Problem 83 - An Introduction to Thermal Physics

Question

Write down the equilibrium condition for each of the following reactions: (a) \(2 \mathrm{H} \leftrightarrow \mathrm{H}_{2}\) (b) \(2 \mathrm{CO}+\mathrm{O}_{2} \leftrightarrow 2 \mathrm{CO}_{2}\) (c) \(\mathrm{CH}_{4}+2 \mathrm{O}_{2} \leftrightarrow 2 \mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2}\) (d) \(\mathrm{H}_{2} \mathrm{SO}_{4} \leftrightarrow 2 \mathrm{H}^{+}+\mathrm{SO}_{4}^{2-}\) (e) \(2 p+2 n \leftrightarrow^{4} \mathrm{He}\)

Step-by-Step Solution

Answer

Equilibrium constants for the reactions are expressions involving the ratio of the concentrations of products to reactants, each raised to the power of their coefficients.

Step 1: Understand the Equilibrium Condition

The equilibrium condition for a reversible chemical reaction is expressed by the reaction quotient \( Q \) reaching a constant value, known as the equilibrium constant \( K \). For a general reaction \( aA + bB \leftrightarrow cC + dD \), the equilibrium constant expression \( K \) is given by \( K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \). At equilibrium, the concentrations of the reactants and products do not change over time.

Step 2: Determine the Equilibrium Condition for Reaction (a)

The reaction is \( 2 \mathrm{H} \leftrightarrow \mathrm{H}_{2} \). The equilibrium expression is: \[ K = \frac{[\mathrm{H}_2]}{[\mathrm{H}]^2} \]

Step 3: Determine the Equilibrium Condition for Reaction (b)

The reaction is \( 2 \mathrm{CO} + \mathrm{O}_{2} \leftrightarrow 2 \mathrm{CO}_{2} \). The equilibrium expression is: \[ K = \frac{[\mathrm{CO}_{2}]^2}{[\mathrm{CO}]^2[\mathrm{O}_2]} \]

Step 4: Determine the Equilibrium Condition for Reaction (c)

The reaction is \( \mathrm{CH}_{4} + 2 \mathrm{O}_{2} \leftrightarrow 2 \mathrm{H}_{2}\mathrm{O} + \mathrm{CO}_{2} \). The equilibrium expression is: \[ K = \frac{[\mathrm{H}_2\mathrm{O}]^2[\mathrm{CO}_2]}{[\mathrm{CH}_4][\mathrm{O}_2]^2} \]

Step 5: Determine the Equilibrium Condition for Reaction (d)

The reaction is \( \mathrm{H}_{2} \mathrm{SO}_{4} \leftrightarrow 2 \mathrm{H}^{+} + \mathrm{SO}_{4}^{2-} \). The equilibrium expression is: \[ K = \frac{[\mathrm{H}^+]^2[\mathrm{SO}_4^{2-}]}{[\mathrm{H}_2\mathrm{SO}_4]} \]

Step 6: Determine the Equilibrium Condition for Reaction (e)

The reaction is \( 2 p + 2 n \leftrightarrow {}^4 \mathrm{He} \). The equilibrium expression, assuming the concentrations of protons (p) and neutrons (n) are relevant, would be: \[ K = \frac{[{}^4\mathrm{He}]}{[p]^2[n]^2} \]

Key Concepts

Equilibrium Constant

The equilibrium constant, often denoted as \( K \), is a vital concept in understanding chemical equilibrium. It represents the ratio of the concentrations of products to reactants at equilibrium, each raised to the power of their respective stoichiometric coefficients. This value is constant for a given reaction at a specific temperature, meaning it provides a measure of the extent of the reaction.

To better understand, consider the reaction:

- \( aA + bB \leftrightarrow cC + dD \)
- The expression for the equilibrium constant \( K \) is \( K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \)

Here, square brackets \([ ]\) denote the concentration of the substances, and exponents correspond to the coefficients from the balanced chemical equation. It is crucial to remember that \( K \) is only affected by temperature, not concentration, hence, it is a temperature-dependent, but concentration-independent constant.

Reaction Quotient

The reaction quotient, \( Q \), serves as a tool to determine the direction in which a reaction will proceed to reach equilibrium. While \( K \) applies specifically to the state of equilibrium, \( Q \) can be calculated at any point during the reaction. It is determined using the same formula as \( K \): the ratio of product concentrations to reactant concentrations, raised to the power of their respective stoichiometric coefficients.

Using \( Q \), one can predict:

• If \( Q = K \), the system is at equilibrium.
• If \( Q < K \), the forward reaction is favored, meaning the reaction will proceed to the right to form more products.
• If \( Q > K \), the reverse reaction is favored, indicating the formation of more reactants.

This concept is essential in predicting the "direction of change" for a reaction that isn't initially at equilibrium, thereby enhancing the understanding and control of chemical reactions.

Reversible Reactions

Reversible reactions are those that can proceed in both forward and backward directions. This type of reactions have a pivotal role in chemical equilibrium as they establish a state where the rates of the forward and reverse reactions are equal, resulting in no net change in the concentration of reactants and products over time.

Key points of reversible reactions include:

• The dynamic nature: even though concentrations don't change at equilibrium, molecules continue to react.
• The double arrows (\(\leftrightarrow\)) in the chemical equation signifying the possibility of both directions.
• The establishment of equilibrium where both reactants and products coexist in a stable ratio defined by the equilibrium constant \( K \).

Understanding reversible reactions is crucial in many practical applications, such as the synthesis of chemicals, metabolic pathways, and industrial processes, which often hinge on controllable equilibria.

Reaction Stoichiometry

Reaction stoichiometry involves the quantitative relationship between reactants and products in a chemical reaction, typically expressed in moles. It is fundamental in formulating both the reaction quotient \( Q \) and the equilibrium constant \( K \), as it dictates the powers to which concentrations are raised.

Considerations in reaction stoichiometry include:

• Balanced equations: It is vital to have the chemical equation balanced before writing the equilibrium constant expression.
• Coefficients in the equation, such as in \( aA + bB \leftrightarrow cC + dD \), are directly used as exponents in the expression for \( K \) or \( Q \).
• This balance forms the basis for calculating quantities like theoretical yields and for executing stoichiometric conversions between different components of a reaction.

Accuracy in stoichiometry is crucial for correct interpretation and application of equilibrium calculations, ensuring that chemical processes are both efficient and predictable.