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4.1. Chemical Equilibrium

Sarah May Sibug-Torres・106 minutes read

The lecture discusses the thermodynamics of multicomponent systems, emphasizing the significance of partial molar quantities in determining extensive properties like Gibbs free energy, which are not simply additive due to component interactions. It also covers the relationships among equilibrium constants, temperature effects, and ionic strength, illustrating how these factors influence solubility and chemical reactions in various systems.

Insights

  • The lecture emphasizes the significance of understanding equilibrium in thermodynamic systems, particularly in multicomponent systems, where the concept of partial molar quantities is introduced to analyze extensive properties like Gibbs free energy.
  • Extensive properties in multicomponent systems are influenced by variables such as temperature, pressure, and the number of moles of each component, with their total differential expressed through a complex mathematical formula that highlights the interdependence of these factors.
  • Partial molar quantities play a crucial role in determining how extensive properties change with variations in the system's composition, and they are represented with a bar notation, underscoring their importance in accurately describing system behavior.
  • The total extensive property of a mixture can be computed using a summation rule, which combines the partial molar volumes of each component weighted by their respective mole quantities, illustrating that these properties are not simply additive due to component interactions.
  • The chemical potential of a component, defined as its partial molar Gibbs free energy, is essential for understanding how each component contributes to the total free energy of the system, impacting chemical reactions and processes significantly.
  • The concept of activity is introduced to represent the effective concentration or pressure of a substance relative to its standard state, leading to a broader application of chemical potential in various systems beyond ideal gases.
  • Le Chatelier's principle is discussed, explaining how equilibrium systems respond to changes in concentration, pressure, or temperature, with specific examples illustrating how adding or removing reactants or products shifts the equilibrium position.
  • The relationship between ionic strength and solubility is highlighted, showing that as ionic strength increases, the activity coefficients decrease, which in turn affects the concentration equilibrium constant and the solubility of ionic compounds, demonstrating the complexity of interactions in solutions.

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Recent questions

  • What is a chemical potential?

    A chemical potential is the change in Gibbs free energy of a system when an additional amount of a substance is introduced, at constant temperature and pressure. It reflects how the energy of a system changes with the addition of a component, indicating the tendency of that component to react or change state. In multicomponent systems, the chemical potential is crucial for understanding the behavior of each component in relation to the overall system. It is mathematically defined as the partial molar Gibbs free energy, which allows for the assessment of how the presence of one substance influences the free energy of the entire mixture. This concept is essential in thermodynamics and plays a significant role in predicting the direction of chemical reactions and phase changes.

  • How do you calculate Gibbs free energy change?

    The Gibbs free energy change (ΔG) for a chemical reaction can be calculated using the formula ΔG = G_products - G_reactants, where G represents the Gibbs free energy of the respective components involved in the reaction. This equation allows for the determination of whether a reaction is spontaneous; if ΔG is negative, the reaction is thermodynamically favorable and will proceed in the forward direction. Additionally, for a multi-component system, the total Gibbs free energy can be determined using the summation rule, which involves multiplying the chemical potential of each component by the number of moles present. This comprehensive approach to calculating ΔG is vital for understanding the energetics of chemical processes and predicting the equilibrium state of reactions.

  • What is the equilibrium constant?

    The equilibrium constant (K) is a numerical value that expresses the ratio of the concentrations of products to reactants at equilibrium for a given chemical reaction, under specific conditions of temperature and pressure. It is derived from the activities of the reactants and products, and for gaseous systems, it can be expressed in terms of partial pressures (K_P) or concentrations (K_C). The equilibrium constant provides insight into the extent of a reaction; a large K value indicates that products are favored at equilibrium, while a small K value suggests that reactants are favored. The relationship between K and the Gibbs free energy change (ΔG) is also significant, as it connects thermodynamic properties with chemical equilibria, allowing for predictions about the direction and extent of reactions.

  • What is Le Chatelier's principle?

    Le Chatelier's principle states that if a stress is applied to a system at equilibrium, the system will adjust to counteract that stress and restore a new equilibrium. This principle applies to changes in concentration, pressure, volume, and temperature. For instance, adding more reactant will shift the equilibrium towards the products, while removing a product will shift it towards the reactants. Similarly, increasing the pressure in a gaseous system will favor the side of the reaction with fewer moles of gas. This principle is fundamental in predicting how changes in conditions affect the position of equilibrium, making it a crucial concept in chemical thermodynamics and reaction dynamics.

  • What is ionic strength?

