## X2 x

Given knowledge of the equilibrium constant, this allows x to be solved for using the quadratic equation.

Exercise 8.2.1 Carry out the algebra to determine x in the above example, and describe how it behaves as keq is varied from very small to very large values. Do you ever completely use up the reactants?

When the activity is something other than simply concentration or density, the reaction rate is no longer the time-derivative of activity, and the forward and back reaction rates should be written abstractly as R± and R_. These are measured in various ways, depending on the nature of the reaction. For example, if the reaction is between a gas or dissolved substance and the surface of a solid, it would be typical to characterized the reaction rate in terms of Moles per unit time, per unit surface area of the reacting solid. The characterization of reaction rate affects how one calculates the approach to equilibrium, but it does not affect the equilibrium itself, since the equilibrium is defined by R± + R_ _ 0, which yields the same equilibrium conditions on activity as before.

Rate and equilibrium equations generalize to reactions involving more reactants and more products in the obvious way. For example, a reaction of the form A+2B+3C has a rate proportional to [A][B]2[C]3. The equations for equilibria are modified correspondingly. Sometimes one of the "reactants" is a molecule that participates in the reaction only to the extent of providing some extra energy by collision. A reaction like that is written, for example, as A + B + M ^ C + M, where M is a generic colliding molecule; in kinetics, its activity [M] would generally be the number density of molecules of any sort available for collision with the other reactants. For example, an atmospheric reaction between substances A and B might proceed more rapidly in a background of N2 even though N2 does not itself react with either A or B. It is also common in atmospheric chemistry for one of the "reactants" to be a photon, usually one within a designated range of frequencies. A photon is designated in reactions by the symbol hv, as in the dissociation reaction AB + hv ^ A+B. The symbol used for photons is meant to be reminiscent of the energy carried by a photon of frequency v. The activity of the photons is usually the number flux of photons having the correct energy range to react. This is obtained by dividing the energy flux at frequency v by the energy of an individual photon hv, then summing up the result over all frequencies involved in the reaction.

When one or more of the reactants is in the form of a pure, solid body - for example a lump of solid substance A reacting with a gaseous substance B within a given volume - the quantity of the solid within the volume is not the limiting factor in determining the availability of molecules of the solid substance to react. Rather, in this case it is the area of solid exposed to other reactants that counts. In such a case, as a matter of convention the activity of the pure solid substance is set to unity,and the effect of available surface area for reaction is taken into account by expressing the reaction rate as Moles per unit area of contact per unit time. If the pure phase is reacting with a gas, the reaction rate expressed this way will depend on the partial pressure of the gas, but not on the amount of solid present (so long as there is some present and it is in contact with the gas). Equilibrium at any given temperature yields a unique value of partial pressure, just as Clausius-Clapeyron yields a unique vapor pressure in equilibrium with the solid or liquid form of a substance, regardless of how much of the solid or liquid form is present. The same behaviour applies when a pure solid phase reacts with a substance B dissolved in a fluid in contact with the solid - the reaction rate per unit area will depend on the concentration of B, but not on the amount of solid present, and equilibrium at any given temperature will yield a unique value of [B], regardless of how much solid phase is present, as long as there is some. We will encounter this situation soon, in our study of weathering reactions.

For all reactions considered in this chapter, the activity will be either a concentration (or its equivalent, such as a partial pressure), or it will be unity for pure condensed phase reactants. The reader will not need to be concerned with more exotic expressions for activity, though it is good to be aware that they exist.

The equilibrium and rate constants depend strongly on temperature. Most of the temperature dependence of the rate constants k+ and is captured by the Arrhenius law, which states that k(T) = A ■ exp(-E/R*T) where A is a constant and E is a quantity known as the activation energy, measured in J/Mole when the Arrhenius law is written in this form. The temperature dependence can be fit a bit more accurately if the constant A is replaced by a power law in T, but for most of our purposes the unmodified Arrhenius law suffices. Since the equilibrium constant is k+/k-, it follows that the equilibrium constant has temperature dependence Aeq exp(-AE/R*T), where AE is the difference in the activation energy between the reactants and the products, and Aeq is the ratio of the prefactors of the forward and back reactions. While activation energy must be positive, and hence all reactions speed up with temperature, AE may be either positive or negative; therefore the equilibrium constant may either increase or decrease with temperature, accordingly as whether AE is positive or negative.

