Equation that connect the partial derivatives of thermodynamic potential (like enthalpy, entropy, etc.).

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    Spontaneous Change

    A process that occurs naturally without any external influence or intervention.

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    Constant Volume

    A condition in thermodynamic processes where the volume of the system remains unchanged (dV=0).

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    Helmholtz Energy

    A thermodynamic function defined as A = U - TS, where U is internal energy, T is temperature, and S is entropy. Changes in A reflect maximum work obtainable at constant temperature and volume.

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    Gibbs Energy

    A thermodynamic function defined as G = H - TS, where H is enthalpy, T is temperature, and S is entropy. Spontaneous changes at constant temperature and pressure are associated with decreasing Gibbs energy.

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    Spontaneous Change (Constant T & P)

    A change that occurs without external intervention at constant temperature and pressure, indicated by a decrease in Gibbs Free Energy (ΔG < 0).

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    Maximum Non-expansion Work

    The maximum amount of work a system can perform at constant temperature and pressure, excluding the work done by expansion or contraction (e.g., pushing a piston).

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    ∆G and spontaneity

    Spontaneous changes at constant temperature and pressure are characterized by a negative Gibbs free energy change (∆G < 0).

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    Maximum Work (A)

    The maximum work attainable in a process at constant temperature and volume is equal to the negative of the change in Helmholtz Energy.

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    Enthalpy Driven/Entropy Driven

    Spontaneous processes can be driven by changes in enthalpy (heat) or entropy (disorder). This fact can be discerned by examining Gibbs energy changes at constant temperature and pressure.

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    What is the Fundamental Equation?

    The Fundamental Equation is a core thermodynamic relationship that expresses the change in internal energy (dU) of a closed system undergoing reversible changes with constant composition. It's based on the First and Second Laws of Thermodynamics.

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    What does the Fundamental Equation apply to?

    The Fundamental Equation holds true for any change in a closed system with constant composition that performs no additional work, such as only heat transfer.

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    Why is dU an exact differential?

    Since internal energy (U) is a state function, its change (dU) depends only on the initial and final states of the system, regardless of the path taken. This means dU is an exact differential, meaning its value is path-independent.

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    What are Maxwell Relations?

    Maxwell Relations are a set of mathematical equations derived from the Fundamental Equation. These equations establish relationships between various thermodynamic variables by connecting their partial derivatives.

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    What is the key to deriving Maxwell Relations?

    The Fundamental Equation, defining dU as a combination of partial derivatives, combined with the fact that dU is an exact differential, leads to the Maxwell Relations.

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    What is the use of Maxwell Relations?

    Maxwell Relations help derive unusual and useful connections between seemingly unrelated thermodynamic quantities. They provide flexibility in analyzing system changes, even when selecting convenient paths for calculation.

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    How are Maxwell Relations derived?

    By exploiting the fact that dU is an exact differential, we can equate the mixed partial derivatives of the coefficients in the equation for dU, leading to the Maxwell Relations. For example, dU = TdS - pdV yields the Maxwell Relation relating partial derivatives of T and p.

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    Can you derive Maxwell Relations for other thermodynamic potentials?

    Yes, similar to the derivation for internal energy (U), we can apply the same logic to other thermodynamic potentials like enthalpy (H), Gibbs free energy (G), and Helmholtz free energy (A) to derive their respective Maxwell Relations

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    Study Notes

    Maximum Work and Free Energy

    • Maximum work and free energy are derived from combining the first and second laws of thermodynamics.
    • Maxwell relations are also involved.

    Overview

    • Derive the Clausius inequality, expressing it in terms of the system only.
    • Define Gibbs (G) and Helmholtz (H) state functions.
    • Derive that Gibbs energy measures maximum non-expansion work.
    • Derive the fundamental equation by combining the first and second laws.
    • Use this equation to derive Maxwell relations and discuss their importance.