    Ionic strength is a measure of the total concentration of ions in a solution, which affects the behavior of ions and their interactions. It is calculated using the formula: Ionic Strength = 0.5 * Σ(ci * zi²), where ci is the concentration of each ion and zi is its charge. Ionic strength plays a significant role in determining the activity coefficients of ions, which account for non-ideal behavior in solutions. As ionic strength increases, the activity coefficients typically decrease, leading to changes in solubility and reaction equilibria. Understanding ionic strength is essential for accurately predicting the behavior of ionic compounds in solution, particularly in contexts such as solubility equilibria and precipitation reactions.

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Summary

00:00

Understanding Partial Molar Quantities in Thermodynamics

  • The lecture continues the discussion on equilibrium in thermodynamic systems, focusing on the thermodynamics of multicomponent systems and introducing the concept of partial molar quantities, which are essential for understanding extensive properties like Gibbs free energy in these systems.
  • Extensive properties in multicomponent systems depend on independent variables such as temperature, pressure, and the number of moles of each component, with the total differential expressed as \( dX = \left( \frac{\partial X}{\partial T} \right)_{P,n} dT + \left( \frac{\partial X}{\partial P} \right)_{T,n} dP + \sum \left( \frac{\partial X}{\partial n_i} \right)_{T,P,n_{j \neq i}} dn_i \).
  • Partial molar quantities indicate how an extensive property varies with changes in the system's molar composition at constant temperature and pressure, represented by a bar notation over the extensive property symbol.
  • The importance of partial molar quantities lies in their ability to determine the extensive property of the entire system, as extensive properties in multicomponent systems are not simply additive due to interactions between components, exemplified by the non-additive nature of volumes when mixing ethanol and water.
  • The total extensive property can be calculated using the summation rule, which states that the total volume of a mixture is the sum of the partial molar volumes of each component multiplied by their respective number of moles, expressed as \( V = \sum \bar{V}_i n_i \).
  • The total Gibbs free energy of a multicomponent system is similarly calculated using the summation of partial molar Gibbs free energies, given by \( G = \sum \bar{G}_i n_i \), where \( \bar{G}_i \) is the partial molar Gibbs free energy of component \( i \).
  • The chemical potential of a component in a mixture is defined as the partial molar Gibbs free energy, which indicates each component's contribution to the total free energy, and is crucial for understanding chemical reactions and processes.
  • To determine the chemical potential of a component in a mixture, one starts from its reference state (pure state) and uses the relationship \( \mu_i = \mu_{i}^{\text{pure}} + RT \ln \left( \frac{P_i}{P_{i}^{\text{standard}}} \right) \), where \( P_i \) is the partial pressure of the component in the mixture.
  • For real gases, the concept of fugacity is introduced, which accounts for deviations from ideal gas behavior, expressed as \( f_i = P_i \phi_i \), where \( \phi_i \) is the fugacity coefficient, and the chemical potential is adjusted to \( \mu_i = \mu_{i}^{\text{pure}} + RT \ln \left( \frac{f_i}{P_{i}^{\text{standard}}} \right) \).
  • The concept of activity is generalized to represent the effective concentration or pressure of a substance relative to its standard state, leading to the expression \( \mu_i = \mu_{i}^{\text{pure}} + RT \ln a_i \), where \( a_i \) is the activity of component \( i \), applicable to various systems beyond gases.

18:26

Understanding Gibbs Free Energy and Equilibrium

  • The fugacity coefficient of a component (I) can be expressed as the ratio of its pressure (P_I) to the standard pressure (P°), or alternatively, as the activity coefficient (γ_I) times the pressure of I over the standard pressure, indicating deviations from ideality for gases and solutes.
  • The activity of a solute can be defined as the activity coefficient (γ_I) multiplied by the concentration (C_I) over the standard state concentration, which is 1 M for solutes, while for gases, the standard state is 1 atmosphere.
  • To ensure that activity is unitless, pressures must be expressed in atmospheres and concentrations in molarity, maintaining consistency with standard state units.
  • The activities of pure solids and liquids are generally considered to be equal to one, simplifying calculations for condensed phases in a system.
  • The Gibbs free energy change (ΔG) for a chemical reaction can be calculated using the expression ΔG = G_products - G_reactants, where G represents the Gibbs free energy of the respective components.
  • For a multi-component system, the total Gibbs free energy can be determined using the summation rule, which involves multiplying the chemical potential of each component by the number of moles present.
  • The chemical potential of a component can be expressed as the pure chemical potential plus RT * ln(activity), where R is the gas constant and T is the temperature in Kelvin.
  • The final expression for Gibbs free energy change can be rewritten as ΔG = ΔG°_reaction + RT ln(Q_A), where Q_A is the reaction quotient in terms of activities, allowing for the assessment of the system's free energy based on the actual amounts of reactants and products.
  • At equilibrium, ΔG equals zero, and the reaction quotient (Q_A) equals the equilibrium constant (K_A), leading to the relationship ΔG°_reaction = -RT ln(K_A), which connects standard Gibbs free energy change to the equilibrium constant.
  • The equilibrium constant (K_A) can be expressed in terms of activities for gases, where K_A = (activity of products) / (activity of reactants), and for ideal gases, K_A can be approximated by K_P, which is based on partial pressures, reinforcing the assumption of ideal behavior in gas systems.