The activation energy in the Arrhenius law is a generalization of the concept of latent heat, which we have discussed in connection with phase transitions, and which appears in the Clausius-Clapeyron equation in a manner similar to the way activiation energy appears in the Arrhenius law. Indeed, a phase transition between, say, the gas and liquid forms of a substance can be considered as a binary reaction in which two molecules of "gas" substance collide and react to form one molecule of "liquid" reactant. To relate the latent heat L to the activation energy in the Arrhenius law, we need only observe that Clausius-Clapeyron was written using the gas constant specific to the gas in question, rather than the universal gas constant. Rewriting in terms of the universal gas constant, the temperature dependence of saturation vapor pressure becomes proportional to exp(-(L • M)/R*T), where M is the molecular weight of the gas. Thus, the activation energy for condensation of water vapor into liquid water is 4.49 • 107 J /Mole.

The constants also depend on the total pressure at which the reaction takes place, but the pressure dependence is usually weak over the range of pressures of interest in atmospheric problems.
The passage primarily discusses chemical equilibrium and reaction rates, including the role of the equilibrium constant, the quadratic equation for solving for \( x \), and the effects of temperature and pressure on reaction rates. It also includes an exercise that asks the reader to solve for \( x \) and analyze its behavior as the equilibrium constant \( K_{eq} \) changes.
single_question

the passage contains a single mathematical question in the form of an exercise, asking the reader to carry out the algebra to determine \( x \) and describe its behavior as \( k_{eq} \) varies. however, no answer to this question is provided in the text.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
## X2 x

Given knowledge of the equilibrium constant, this allows x to be solved for using the quadratic equation.

Exercise 8.2.1 Carry out the algebra to determine x in the above example, and describe how it behaves as keq is varied from very small to very large values. Do you ever completely use up the reactants?

When the activity is something other than simply concentration or density, the reaction rate is no longer the time-derivative of activity, and the forward and back reaction rates should be written abstractly as R± and R_. These are measured in various ways, depending on the nature of the reaction. For example, if the reaction is between a gas or dissolved substance and the surface of a solid, it would be typical to characterized the reaction rate in terms of Moles per unit time, per unit surface area of the reacting solid. The characterization of reaction rate affects how one calculates the approach to equilibrium, but it does not affect the equilibrium itself, since the equilibrium is defined by R± + R_ _ 0, which yields the same equilibrium conditions on activity as before.

Rate and equilibrium equations generalize to reactions involving more reactants and more products in the obvious way. For example, a reaction of the form A+2B+3C has a rate proportional to [A][B]2[C]3. The equations for equilibria are modified correspondingly. Sometimes one of the "reactants" is a molecule that participates in the reaction only to the extent of providing some extra energy by collision. A reaction like that is written, for example, as A + B + M ^ C + M, where M is a generic colliding molecule; in kinetics, its activity [M] would generally be the number density of molecules of any sort available for collision with the other reactants. For example, an atmospheric reaction between substances A and B might proceed more rapidly in a background of N2 even though N2 does not itself react with either A or B. It is also common in atmospheric chemistry for one of the "reactants" to be a photon, usually one within a designated range of frequencies. A photon is designated in reactions by the symbol hv, as in the dissociation reaction AB + hv ^ A+B. The symbol used for photons is meant to be reminiscent of the energy carried by a photon of frequency v. The activity of the photons is usually the number flux of photons having the correct energy range to react. This is obtained by dividing the energy flux at frequency v by the energy of an individual photon hv, then summing up the result over all frequencies involved in the reaction.

