Physical Chemistry 1 Quiz: Gibbs Duhem Relationship

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Gibbs Duhem RelationshipQuestion 1 of 11

In a binary system where component 1 obeys Raoult's law ( $a_1 = x_1$ ) across the entire composition range, what does the Gibbs-Duhem relationship predict about the activity coefficient of component 2?

Component 2 must also obey Raoult's law with

\gamma_2 = 1

across the entire composition range to satisfy the Gibbs-Duhem constraint.

Component 2 must exhibit positive deviations from Raoult's law with

\gamma_2 > 1

to compensate for component 1's ideal behavior.

Component 2 must exhibit negative deviations from Raoult's law with

\gamma_2 < 1

to maintain thermodynamic consistency.

Component 2 can exhibit any activity coefficient behavior since the Gibbs-Duhem relationship only constrains the differential changes, not absolute values.

Component 2 must have

\gamma_2 = x_1/x_2

to satisfy the integral constraint imposed by the Gibbs-Duhem equation.

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Physical Chemistry 1 Quiz

Physical Chemistry 1 Quiz: Gibbs Duhem Relationship

Practice Gibbs Duhem Relationship in Physical Chemistry 1 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Gibbs Duhem Relationship, giving you a quick way to practice the rules, question types, and explanations that matter most for Physical Chemistry 1.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Component 2 must also obey Raoult's law with $\gamma_2 = 1$ across the entire composition range to satisfy the Gibbs-Duhem constraint. (correct answer)
Component 2 must exhibit positive deviations from Raoult's law with $\gamma_2 > 1$ to compensate for component 1's ideal behavior.
Component 2 must exhibit negative deviations from Raoult's law with $\gamma_2 < 1$ to maintain thermodynamic consistency.
Component 2 can exhibit any activity coefficient behavior since the Gibbs-Duhem relationship only constrains the differential changes, not absolute values.
Component 2 must have $\gamma_2 = x_1/x_2$ to satisfy the integral constraint imposed by the Gibbs-Duhem equation.

Explanation: When you encounter problems involving one component following Raoult's law perfectly, you need to apply the Gibbs-Duhem relationship, which ensures thermodynamic consistency between components in a mixture. The Gibbs-Duhem equation for a binary system at constant temperature and pressure states:

x_1 d\ln a_1 + x_2 d\ln a_2 = 0

. Since component 1 obeys Raoult's law (

a_1 = x_1

), we have

d\ln a_1 = d\ln x_1 = \frac{dx_1}{x_1}

. For component 2,

d\ln a_2 = d\ln x_2 + d\ln \gamma_2

. Since

x_1 + x_2 = 1

, we get

dx_1 = -dx_2

. Substituting into Gibbs-Duhem:

x_1 \frac{dx_1}{x_1} + x_2 \left(\frac{dx_2}{x_2} + d\ln \gamma_2\right) = 0

. This simplifies to

dx_1 + dx_2 + x_2 d\ln \gamma_2 = 0

. Since

dx_1 + dx_2 = 0

, we get

x_2 d\ln \gamma_2 = 0

. For this to hold across all compositions where

x_2 \neq 0

, we must have

d\ln \gamma_2 = 0

, meaning

\gamma_2 = \text{constant} = 1

. Answer A is correct because thermodynamic consistency requires component 2 to also follow Raoult's law. Answer B incorrectly assumes positive deviations are needed for "compensation." Answer C wrongly suggests negative deviations maintain consistency. Answer D misunderstands that while Gibbs-Duhem constrains differentials, integrating these constraints determines absolute behavior. Remember: when one component in a binary mixture follows Raoult's law perfectly across the entire range, the Gibbs-Duhem relationship forces the other component to also behave ideally—this defines an ideal solution.

Question 2

In studying a binary mixture, a researcher finds that the activity of component 1 follows $\ln a_1 = \ln x_1 + 2Cx_2^2$ where C is a positive constant. If the Gibbs-Duhem relationship is to be satisfied, what is the minimum number of additional parameters needed to completely specify $\ln a_2$ ?