36:19

Understanding Gas Equilibrium Constants and Calculations

  • The concentration of a gas can be expressed as \( C = \frac{n}{V} \), where \( n \) is the number of moles and \( V \) is the volume, leading to the equation \( P = CR \cdot T \), where \( R \) is the gas constant and \( T \) is the temperature in Kelvin.
  • The equilibrium constant \( K_C \) for a gaseous system can be derived by substituting partial pressures with concentrations, resulting in \( K_C = \frac{[NO_2] \cdot RT/P^0}{[N_2]^{1/2} \cdot RT/P^0 \cdot [O_2]^{1/2}} \), where \( P^0 \) is the standard pressure.
  • The relationship between the equilibrium constants \( K_P \) (in terms of partial pressures) and \( K_C \) (in terms of concentrations) is given by \( K_P = K_C \cdot (RT)^{\Delta n} \), where \( \Delta n \) is the change in the number of moles of gas in the balanced equation.
  • \( K_P \) is unitless because all partial pressures are divided by the standard pressure, while \( K_C \) has units of \( \text{M}^{-\Delta n} \), where \( \Delta n \) is the change in moles.
  • For solution systems, the equilibrium constant \( K_a \) for the dissociation of acetic acid in water is expressed as \( K_a = \frac{[H_3O^+] \cdot [CH_3COO^-]}{[CH_3COOH] \cdot [H_2O]} \), with the activity of water considered equal to one.
  • In dilute ideal solutions, the activity coefficients approach one, allowing \( K_a \) to be approximated by \( K_{Q'} \), which is the equilibrium constant expressed in terms of molar concentrations.
  • For mixed systems, such as the reaction of carbon dioxide with water to form carbonic acid, the equilibrium constant can be expressed as \( K_a = \frac{[H_2CO_3]}{[CO_2] \cdot [H_2O]} \), with water's activity set to one.
  • The properties of equilibrium constants include that reversing a reaction inverts the constant, multiplying the equation by a coefficient raises the constant to that power, and adding reactions results in the product of their constants.
  • The equilibrium constant can be determined from equilibrium composition, changes in pressure, or degree of dissociation, using an ICE table to track initial, change, and equilibrium states.
  • For a reaction starting with 1 part nitrogen and 3 parts hydrogen, if the equilibrium composition contains 20% mole ammonia at a total pressure of 10 atmospheres, the partial pressures can be calculated to find \( K_P \) using the relationship between the components at equilibrium.

