Question

Write the \(K\) p expression for each reaction. a) \(\mathrm{CH}_{4}(\mathrm{~g})+2 \mathrm{O}_{2}(\mathrm{~g}) \rightleftarrows \mathrm{CO}_{2}(\mathrm{~g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) b) \(\mathrm{CH}_{4}(\mathrm{~g})+4 \mathrm{Cl}_{2}(\mathrm{~g}) \rightleftarrows \mathrm{CCl}_{4}(\mathrm{~g})+4 \mathrm{HCl}(\mathrm{g})\)

Short answer

a) \( K_p = \frac{{P_{\mathrm{CO}_2} P_{\mathrm{H}_2\mathrm{O}}^2}}{{P_{\mathrm{CH}_4} P_{\mathrm{O}_2}^2}} \); b) \( K_p = \frac{{P_{\mathrm{CCl}_4} P_{\mathrm{HCl}}^4}}{{P_{\mathrm{CH}_4} P_{\mathrm{Cl}_2}^4}} \).

Step-by-step solution

Understanding Kp Expression

The equilibrium constant for a reaction in terms of partial pressures is denoted as \( K_p \). For a general reaction \( aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g) \), the expression for \( K_p \) is formulated as: \[ K_p = \frac{{P_C^c P_D^d}}{{P_A^a P_B^b}} \] where \( P \) represents partial pressures and the exponents match the stoichiometric coefficients from the balanced equation.

Formulate Kp for Reaction (a)

For the reaction \( \mathrm{CH}_4(g) + 2 \mathrm{O}_2(g) \rightleftharpoons \mathrm{CO}_2(g) + 2 \mathrm{H}_2\mathrm{O}(g) \), apply the formula: \[ K_p = \frac{{P_{\mathrm{CO}_2} P_{\mathrm{H}_2\mathrm{O}}^2}}{{P_{\mathrm{CH}_4} P_{\mathrm{O}_2}^2}} \] Here, \( P_{\mathrm{CO}_2} \), \( P_{\mathrm{H}_2\mathrm{O}} \), \( P_{\mathrm{CH}_4} \), and \( P_{\mathrm{O}_2} \) denote the partial pressures of carbon dioxide, water vapor, methane, and oxygen, respectively.

Formulate Kp for Reaction (b)

For the reaction \( \mathrm{CH}_4(g) + 4 \mathrm{Cl}_2(g) \rightleftharpoons \mathrm{CCl}_4(g) + 4 \mathrm{HCl}(g) \), again apply the formula: \[ K_p = \frac{{P_{\mathrm{CCl}_4} P_{\mathrm{HCl}}^4}}{{P_{\mathrm{CH}_4} P_{\mathrm{Cl}_2}^4}} \] Here, \( P_{\mathrm{CCl}_4} \), \( P_{\mathrm{HCl}} \), \( P_{\mathrm{CH}_4} \), and \( P_{\mathrm{Cl}_2} \) denote the partial pressures of carbon tetrachloride, hydrogen chloride, methane, and chlorine, respectively.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a state in a chemical reaction where the concentrations of reactants and products remain constant over time. It occurs when the rate of the forward reaction equals the rate of the reverse reaction. This balance creates a dynamic but stable system. Even though the individual molecules continue to react, there is no net change in the concentration of substances. In equations, chemical equilibrium is represented with the double arrow symbol \( \rightleftharpoons \). This symbol indicates that both forward and reverse reactions are happening simultaneously.
Equilibrium can be described quantitatively using the equilibrium constant, which varies depending on the form of the reaction. For reactions involving gases, we use \( K_p \), which refers to the equilibrium constant expressed in terms of partial pressures. Understanding chemical equilibrium helps predict the outcome of reactions and know how different conditions can drive the reactions toward products or reactants.
**Factors Affecting Equilibrium:**
- **Concentration**: Changing the concentration of substances can shift the equilibrium to favor either the reactants or products.
- **Temperature**: Increasing or decreasing temperature will affect the rates of the forward and reverse reactions differently.
- **Pressure**: For reactions involving gases, pressure change will shift equilibrium towards the side with fewer or more moles of gas, depending on whether pressure is increased or decreased.
These changes are well explained by Le Chatelier’s Principle, which predicts how a change in conditions will affect chemical equilibrium.

Partial Pressure

Partial pressure is the pressure exerted by an individual gas component in a mixture of gases. It is a crucial concept in chemical reactions involving gases, especially when dealing with \( K_p \, \) the equilibrium constant in terms of partial pressures. In a gaseous equilibrium state, each component of the gas mixture contributes to the total pressure.
**Calculating Partial Pressure:**
The partial pressure \( P_i \) of a gas in a mixture can be calculated using the formula:
\( P_i = \chi_i \times P_{total} \)
where \( \chi_i \) is the mole fraction of the gas and \( P_{total} \) is the total pressure of the gas mixture.
Mole fraction, \( \chi_i \), is calculated by dividing the number of moles of a particular gas by the total number of moles of all gases present.
**Importance in Chemical Reactions:**
- In gas-phase reactions, knowing the partial pressures of reactants and products helps to write the \( K_p \) expression.
- Allows for the determination of equilibrium constants using measurable quantities such as pressure.
- It helps identify the direction in which a reaction needs to shift to reach equilibrium.

Stoichiometry

Stoichiometry is the quantitative study of reactants and products in chemical reactions. It involves the use of balanced chemical equations to calculate the masses, moles, and volumes of substances involved. Understanding stoichiometry is essential for formulating the \( K_p \) expressions, as it directly involves the stoichiometric coefficients that appear as exponents in these expressions.
In a balanced chemical equation, the number of atoms of each element must be the same on both sides of the equation, reflecting the conservation of mass.
**Stoichiometric Coefficients:**
- These are the numerical values in front of molecules in a chemical equation, indicating the ratio in which substances react and are produced.
- In the \( K_p \) expression, the stoichiometric coefficients determine the power to which the partial pressure of each gas is raised.
For example, in the reaction \( \mathrm{CH}_4 + 2 \mathrm{O}_2 \rightleftharpoons \mathrm{CO}_2 + 2 \mathrm{H}_2\mathrm{O} \), the coefficients show that one mole of methane reacts with two moles of oxygen to produce one mole of carbon dioxide and two moles of water vapor. Hence, the \( K_p \) expression arises from these stoichiometric relationships.
These coefficients ensure that calculations and predictions based on the \( K_p \) expression accurately reflect the behavior of the chemical system.