Zero additional parameters are needed since the Gibbs-Duhem relationship completely determines $\ln a_2$ from the given information about component 1. (correct answer)
One additional parameter is required to specify the integration constant when applying the Gibbs-Duhem constraint to determine $\ln a_2$ .
Two additional parameters are needed to account for both the magnitude and composition dependence of the interaction effects in component 2.
Three additional parameters are required since the activity expression for component 2 must be independent of the form chosen for component 1.
No finite number of parameters can specify $\ln a_2$ because the given expression for component 1 violates fundamental thermodynamic constraints.

Explanation: When you encounter activity expressions in binary mixtures, the Gibbs-Duhem relationship is your key constraint. This fundamental thermodynamic relationship requires that

x_1 d\ln a_1 + x_2 d\ln a_2 = 0

at constant temperature and pressure. Starting with the given expression

\ln a_1 = \ln x_1 + 2Cx_2^2

, you can find

d\ln a_1

by taking the total differential. This gives you

d\ln a_1 = \frac{dx_1}{x_1} + 4Cx_2 dx_2

. Since

dx_1 = -dx_2

in a binary system, this becomes

d\ln a_1 = -\frac{dx_2}{x_1} + 4Cx_2 dx_2

. Substituting into the Gibbs-Duhem equation and solving for

d\ln a_2

, you get a differential equation that integrates directly to give

\ln a_2 = \ln x_2 + 2Cx_1^2

. The integration constant vanishes when you apply the boundary condition that activities approach mole fractions as the mixture approaches ideal behavior. Answer A is correct because the Gibbs-Duhem relationship, combined with the given activity expression and proper boundary conditions, completely determines

\ln a_2

without any additional parameters. Answer B incorrectly suggests an integration constant remains—the boundary conditions eliminate this. Answer C misunderstands that the same parameter C appears in both expressions due to thermodynamic consistency. Answer D wrongly implies that component activities are independent—they're intimately connected through the Gibbs-Duhem constraint. Remember: In binary mixtures, once you specify one component's activity expression, thermodynamic consistency demands a specific form for the other component's activity.

Question 3

For a three-component system in equilibrium, the partial molar volumes are observed to follow: $\bar{V_1} = V_1^* + \alpha x_2 x_3$ , $\bar{V_2} = V_2^* + \beta x_1 x_3$ , and $\bar{V_3} = V_3^* + \gamma x_1 x_2$ . What constraint does the Gibbs-Duhem relationship impose on the parameters $\alpha$ , $\beta$ , and $\gamma$ ?

$\alpha + \beta + \gamma = 0$ to ensure that the total volume change upon mixing equals the sum of individual component contributions. (correct answer)
$\alpha = \beta = \gamma$ required by the symmetry conditions that must be satisfied in any thermodynamically consistent ternary mixture.
$x_1\alpha + x_2\beta + x_3\gamma = 0$ for all compositions, which can only be satisfied if each parameter individually equals zero.
$\alpha\beta\gamma = 1$ following from the multiplicative constraint imposed by the cross-derivative relationships in ternary systems.
No constraint is imposed since partial molar volumes are intensive properties that are independent of the Gibbs-Duhem relationship.

Explanation: When you encounter partial molar properties in multi-component systems, the Gibbs-Duhem equation is your key constraint. This fundamental relationship ensures thermodynamic consistency by requiring that changes in intensive properties across all components must sum to zero when weighted by their amounts. For this three-component system, apply the Gibbs-Duhem equation at constant temperature and pressure:

x_1 d\bar{V_1} + x_2 d\bar{V_2} + x_3 d\bar{V_3} = 0

. Taking differentials of the given expressions and substituting into this constraint yields relationships between the parameters. When you work through the mathematics for all possible composition changes, you find that

\alpha + \beta + \gamma = 0

must hold. Choice A is correct because this constraint ensures the partial molar volumes are thermodynamically consistent. The sum equaling zero reflects that excess properties in ternary mixtures must balance—positive deviations in some interactions must be offset by negative deviations in others. Choice B incorrectly assumes symmetry requires equal parameters. While some mixing models have symmetric forms, thermodynamic consistency doesn't demand identical interaction parameters between different component pairs. Choice C misapplies the constraint. The composition-weighted sum

x_1\alpha + x_2\beta + x_3\gamma = 0

would indeed require each parameter to be zero individually, but this isn't the actual Gibbs-Duhem constraint for this system. Choice D invents a multiplicative relationship that has no basis in thermodynamic theory. The Gibbs-Duhem equation involves linear combinations, not products of parameters. Remember: whenever you see partial molar properties, immediately think Gibbs-Duhem—it's the universal consistency check for intensive properties in mixtures.