    The Clausius Inequality

    • For a system in thermal and mechanical contact with surroundings (not necessarily in equilibrium), undergoing spontaneous change, dSsys + dSsurr ≥ 0.
    • This implies dSsys ≥ -dSsurr.
    • Recall dSsurr= - dqsys / T.
    • Therefore, dSsys ≥ dqsys / T.
    • This is the Clausius inequality.
    • Consider what happens if the system is isolated.

    Concentrating on the System Only

    • Entropy (S) measures spontaneity, but simultaneously considering the system and surroundings can be complex.
    • System-only criteria for spontaneity are needed.
    • For a system in thermal equilibrium with surroundings at temperature T, consider the Clausius inequality: dS - dq/T ≥ 0.

    System-Only Parameters

    • When heat transfer occurs at constant volume, dU = dq, since no work is done (dw = 0).
    • This implies dS ≥ dU/T, which rearranges to TdS ≥ dU.
    • System-only parameters are used to describe spontaneous change.
    • Consider constant U and constant S.
    • For heat transfer occurring at constant pressure (dU = dH), the Clausius inequality gives TdS ≥ dH.

    Gibbs and Helmholtz Functions

    • Two ways to describe spontaneous change without including surroundings.
    • dU - TdS ≤ 0 and dH - TdS ≤ 0
    • Helmholtz energy (A) is defined as A = U - TS, with dA = dU - TdS or ΔA = ΔU - TΔS.
    • Gibbs energy (G) is defined as G = H - TS, with dG = dH - TdS or ΔG = ΔH - TΔS.
    • Now spontaneity can be evaluated as dA ≤ 0 or dG ≤ 0, at constant T, respectively.

    Why Focus on Gibbs Energy?

    • Gibbs energy (G) is more popular than Helmholtz energy (A) at constant T and P.
    • Spontaneous changes and reactions occur when Gibbs energy decreases.
    • Gibbs energy can be used to determine if a change is enthalpy or entropy driven.

    ΔG and Maximum Non-expansion Work

    • Non-expansion work is additional work.
    • Examples include moving electrons or raising a mass.
    • ΔG is the maximum possible non-expansion work a system can perform at constant temperature and pressure. -ΔG = Wadd, max

    Justification for Maximum Non-Expansion Work

    • Change in enthalpy considers conditions and dH=dq+dw+d(pV)
    • Corresponding change of Gibbs energy, dG = dH-TdS.
    • At constant temperature and reversible change, ΔG = Wadd,max.
    • This work can occur in any process occurring reversibly at constant P and T.
    • Work can include electrical work or pushing electrons through a circuit.

    ΔA and Maximum Work

    • The change in Helmholtz energy is equal to the maximum work in a process.
    • dA = dwmax
    • Helmholtz functions are sometimes referred to as maximum work functions.

    Justification for Maximum Work

    • To demonstrate that maximum work = the change in Helmholtz energy, combine Clausius inequality (TdS ≥ dq) and first law (dU = dq + dw).
    • dU ≤ TdS + dw.
    • dw ≥ dU - TdS.
    • Maximum work = the most negative value of dw = dU - TdS, or dwmax = dA for constant T.

    Combining First and Second Laws

    • First law states dU = dq + dw.
    • Second law states dqrev = TdS.
    • Also, dwrev = -pdV.
    • Combining these gives dU = TdS - pdV.

    Maxwell Relations

    • The fundamental equation applies to enclosed systems with constant composition that do no additional work.
    • Since U is a state function, dU is an exact differential, thus dU= (∂U/∂S)V dS + (∂U/∂V)S dV.
    • This implies T = (∂U/∂S)V and -p = (∂U/∂V)S .
    • Maxwell relations allow to derive relationships between quantities that might not seem related allowing for changes in the system in convenient paths.
    • Maxwell relations can be derived for H, G and A.
    • Table 3.5 includes derived Maxwell relations.

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    Description

    This quiz covers the concepts of maximum work and free energy in thermodynamics, integrating the first and second laws. You will explore the Clausius inequality, Gibbs and Helmholtz state functions, and derive Maxwell relations, highlighting their significance in thermodynamic processes.

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