53:52

Equilibrium Constants and Partial Pressures Explained

  • The partial pressure of ammonia at equilibrium is 2 atmospheres, leading to a total sum of the partial pressures of nitrogen and hydrogen being 8 atmospheres, with nitrogen's partial pressure also calculated to be 2 atmospheres.
  • The partial pressure of hydrogen is determined to be 6 atmospheres, calculated as three times the partial pressure of nitrogen.
  • The equilibrium constant \( K_P \) is calculated using the formula \( K_P = \frac{(P_{NH_3})^2}{(P_{N_2})(P_{H_2})^3} \), resulting in a value of \( 9.26 \times 10^{-3} \), which is unitless due to the cancellation of pressure units.
  • The next problem involves calculating \( K_P \) from changes in pressure, starting with 300 mmHg of nitrogen and 250 mmHg of chlorine gas, with a total equilibrium pressure of 420 mmHg.
  • The equilibrium partial pressures are expressed as \( 300 - x \) for nitrogen, \( 250 - \frac{1}{2}x \) for chlorine, and \( x \) for the product, leading to the equation \( (300 - x) + (250 - \frac{1}{2}x) + x = 420 \) mmHg.
  • Solving for \( x \) gives a value of 260 mmHg for the product, with the equilibrium partial pressures for nitrogen and chlorine calculated as 40 mmHg and 120 mmHg, respectively.
  • The expression for \( K_P \) is then formulated as \( K_P = \frac{(P_{NOCl})}{(P_{NO})(P_{Cl_2})^2} \), resulting in a calculated value of 1636, which is also unitless.
  • To find \( K_C \), the relationship \( K_C = \frac{K_P}{(RT/P)^{\Delta n}} \) is used, with \( R = 0.08206 \) L·atm/(mol·K) and temperature converted to Kelvin (303.15 K), yielding \( K_C = 81.66 \) M².
  • The degree of dissociation \( \alpha \) is defined as the number of moles dissociated over the initial number of moles, and expressions for \( K_P \) are derived in terms of \( \alpha \) and total pressure \( P_T \).
  • Le Chatelier's principle is introduced, stating that if a stress is applied to an equilibrium system, the system will shift to counteract the stress, with various stresses including changes in reactant/product amounts, pressure, volume, and temperature affecting the equilibrium state.

01:12:51

Equilibrium Shifts in Gas Reactions Explained

  • The equilibrium constant (Kc) for the reaction involving hydrogen iodide gas and hydrogen sulfide gas is expressed as the concentration of hydrogen iodide over the concentration of hydrogen sulfide, omitting solids since their activities equal one when present.
  • Adding reactant (hydrogen sulfide) increases its concentration, causing the reaction quotient (Q) to become less than Kc, resulting in a forward shift in equilibrium to produce more product.
  • Conversely, adding product increases the numerator in the Q expression, making Q greater than Kc, which leads to a backward shift in equilibrium to produce more reactant.
  • Removing reactant decreases the denominator in Q, causing Q to exceed Kc, resulting in a backward shift to restore reactant concentration.
  • Removing product decreases the numerator in Q, making Q less than Kc, which causes a forward shift to restore product concentration.
  • Changes in the amounts of condensed phase reactants or products (like iodine and sulfur solids) do not affect the equilibrium since their activities remain equal to one, provided they are present.
  • If a solid reactant is completely removed, its activity becomes zero, leading to Q becoming infinite, which results in a backward shift to reform the lost reactant.
  • Removing solid sulfur results in its activity becoming zero, making Q equal to zero, which leads to a forward shift to reform the lost product.
  • Increasing pressure or decreasing volume in a gaseous system causes Q to become larger than Kc, resulting in a backward shift towards the side with fewer moles of gas.
  • Adding inert gases at constant pressure increases the volume, leading to a shift towards the side with more moles of gas, while adding inert gases at constant volume does not change the equilibrium since partial pressures remain unchanged.

01:30:06

Temperature Effects on Reaction Equilibrium Constants

  • Increasing the temperature in an exothermic reaction causes a backward shift in equilibrium, leading to the formation of more reactants and a decrease in the equilibrium constant (K).
  • Conversely, decreasing the temperature in an exothermic reaction results in a forward shift, producing more products and increasing the equilibrium constant (K).
  • In an endothermic reaction, heat is treated as a reactant; thus, increasing the temperature causes a forward shift, increasing the equilibrium constant (K), while decreasing the temperature results in a backward shift and a decrease in K.
  • The relationship between changes in temperature and the equilibrium constant can be expressed mathematically using the Gibbs-Helmholtz equation, which connects the change in Gibbs free energy (ΔG) to the equilibrium constant (K).
  • The derivative of the natural logarithm of the equilibrium constant (d ln K) with respect to temperature (T) can be expressed as ΔH/(R*T²), where ΔH is the change in enthalpy and R is the gas constant.
  • The Van 't Hoff equation relates the change in equilibrium constants (K) with temperature, allowing for the calculation of ΔH when K values at two different temperatures are known.
  • For the dissociation of ammonium chloride, the balanced chemical equation is NH4Cl(s) ⇌ NH3(g) + HCl(g), and the equilibrium constant (Kp) can be calculated using the partial pressures of the products.
  • At 700 K, the total pressure of the system is 608 kPa, leading to a calculated Kp of 9, while at 732 K, the total pressure is 1115 kPa, resulting in a Kp of 3027.
  • The standard Gibbs free energy (ΔG) can be calculated using the formula ΔG = -RT ln Kp, yielding values of -1279 kJ/mol at 700 K and -2.75 kJ/mol at 732 K, indicating an endothermic reaction.
  • The positive enthalpy change (ΔH) calculated using the Van 't Hoff equation confirms that the reaction is endothermic, as increasing the temperature results in an increase in the equilibrium constant (K).