Question 4

For a binary system, the excess Gibbs energy per mole is given by $g^E = \frac{\alpha x_1 x_2}{x_1 + \beta x_2}$ where $\alpha$ and $\beta$ are constants. What relationship between $\alpha$ and $\beta$ is required for this expression to be consistent with the Gibbs-Duhem equation?

$\beta = 1$ is required since only symmetric excess Gibbs energy functions can satisfy the Gibbs-Duhem relationship in binary systems.
$\alpha = \beta$ is necessary to ensure that both activity coefficients approach unity at their respective infinite dilution limits.
$\beta = 2\alpha$ is required by the cross-derivative relationship that emerges from the thermodynamic consistency conditions.
No specific relationship between $\alpha$ and $\beta$ is required since any excess Gibbs energy function automatically satisfies Gibbs-Duhem. (correct answer)
$\alpha + \beta = 1$ is necessary to maintain the proper scaling behavior required by the fundamental thermodynamic relationships.

Explanation: When you encounter questions about excess Gibbs energy functions and thermodynamic consistency, remember that the Gibbs-Duhem equation is automatically satisfied by any properly defined excess function - it's a fundamental thermodynamic relationship, not an additional constraint. The key insight is that any excess Gibbs energy function that can be differentiated to obtain activity coefficients will inherently satisfy the Gibbs-Duhem equation. This is because the Gibbs-Duhem relationship emerges directly from the mathematical properties of partial molar quantities, not from specific functional forms or parameter relationships. For the given function

g^E = \frac{\alpha x_1 x_2}{x_1 + \beta x_2}

, you can derive activity coefficients by taking appropriate partial derivatives, and these will automatically satisfy

x_1 d\ln\gamma_1 + x_2 d\ln\gamma_2 = 0

regardless of the values of

\alpha

and

\beta

. Option A is wrong because asymmetric functions can perfectly satisfy Gibbs-Duhem - symmetry isn't required. Option B incorrectly assumes infinite dilution behavior determines thermodynamic consistency, when these are separate considerations. Option C suggests a specific mathematical relationship that simply doesn't exist - there's no cross-derivative requirement linking

\alpha

and

\beta

for consistency. Study tip: Remember that thermodynamic consistency (Gibbs-Duhem compliance) is built into the mathematical framework of excess functions. Don't confuse this with other requirements like fitting experimental data or achieving desired limiting behavior - those might constrain parameters, but consistency doesn't.

Question 5

In a three-component system at equilibrium, if the chemical potentials of components 1 and 2 are held constant while the system composition changes, what constraint does the Gibbs-Duhem relationship place on component 3?

The chemical potential of component 3 must remain constant, making the system invariant according to the Gibbs phase rule.
The chemical potential of component 3 can vary freely since only two components are constrained by the Gibbs-Duhem equation.
The chemical potential of component 3 must change in proportion to its mole fraction to satisfy the constraint equation.
The chemical potential of component 3 must also remain constant, as all three chemical potentials are interdependent through the Gibbs-Duhem relation. (correct answer)
The chemical potential of component 3 must vary inversely with the sum of the other two chemical potentials to maintain thermodynamic consistency.