When one or more of the reactants is in the form of a pure, solid body - for example a lump of solid substance A reacting with a gaseous substance B within a given volume - the quantity of the solid within the volume is not the limiting factor in determining the availability of molecules of the solid substance to react. Rather, in this case it is the area of solid exposed to other reactants that counts. In such a case, as a matter of convention the activity of the pure solid substance is set to unity,and the effect of available surface area for reaction is taken into account by expressing the reaction rate as Moles per unit area of contact per unit time. If the pure phase is reacting with a gas, the reaction rate expressed this way will depend on the partial pressure of the gas, but not on the amount of solid present (so long as there is some present and it is in contact with the gas). Equilibrium at any given temperature yields a unique value of partial pressure, just as Clausius-Clapeyron yields a unique vapor pressure in equilibrium with the solid or liquid form of a substance, regardless of how much of the solid or liquid form is present. The same behaviour applies when a pure solid phase reacts with a substance B dissolved in a fluid in contact with the solid - the reaction rate per unit area will depend on the concentration of B, but not on the amount of solid present, and equilibrium at any given temperature will yield a unique value of [B], regardless of how much solid phase is present, as long as there is some. We will encounter this situation soon, in our study of weathering reactions.

For all reactions considered in this chapter, the activity will be either a concentration (or its equivalent, such as a partial pressure), or it will be unity for pure condensed phase reactants. The reader will not need to be concerned with more exotic expressions for activity, though it is good to be aware that they exist.

The equilibrium and rate constants depend strongly on temperature. Most of the temperature dependence of the rate constants k+ and is captured by the Arrhenius law, which states that k(T) = A ■ exp(-E/R*T) where A is a constant and E is a quantity known as the activation energy, measured in J/Mole when the Arrhenius law is written in this form. The temperature dependence can be fit a bit more accurately if the constant A is replaced by a power law in T, but for most of our purposes the unmodified Arrhenius law suffices. Since the equilibrium constant is k+/k-, it follows that the equilibrium constant has temperature dependence Aeq exp(-AE/R*T), where AE is the difference in the activation energy between the reactants and the products, and Aeq is the ratio of the prefactors of the forward and back reactions. While activation energy must be positive, and hence all reactions speed up with temperature, AE may be either positive or negative; therefore the equilibrium constant may either increase or decrease with temperature, accordingly as whether AE is positive or negative.

The activation energy in the Arrhenius law is a generalization of the concept of latent heat, which we have discussed in connection with phase transitions, and which appears in the Clausius-Clapeyron equation in a manner similar to the way activiation energy appears in the Arrhenius law. Indeed, a phase transition between, say, the gas and liquid forms of a substance can be considered as a binary reaction in which two molecules of "gas" substance collide and react to form one molecule of "liquid" reactant. To relate the latent heat L to the activation energy in the Arrhenius law, we need only observe that Clausius-Clapeyron was written using the gas constant specific to the gas in question, rather than the universal gas constant. Rewriting in terms of the universal gas constant, the temperature dependence of saturation vapor pressure becomes proportional to exp(-(L • M)/R*T), where M is the molecular weight of the gas. Thus, the activation energy for condensation of water vapor into liquid water is 4.49 • 107 J /Mole.

The constants also depend on the total pressure at which the reaction takes place, but the pressure dependence is usually weak over the range of pressures of interest in atmospheric problems.
user
The passage primarily discusses chemical equilibrium and reaction rates, including the role of the equilibrium constant, the quadratic equation for solving for \( x \), and the effects of temperature and pressure on reaction rates. It also includes an exercise that asks the reader to solve for \( x \) and analyze its behavior as the equilibrium constant \( K_{eq} \) changes.

TYPE: SINGLE_QUESTION

The passage contains a single mathematical question in the form of an exercise, asking the reader to carry out the algebra to determine \( x \) and describe its behavior as \( K_{eq} \) varies. However, no answer to this question is provided in the text.