01:48:29

Ionic Interactions and Equilibrium Shifts Explained

  • Adding equal amounts of ferric nitrate solution increases the amount of reactant, leading to a forward shift in equilibrium, which is observed as a darker solution.
  • The addition of silver nitrate introduces silver ions that react with diacetate ions to form a precipitate, decreasing the amount of reactant and causing a backward shift in equilibrium, resulting in a lighter solution and the formation of a white precipitate.
  • When potassium nitrate is added, it does not react directly with any species but still results in a lighter solution, indicating a backward shift in equilibrium; this phenomenon is known as the "uncommon ion effect."
  • The addition of potassium nitrate to a saturated solution of silver chloride increases solubility, demonstrating a forward shift in equilibrium, contrasting with the previous example involving complexation.
  • The concept of ionic interactions is crucial in equilibrium systems, as the presence of other ions can affect the attraction between cations and anions, leading to changes in solubility and precipitation.
  • In dilute solutions, ions interact fully, but in non-ideal solutions with higher ion concentrations, a counter-ion atmosphere forms around ions, reducing their effective concentration and attraction, which is described by the Debye-Hückel theory.
  • Increasing the concentration of solutes, such as potassium nitrate, enhances the ionic atmosphere, promoting dissociation into ions and leading to a forward shift in equilibrium for precipitation reactions.
  • The ionic strength of a solution, which measures the total concentration of ionic charge, can be calculated using the formula: Ionic Strength = 0.5 * Σ(ci * zi²), where ci is the concentration of each ion and zi is its charge.
  • For example, a 0.01 M sodium chloride solution has an ionic strength of 0.01, while a mixture of sodium chloride and sodium sulfate results in an ionic strength of 0.04, demonstrating that ionic strengths are additive.
  • The mean activity coefficient, which accounts for non-ideality in solutions, decreases with increasing ionic strength, affecting the concentration equilibrium constant and solubility of ionic systems, as described by the Debye-Hückel limiting law.

02:05:47

Ionic Strength Effects on Calcium Solubility

  • As ionic strength increases, the activity coefficient decreases, leading to a corresponding increase in the concentration equilibrium constant (KSP Prime), while the thermodynamic equilibrium constant remains constant. This relationship indicates that KSP Prime increases to compensate for the decrease in the activity coefficient (K gamma) as ionic strength rises.
  • A plot of ionic strength versus KSP Prime shows that at zero ionic strength, KSP Prime equals KSP. As ionic strength increases, KSP Prime increases, indicating a rise in the concentrations of silver ions and chloride ions in solution, which shifts the equilibrium forward.
  • To calculate the concentration of calcium ions in equilibrium with a saturated solution of calcium fluoride in water and a 0.1 molar sodium chloride solution, initial assumptions include no other reactions occurring and that KSP Prime equals KSP for simplification.
  • The dissolution of calcium fluoride produces calcium ions (Ca²⁺) and fluoride ions (F⁻), with the equilibrium concentrations represented as X for Ca²⁺ and 2X for F⁻. The thermodynamic equilibrium constant is expressed as the product of the activities of these ions, including their activity coefficients.
  • The mean activity coefficient can be calculated using the ionic strength of the solution, which is determined by the concentrations of calcium and fluoride ions. The initial approximation yields a calcium ion concentration of 2.14 × 10⁻⁴ molar and fluoride ion concentration of 4.27 × 10⁻⁴ molar, leading to an ionic strength of 6.42 × 10⁻⁴ molar.
  • Using the Debye-Hückel limiting law, the mean activity coefficient is calculated as 0.942, which is then used to find the real KSP Prime, resulting in a value of 4.666 × 10¹¹. This indicates an increase in the solubility of calcium fluoride when ionic strength is considered.
  • In a second scenario, when sodium chloride is added to the solution, the new ionic strength is calculated to be 1.642 × 10⁻³ molar. The mean activity coefficient for this ionic strength is found to be 0.909, which is then used to update KSP, yielding a calcium ion concentration of 2.4 × 10⁻⁴ molar.
  • The analysis demonstrates that increasing ionic strength further enhances the concentration of calcium ions in solution, indicating a shift in equilibrium towards the products, and highlights the importance of considering ionic strength and activity coefficients in aqueous solutions.
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