Explanation: When you encounter questions about the Gibbs-Duhem relationship, you're dealing with a fundamental thermodynamic constraint that connects the chemical potentials of all components in a system. This relationship states that at constant temperature and pressure, the sum of each component's mole fraction times its chemical potential change must equal zero:

\sum x_i d\mu_i = 0

. The correct answer is D because the Gibbs-Duhem equation creates an interdependence among all chemical potentials. If

d\mu_1 = 0

and

d\mu_2 = 0

(components 1 and 2 held constant), then the equation becomes

x_1(0) + x_2(0) + x_3 d\mu_3 = 0

, which simplifies to

x_3 d\mu_3 = 0

. Since

x_3 \neq 0

(component 3 is present), we must have

d\mu_3 = 0

, meaning the chemical potential of component 3 must also remain constant. Answer A incorrectly focuses on the phase rule rather than the Gibbs-Duhem constraint itself. Answer B fundamentally misunderstands the relationship—the Gibbs-Duhem equation constrains all components simultaneously, not just two. Answer C suggests a proportional relationship with mole fraction, which isn't what the Gibbs-Duhem equation requires under these specific conditions. Remember this key insight: the Gibbs-Duhem relationship means you can never independently vary all chemical potentials in a multicomponent system. If some are fixed, the constraint propagates to determine the behavior of the others—there's always one less degree of freedom than you might initially expect.

Question 6

A researcher observes that in a binary liquid mixture, when the partial molar volume of component A decreases by 2.5 cm³/mol upon increasing pressure, the partial molar volume of component B increases by 1.8 cm³/mol. If the mole fraction of A is 0.6, what can be concluded about this experimental observation?

The observation is thermodynamically consistent since the weighted sum of partial molar volume changes equals zero within experimental error.
The observation violates the Gibbs-Duhem relationship and indicates experimental error or non-equilibrium conditions in the measurement. (correct answer)
The observation is consistent only if the temperature was simultaneously varied to compensate for the pressure-induced volume changes.
The observation is valid since the Gibbs-Duhem equation does not apply to intensive properties like partial molar volumes under pressure changes.
The observation indicates that the mixture exhibits ideal behavior where partial molar volumes can change independently of each other.

Explanation: When you encounter partial molar volume changes in mixtures, you need to check whether they satisfy the Gibbs-Duhem relationship, which constrains how intensive properties can change simultaneously in equilibrium systems. The Gibbs-Duhem equation for a binary mixture at constant temperature states that:

n_A d\bar{V}_A + n_B d\bar{V}_B = 0

, where

n_A

and

n_B

are the number of moles and

d\bar{V}_A

and

d\bar{V}_B

are the changes in partial molar volumes. This can be rewritten using mole fractions:

x_A d\bar{V}_A + x_B d\bar{V}_B = 0

. Let's check this experimental data:

x_A = 0.6

, so

x_B = 0.4

. The changes are

d\bar{V}_A = -2.5

cm³/mol and

d\bar{V}_B = +1.8

cm³/mol. Substituting:

(0.6)(-2.5) + (0.4)(1.8) = -1.5 + 0.72 = -0.78

cm³/mol. Since this doesn't equal zero, the observation violates the Gibbs-Duhem relationship, confirming answer B. Answer A incorrectly suggests the weighted sum equals zero—our calculation shows it's -0.78 cm³/mol, not zero. Answer C wrongly implies temperature variation could fix this violation; the Gibbs-Duhem constraint applies regardless. Answer D makes a false claim—the Gibbs-Duhem equation absolutely applies to partial molar volumes under pressure changes. Study tip: Always verify that partial molar property changes satisfy the Gibbs-Duhem relationship. Calculate

\sum x_i d\bar{Y}_i = 0

for any intensive property Y—if it's not zero, the data indicates experimental error or non-equilibrium conditions.

Question 7

When applying the Gibbs-Duhem equation to analyze phase equilibria, which statement correctly describes the relationship between intensive properties in a two-component system undergoing isothermal compression?

The chemical potentials must change proportionally to maintain equilibrium, with the ratio determined by molecular weights
The sum of chemical potential changes weighted by mole fractions must equal zero, independent of pressure effects
The chemical potential changes must satisfy $x_1 d\mu_1 + x_2 d\mu_2 = V_m dP$ , linking composition and pressure dependencies (correct answer)
The individual chemical potentials remain constant during compression, while only their difference changes with pressure

Explanation: The complete Gibbs-Duhem equation for a binary system is

x_1 d\mu_1 + x_2 d\mu_2 = -S_m dT + V_m dP

. Under isothermal conditions (

dT = 0

), this becomes

x_1 d\mu_1 + x_2 d\mu_2 = V_m dP

, showing that the weighted sum of chemical potential changes equals the molar volume times pressure change. Choice A incorrectly invokes molecular weights rather than thermodynamic constraints. Choice B ignores pressure effects entirely. Choice D incorrectly states that individual chemical potentials are constant during compression.

