Problem 118 - Introductory Chemistry Atoms First

Question

Which of the following reactions is described by the equilibrium constant expression \(K_{\mathrm{eq}}=\frac{[\mathrm{A}]^{2} \times[\mathrm{B}]^{3}}{[\mathrm{C}]^{3} \times[\mathrm{D}]^{2}}\) (a) \(\mathrm{A}_{2}+\mathrm{B}_{3} \rightleftarrows \mathrm{C}_{3}+\mathrm{D}_{2}\) (b) \(2 \mathrm{~A}+3 \mathrm{~B} \rightleftarrows 3 \mathrm{C}+2 \mathrm{D}\) (c) \(3 \mathrm{C}+2 \mathrm{D} \rightleftarrows 2 \mathrm{~A}+3 \mathrm{~B}\) (d) \(A^{2}+B^{3} \rightleftarrows C^{3}+D^{2}\) (e) \(2 \mathrm{C}+3 \mathrm{D} \rightleftarrows 3 \mathrm{~A}+2 \mathrm{~B}\)

Step-by-Step Solution

Answer

The correct balanced chemical equation that matches the given equilibrium constant expression \(K_{\mathrm{eq}}=\frac{[\mathrm{A}]^{2} \times[\mathrm{B}]^{3}}{[\mathrm{C}]^{3} \times[\mathrm{D}]^{2}}\) is option (b): \(2 \mathrm{~A}+3 \mathrm{~B} \rightleftarrows 3 \mathrm{C}+2 \mathrm{D}\).

Step 1: Analyze Option (a)

We can compare the equilibrium expression for option (a) with the given equilibrium constant expression: $K_{\mathrm{eq}}=\frac{[\mathrm{A}]^{2} \times[\mathrm{B}]^{3}}{[\mathrm{C}]^{3} \times[\mathrm{D}]^{2}}$ Option (a): $\mathrm{A}_{2}+\mathrm{B}_{3} \rightleftarrows \mathrm{C}_{3}+\mathrm{D}_{2}$ The stoichiometric coefficients in option (a) are not matching the powers of the concentrations in the given equilibrium expression. Therefore, option (a) is incorrect.

Step 2: Analyze Option (b)

We can compare the equilibrium expression for option (b) with the given equilibrium constant expression. $K_{\mathrm{eq}}=\frac{[\mathrm{A}]^{2} \times[\mathrm{B}]^{3}}{[\mathrm{C}]^{3} \times[\mathrm{D}]^{2}}$ Option (b): \(2 \mathrm{~A}+3 \mathrm{~B} \rightleftarrows 3 \mathrm{C}+2 \mathrm{D}\) The stoichiometric coefficients in option (b) match the powers of the concentrations in the given equilibrium expression. Therefore, option (b) is correct.

Step 3: Analyze the Other Options (c), (d), and (e) for Confirmation

We can perform a quick analysis of the other options to confirm that option (b) is the correct answer. Option (c): \(3 \mathrm{C}+2 \mathrm{D} \rightleftarrows 2 \mathrm{~A}+3 \mathrm{~B}\) Option (d): \(A^{2}+B^{3} \rightleftarrows C^{3}+D^{2}\) Option (e): \(2 \mathrm{C}+3 \mathrm{D} \rightleftarrows 3 \mathrm{~A}+2 \mathrm{~B}\) None of these options have the stoichiometric coefficients matching the powers of the concentrations in the given equilibrium expression. This confirms that option (b) is the correct answer.

Step 4: Conclusion

Based on our analysis, the equilibrium constant expression \(K_{\mathrm{eq}}=\frac{[\mathrm{A}]^{2} \times[\mathrm{B}]^{3}}{[\mathrm{C}]^{3} \times[\mathrm{D}]^{2}}\) corresponds to the balanced chemical equation given in option (b): \(2 \mathrm{~A}+3 \mathrm{~B} \rightleftarrows 3 \mathrm{C}+2 \mathrm{D}\).

Key Concepts

Chemical Reactions

Chemical reactions involve the transformation of substances through breaking and forming of chemical bonds. During this process, reactants are turned into products. This can be represented by a chemical equation which provides vital information about the participating substances.

• Reactants: The starting substances present before the reaction takes place.
• Products: The new substances formed as a result of the reaction.
• Reaction Conditions: Conditions like temperature, pressure, and the presence of a catalyst can affect the reaction.

Understanding the reaction is crucial because it helps predict how substances will behave under different conditions. For students, grasping the basics of chemical reactions is essential for mastering chemistry. It also lays the groundwork for more complex topics such as reaction kinetics and equilibrium.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It is the bridge that allows chemists to predict how much product can be formed from known amounts of reactants. It can also help determine the amount of reactants needed to produce a desired quantity of product. Stoichiometry involves using balanced chemical equations to ensure mass conservation.

• Coefficients in a Balanced Equation: Represent the molar ratio of substances involved in the reaction.
• Mole Concept: Central to stoichiometry, it allows conversion between mass and the number of particles.
• Limiting Reactant: The reactant that is completely consumed, limiting the amount of product formed.

Mastery of stoichiometry enables students to solve practical problems like calculating reagent quantities for chemical synthesis and understanding nutritional content through chemical formulas.

Balanced Chemical Equation

A balanced chemical equation represents the equality of mass in a chemical reaction. It is achieved when the number of atoms for each element is the same on both sides of the equation. This reflects the law of conservation of mass, which states that matter is neither created nor destroyed in a chemical reaction.

• Identifying Reactants and Products: Ensures all reactants and products are correctly listed.
• Adjusting Coefficients: The stoichiometric coefficients are changed until each element has the same number of atoms on both sides of the equation.
• Verification: Double-checking the equation ensures accuracy and helps avoid mistakes.

A balanced chemical equation is crucial because it is the foundation for calculating equilibrium constants and performing stoichiometric calculations. Without a balanced equation, predictions about the reaction are inaccurate and unreliable.