Question

Unit of \( K_C \) for equilibrium:

• A. \( \text{mol L}^{-1} \)
• B. \( \text{L mol}^{-1} \)
• C. \( \text{mol}^2 \text{L}^{-2} \)
• D. Dimensionless

Hint:

For equilibrium constants:
- \( K_C \) units depend on the reaction stoichiometry.
- If the sum of product coefficients equals the sum of reactant coefficients, \( K_C \) is dimensionless.
- Always write the expression for \( K_C \) and check the powers of concentration units.

Answer 1

The Correct Option is D

The equilibrium constant \( K_C \) for a chemical reaction is defined in terms of molar concentrations of products and reactants at equilibrium. For a general reaction: \[ aA + bB \rightleftharpoons cC + dD \] The equilibrium constant \( K_C \) is: \[ K_C = \frac{[C]^c [D]^d}{[A]^a [B]^b} \] where \([X]\) represents the molar concentration of species \(X\) in \(\text{mol L}^{-1}\).
To find the units of \( K_C \), we analyze the units of the concentrations: - Concentration \([X] = \text{mol L}^{-1}\). - For each term \([X]^n\), the unit is \((\text{mol L}^{-1})^n = \text{mol}^n \text{L}^{-n}\).
The units of \( K_C \) are: \[ \text{Units of } K_C = \frac{(\text{mol L}^{-1})^c (\text{mol L}^{-1})^d}{(\text{mol L}^{-1})^a (\text{mol L}^{-1})^b} = \frac{\text{mol}^{c+d} \text{L}^{-(c+d)}}{\text{mol}^{a+b} \text{L}^{-(a+b)}} \] Simplify: \[ = \text{mol}^{(c+d)-(a+b)} \text{L}^{-(c+d)+ (a+b)} = \text{mol}^{\Delta n} \text{L}^{-\Delta n} \] where \(\Delta n = (c+d) - (a+b)\) is the change in the number of moles of gas (for gas-phase reactions, but here we consider concentrations for \( K_C \)).
However, the question does not specify a particular reaction, suggesting a case where \( K_C \) is dimensionless. This occurs when \(\Delta n = 0\), i.e., the sum of the stoichiometric coefficients of products equals that of reactants (\(a+b = c+d\)). For example, in the reaction: \[ N_2(g) + O_2(g) \rightleftharpoons 2NO(g) \] \[ K_C = \frac{[NO]^2}{[N_2]^1 [O_2]^1} = \frac{(\text{mol L}^{-1})^2}{(\text{mol L}^{-1})^1 (\text{mol L}^{-1})^1} = \text{mol}^{2-1-1} \text{L}^{-2+1+1} = \text{mol}^0 \text{L}^0 = 1 \] Here, \( K_C \) is dimensionless (\(\Delta n = 2 - (1+1) = 0\)).
Among the options, “dimensionless” fits this case. Without a specific reaction, we assume the question refers to an equilibrium where \( K_C \) has no units, which is common in standard problems when \(\Delta n = 0\).