Question 8

For a binary solution where the excess Gibbs energy is given by $G^E = x_1 x_2 [A + B(x_1 - x_2)]$ , what does the Gibbs-Duhem relationship predict about the behavior of the individual activity coefficients at the equimolar composition?

Both $\ln\gamma_1$ and $\ln\gamma_2$ must have zero derivatives with respect to mole fraction changes
The activity coefficients must be equal, and their logarithmic derivatives must have opposite signs
The derivatives $d\ln\gamma_1/dx_1$ and $d\ln\gamma_2/dx_1$ must sum to $-2A/RT$ at this composition
$\ln\gamma_1 = \ln\gamma_2 = A/4RT$ with symmetric behavior around the equimolar point (correct answer)

Explanation: When you encounter excess Gibbs energy expressions for binary solutions, you need to connect them to activity coefficients through thermodynamic relationships. The key is recognizing that activity coefficients derive directly from the excess Gibbs energy. For the given expression

G^E = x_1 x_2 [A + B(x_1 - x_2)]

, you can find individual activity coefficients using

RT \ln\gamma_i = \left(\frac{\partial G^E}{\partial x_i}\right)_{T,P,x_j}

. Taking the partial derivative with respect to

x_1

(remembering that

x_2 = 1-x_1

RT \ln\gamma_1 = x_2^2[A + B(3x_1 - x_2)]

At the equimolar composition where

x_1 = x_2 = 0.5

RT \ln\gamma_1 = (0.5)^2[A + B(1.5 - 0.5)] = 0.25[A + B] = A/4

(since the

B

term vanishes when

3x_1 - x_2 = 1

) By symmetry,

\ln\gamma_2 = A/4RT

as well. The Gibbs-Duhem relationship ensures this symmetric behavior around the equimolar point for this particular functional form. Option A is incorrect because the derivatives are not necessarily zero at this composition. Option B is wrong because while the activity coefficients are equal, their derivatives don't have opposite signs at the equimolar point. Option C incorrectly assumes a specific sum relationship that doesn't hold for this expression. Study tip: For excess Gibbs energy problems, always check the equimolar composition first—it often reveals symmetric behavior and simplifies complex expressions due to the

x_1 = x_2

condition.

Question 9

In applying the Gibbs-Duhem relationship to test experimental data, a researcher finds that for a proposed activity coefficient model, the integral $\int_0^1 \ln(\gamma_1/\gamma_2) dx_1$ does not equal zero. What does this indicate about the thermodynamic validity of the model?

The model is thermodynamically inconsistent and violates fundamental equilibrium principles (correct answer)
The model is acceptable since the Gibbs-Duhem test applies only to differential changes, not integrals
The model requires normalization by dividing each activity coefficient by the integral value
The model is valid for practical calculations but may have systematic errors at composition extremes

Explanation: The Gibbs-Duhem equation for a binary system at constant T and P gives

x_1 d\ln\gamma_1 + x_2 d\ln\gamma_2 = 0

. Rearranging:

d\ln\gamma_1 - d\ln\gamma_2 = d\ln(\gamma_1/\gamma_2) = -\frac{d\ln\gamma_2}{x_2} + \frac{d\ln\gamma_1}{x_1}

. Integration from

x_1 = 0

x_1 = 1

with proper boundary conditions (

\gamma_i \to 1

x_i \to 1

) must give

\int_0^1 \ln(\gamma_1/\gamma_2) dx_1 = 0

. Violation of this integral test indicates fundamental thermodynamic inconsistency. Choice B incorrectly distinguishes differential and integral consistency. Choice C suggests an invalid normalization. Choice D incorrectly accepts inconsistent models.

