Question

In \(\mathrm{K}_{\mathrm{P}}=\mathrm{K}_{\mathrm{C}}[R T]^{n+}, \Delta n\) may have (1) +ve values (2) - ve values (3) integer or fractional values (4) either of these

Short answer

The correct answer is (4) either of these.

Step-by-step solution

Understand the given formula

The formula given is \(\mathrm{K}_{\mathrm{P}} = \mathrm{K}_{\mathrm{C}}[RT]^{\Delta n}\). Here, \(\mathrm{K}_{\mathrm{P}}\) is the equilibrium constant for pressure, \(\mathrm{K}_{\mathrm{C}}\) is the equilibrium constant for concentration, and \(\Delta n\) is the change in the number of moles of gas, calculated as the difference between the sum of the stoichiometric coefficients of the products and the reactants.

Interpretation of \(\Delta n\)

\(\Delta n\) can take different values depending on the specific chemical reaction. It represents the net change in the number of moles of gas during the reaction.

Positive values of \(\Delta n\)

\(\Delta n\) can be positive if the number of moles of gaseous products is greater than the number of moles of gaseous reactants.

Negative values of \(\Delta n\)

\(\Delta n\) can be negative if the number of moles of gaseous products is less than the number of moles of gaseous reactants.

Integer or fractional values

\(\Delta n\) can also be an integer or a fractional value. It is an integer if the reaction involves whole numbers of molecules. It can be fractional if the reaction involves fractional stoichiometric coefficients.

Conclusion

Given the different possible values for \(\Delta n\), the correct answer is that \(\Delta n\) may have any of these values (positive, negative, integer, or fractional values).

Key concepts

Equilibrium Constant

An equilibrium constant, denoted as either \(\text{K}_{\text{P}}\) for pressure or \(\text{K}_{\text{C}}\) for concentration, is a measure of the ratio of the concentrations of products to reactants at equilibrium. The equilibrium constant helps us understand the extent of a reaction, i.e., how far the reaction proceeds before reaching equilibrium. It's calculated using the equilibrium concentrations (or partial pressures) of the chemical species involved in the reaction. A high equilibrium constant means the reaction favors products, while a low one favors reactants. The formula \(\text{K}_{\text{P}} = \text{K}_{\text{C}}[RT]^{\text{Δ} n}\) further indicates the relationship between \(\text{K}_{\text{P}}\) and \(\text{K}_{\text{C}}\), where \(\text{Δ} n\)\refers to the change in the number of moles of gas, and \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the ratios in which substances react. It involves calculations based on the laws of conservation of mass and energy. In a balanced chemical equation, stoichiometric coefficients indicate the proportions of reactants and products. These coefficients are essential for determining \(\text{Δ} n\), the change in moles of gas. For example, in the reaction \(2H_2 + O_2 \rightarrow 2H_2O\), the coefficients 2, 1, and 2 represent the stoichiometric amounts of hydrogen, oxygen, and water. \(\text{Δ} n\) is the difference in the sums of these coefficients for products and reactants. Understanding stoichiometry ensures accurate and meaningful equilibrium constant calculations.

Reaction Quotient

The reaction quotient, \(Q\), measures the relative amounts of products and reactants at any point in a reaction before reaching equilibrium. It's calculated similarly to the equilibrium constant but using initial or arbitrary concentrations or partial pressures. By comparing \(Q\) with \(\text{K}\), we can predict the direction in which the reaction will shift to reach equilibrium:

• If \(Q < \text{K}\), the reaction will proceed forward, favoring products.
• If \(Q > \text{K}\), the reaction will shift backward, favoring reactants.
• If \(Q = \text{K}\), the reaction is already at equilibrium.

Understanding \(Q\) is crucial in determining reaction shifts and calculating changes during a reaction.

Gas Laws in Chemistry

Gas laws describe the behavior of gases in relation to pressure, volume, temperature, and quantity. They are fundamental in understanding equilibrium involving gaseous reactants and products. Key gas laws include:

• Boyle's Law: \(P \times V = \text{constant}\) at constant temperature.
• Charles's Law: \(\frac{V}{T} = \text{constant}\) at constant pressure.
• Avogadro's Law: \(\frac{V}{n} = \text{constant}\) at constant temperature and pressure.
• Ideal Gas Law: \(PV = nRT\), which combines the above laws. Here, \(R\) is the gas constant, \(n\) is the number of moles, \(P\) is pressure, \(V\) is volume, and \(T\) is temperature.

These laws help us understand the \(RV\) and \(P\) relationships in the formula for the equilibrium constant, linking the macroscopic properties of gases to chemical equilibria.