Question

Using molar concentrations, what is the unit of \(K_{c}\) for the reaction? $$ \mathrm{CH}_{3} \mathrm{OH}(g) \rightleftharpoons \mathrm{CO}(g)+2 \mathrm{H}_{2}(g) $$ (a) \(M^{-2}\) (b) \(M^{2}\) (c) \(M^{-1}\) (d) \(M\)

Short answer

The unit of \(K_{c}\) for the given reaction is M or \(M^{1}\).

Step-by-step solution

Write the expression for the equilibrium constant Kc

Based on the balanced chemical equation, the expression for the equilibrium constant, Kc, is given by the products raised to their stoichiometric coefficients divided by the reactants raised to their stoichiometric coefficients. For the reaction provided, the equilibrium expression is: \[ K_{c} = \frac{[CO][H_{2}]^{2}}{[CH_{3}OH]} \].

Determine the units of Kc

The molar concentration, [CO], and [H_{2}] have units of M (molarity, which is moles per liter). Using the equilibrium expression, calculate the units for Kc by substituting the units of concentration into the expression: \[ \text{Units of } K_{c} = \frac{M \times M^2}{M} = M^2 \]

Simplify the units

After substituting, the units of Kc can be simplified by cancelling out the units present in both the numerator and the denominator. Hence, the units for Kc simplify to: \[ \text{Units of } K_{c} = M^2 \cdot M^{-1} = M^{(2-1)} = M^{1} \].

Key concepts

Chemical Equilibrium

Chemical equilibrium occurs when a chemical reaction and its reverse reaction proceed at the same rate, resulting in no net change in the composition of the system. This dynamic state means that the concentrations of reactants and products remain constant over time. It's crucial to understand that reaching equilibrium doesn't mean the reactants and products are present in equal amounts; rather, their ratios are fixed according to the equilibrium constant for that reaction.

Understanding the behavior of a system at equilibrium is essential for predicting how the system will respond to changes in conditions, a principle known as Le Chatelier's Principle. This concept is foundational in chemical kinetics and thermodynamics, affecting various real-world applications such as chemical production, pharmacology, and environmental science.

Molarity

Molarity, denoted by the unit M (molar), is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per liter of solution and is expressed mathematically as: \[ \text{Molarity (M)} = \frac{\text{moles of solute}}{\text{liters of solution}} \.\]

Molarity is particularly useful in chemical reactions as it allows chemists to relate the volumes of different solutions with their respective stoichiometric coefficients in reactions. Furthermore, it is used in preparing solutions with precise concentrations, essential for laboratory experiments, and industrially in controlling the specifications of chemical products.

Stoichiometry

Stoichiometry is the section of chemistry that pertains to the quantitative relationship and calculations based on the laws of conserved mass and balanced chemical equations. It involves using the coefficients from balanced chemical equations to determine the relative amounts of substances involved in reactions.

In terms of a balanced chemical equation, the coefficients represent the molar ratios, which are crucial for calculating how much reactants are needed or how much products are formed. This concept is a cornerstone for many chemical processes, guiding chemists in predictive and quantitative analysis and ensures the efficiency and cost-effectiveness of reactions in industrial applications.

Equilibrium Expressions

Equilibrium expressions are mathematical representations of a chemical equilibrium condition, quantitatively describing the ratio of concentrations of products to reactants, each raised to the power of its stoichiometric coefficient in the balanced chemical equation. The equilibrium constant, represented as \(K\) with various subscripts, indicates the extent of the reaction when it reaches equilibrium.

For example, in the given reaction \[ \mathrm{CH}_{3} \mathrm{OH}(g) \rightleftharpoons \mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \.\], the equilibrium constant in terms of molarity (\(K_{c}\)) would be expressed as shown in the solution. The units of \(K_{c}\) provide important insight into the reaction's behavior under different conditions and are derived by considering the molarity units of the reactants and products. The proper understanding and calculation of equilibrium expressions are vital for controlling reactions, optimizing product yields, and conducting accurate thermodynamic calculations in chemistry and chemical engineering.