Question 10

In a study of liquid-vapor equilibrium, the vapor pressures of components A and B above their binary solution are found to follow $P_A = x_A P_A^* \exp(\alpha x_B^2)$ and $P_B = x_B P_B^* \exp(\beta x_A^2)$ . Which relationship between $\alpha$ and $\beta$ is required for thermodynamic consistency?

$\alpha = \beta$ to ensure symmetric behavior in the binary mixture
$\alpha + \beta = 0$ to satisfy the Gibbs-Duhem constraint for vapor pressures
$\alpha = \beta$ to maintain equal activity coefficient relationships for both components (correct answer)
$\alpha - \beta = 1$ to account for the different pure component vapor pressures

Explanation: The activity coefficients are

\gamma_A = \exp(\alpha x_B^2)

and

\gamma_B = \exp(\beta x_A^2)

. For thermodynamic consistency via the Gibbs-Duhem equation:

x_A d\ln\gamma_A + x_B d\ln\gamma_B = 0

. Taking derivatives:

d\ln\gamma_A = 2\alpha x_B dx_B = -2\alpha x_B dx_A

and

d\ln\gamma_B = 2\beta x_A dx_A

. Substituting:

x_A(-2\alpha x_B) + x_B(2\beta x_A) = 0

, which simplifies to

2x_A x_B(\beta - \alpha) = 0

. For this to hold for all compositions,

\alpha = \beta

. Choice A reaches the right conclusion but with incorrect reasoning about symmetry. Choice B incorrectly suggests the parameters sum to zero. Choice D introduces an arbitrary constant.

Question 11

When temperature is varied at constant pressure and composition in a binary mixture, the Gibbs-Duhem equation becomes $x_1 d\mu_1 + x_2 d\mu_2 = -S_m dT$ . If the partial molar entropy of component 1 decreases by $8.5 \text{ J/(mol·K)}$ when temperature increases by $10 \text{ K}$ , and $x_1 = 0.35$ , what change in partial molar entropy of component 2 is required for consistency?

An increase of $12.8 \text{ J/(mol·K)}$ to balance the entropy change through weighted compensation
An increase of $4.6 \text{ J/(mol·K)}$ due to the complementary entropy relationship in binary systems (correct answer)
A decrease of $4.6 \text{ J/(mol·K)}$ to maintain the required thermodynamic balance
A decrease of $8.5 \text{ J/(mol·K)}$ to mirror the behavior of component 1

Explanation: When you encounter Gibbs-Duhem equation problems, you're dealing with the fundamental constraint that chemical potentials in a mixture cannot change independently—they must satisfy specific thermodynamic relationships. Starting with the given Gibbs-Duhem equation at constant pressure:

x_1 d\mu_1 + x_2 d\mu_2 = -S_m dT

. Since

\left(\frac{\partial \mu_i}{\partial T}\right)_P = -S_{m,i}

(the partial molar entropy), we can rewrite this as:

-x_1 S_{m,1} dT - x_2 S_{m,2} dT = -S_m dT

, which simplifies to

x_1 S_{m,1} + x_2 S_{m,2} = S_m

. For changes in partial molar entropies:

x_1 \Delta S_{m,1} + x_2 \Delta S_{m,2} = \Delta S_m

. However, for a binary mixture at constant composition and pressure,

\Delta S_m = 0

(the total molar entropy change depends only on the pure components' entropy changes, which cancel in this constraint). Therefore:

x_1 \Delta S_{m,1} + x_2 \Delta S_{m,2} = 0

Given

x_1 = 0.35

, so

x_2 = 0.65

, and

\Delta S_{m,1} = -8.5 \text{ J/(mol·K)}

0.35(-8.5) + 0.65(\Delta S_{m,2}) = 0

\Delta S_{m,2} = \frac{8.5 \times 0.35}{0.65} = +4.6 \text{ J/(mol·K)}

Answer B is correct—component 2's partial molar entropy must increase by 4.6 J/(mol·K). Answer A uses incorrect weighting factors. Answer C has the wrong sign. Answer D incorrectly assumes equal magnitude changes regardless of mole fractions. Remember: In Gibbs-Duhem problems, changes in intensive properties are weighted by mole fractions and must sum to maintain thermodynamic consistency.