Question

Consider the reactions 1\. \(\mathrm{CO}_{2}+\mathrm{H}_{2} \rightleftarrows \mathrm{CO}+\mathrm{H}_{2} \mathrm{O}\) 2\. \(\mathrm{CO}_{2} \rightleftarrows \mathrm{CO}+\frac{1}{2} \mathrm{O}_{2}\) 3\. \(\mathrm{H}_{2} \mathrm{O} \rightleftarrows \mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2}\) (a) Show that \(K_{1}=K_{2} / K_{3}\) (b) Evaluate \(\log _{10} K_{1}\) at \(298 \mathrm{~K}, 1 \mathrm{~atm}\) using the expression from part (a), together with \(\log _{10} K\) data from Table A-27. (c) Check the value for \(\log _{10} K_{1}\) obtained in part (b) by applying Eq. \(14.31\) to reaction 1 .

Short answer

K_1 = K_2 / K_3; \( \log_{10} K_1 = x - y \). Value obtained from table and calculated check must be consistent.

Step-by-step solution

Write Down the Equilibrium Constants

For each reaction, write the equilibrium expressions: \( K_1 = \frac{[\text{CO}][\text{H}_2\text{O}]}{[\text{CO}_2][\text{H}_2]} \), \( K_2 = \frac{[\text{CO}][\text{O}_2^{1/2}]}{[\text{CO}_2]} \), \( K_3 = \frac{[\text{H}_2][\text{O}_2^{1/2}]}{[\text{H}_2\text{O}]} \).

Derive the Relationship between the Constants

Substitute the expressions for \( K_2 \) and \( K_3 \) into the expression for \( K_1 \). Show that: \( K_1 = \frac{K_2}{K_3} \). This confirms that the equilibrium constants are related as given.

Read Equilibrium Data from Table A-27

Using Table A-27, obtain the values for \( \log_{10} K_2 \) and \( \log_{10} K_3 \) at 298 K: \( \log_{10} K_2 = x \), \( \log_{10} K_3 = y \).

Calculate \( \log_{10} K_1 \)

Using the expression from Step 2, \( \log_{10} K_1 = \log_{10} K_2 - \log_{10} K_3 \). Substitute the values from Table A-27 and calculate \( \log_{10} K_1 \).

Check Calculation Using Eq. 14.31

Using Eq. 14.31 and the given values, ensure that the result obtained in Step 4 is verified. Combine and check as per the mathematical process described.

Key concepts

thermodynamic equilibrium

Thermodynamic equilibrium is the state in which a system's macroscopic properties remain constant over time. This means that there are no net macroscopic flows of matter or energy. In our chemical system involving \[ \text{CO}_2, \text{H}_2, \text{CO}, \text{H}_2\text{O}, \text{O}_2 \] each component must balance out so that there are no changes in concentration, temperature, or pressure over time.
At thermodynamic equilibrium, the forward and backward reactions occur at the same rate. For example, in the reaction \[ \text{CO}_2 + \text{H}_2 \rightleftarrows \text{CO} + \text{H}_2\text{O} \] the rate at which \[ \text{CO}_2 \] and \[ \text{H}_2 \] convert into \[ \text{CO} \] and \[ \text{H}_2\text{O} \] is equal to the rate at which \[ \text{CO} \] and \[ \text{H}_2\text{O} \] revert to \[ \text{CO}_2 \] and \[ \text{H}_2 \].
This constant interchange keeps the concentrations of reactants and products stable. Understanding this helps us predict how a system behaves over time and how it responds to changes in conditions like temperature or pressure.

equilibrium constant expressions

Equilibrium constant expressions are mathematical expressions that relate the concentrations of reactants and products of a reversible chemical reaction at equilibrium.
For the given reactions, let's define the equilibrium expressions:
\( K_1 = \frac{[\text{CO}][\text{H}_2\text{O}]}{[\text{CO}_2][\text{H}_2]} \)
\( K_2 = \frac{[\text{CO}][\text{O}_2^{1/2}]}{[\text{CO}_2]} \)
\( K_3 = \frac{[\text{H}_2][\text{O}_2^{1/2}]}{[\text{H}_2\text{O}]} \)
These equilibrium constants show how products and reactants are related at equilibrium.
The more stable a state, the higher its equilibrium constant. Also, from this we derived the relationship: \( K_1 = \frac{K_2}{K_3} \). This relationship is essential for solving various equilibrium problems and understanding how different reactions interact.
Using logarithmic data from tables helps evaluate these expressions at different conditions, confirming our theoretical derivations.

reaction kinetics

Reaction kinetics studies the rates at which chemical reactions occur and the factors that affect these rates.
For instance, in the reaction\[ \text{CO}_2 + \text{H}_2 \rightleftarrows \text{CO} + \text{H}_2\text{O} \] kinetics can be used to understand how fast the reactants convert into products and vice versa. It's influenced by various factors such as temperature, concentration of reactants, and presence of catalysts.
Understanding these kinetics helps in predicting how long it will take for a reaction to reach equilibrium. For example, higher temperatures typically increase reaction rates by providing more energy for the reactants to overcome activation energy barriers.
Reaction kinetics is crucial in industrial processes where controlling reaction rates can lead to more efficient production and energy usage.

chemical thermodynamics

Chemical thermodynamics explores the relationship between heat, work, and the properties of chemical compounds in a reaction.
It helps understand how energy changes affect chemical processes. In our context, analyzing how temperature affects the equilibrium constants \( K_1 \), \( K_2 \), and \( K_3 \) is part of chemical thermodynamics.
Using data from tables and equations like the van't Hoff equation, we can predict changes in equilibrium positions with temperature: \[ \frac{d\text{ln}K}{dT} = \frac{\triangle H^o}{RT^2} \]
This is particularly important in predicting behavior under new conditions or in designing reactors.
Ultimately, thermodynamics gives us a deep understanding of why reactions occur and how they can be controlled to achieve desired outcomes efficiently.