For the reversible reaction: (g) + 3(g) ⇌ 2(g) at 500°C, the value of Kp is 1.44 × 10^-5 when partia — Chemical Equilibrium Chemistry Question

💡 Solution & Explanation

Step 1 - Identify the Chemical Equation and Determine \(\Delta n_g\) The given reversible gas-phase reaction representing the synthesis of ammonia (Haber's process) is: \[\ce{N2(g) + 3H2(g) <=> 2NH3(g)}\] First, we determine the change in the number of gaseous moles (\(\Delta n_g\)) during the reaction: \[\Delta n_g = \sum n_{\text{g, products}} - \sum n_{\text{g, reactants}}\] For this reaction: * Number of moles of gaseous products (\(n_{\text{g, products}}\)) = \(2\) (from \(\ce{2NH3(g)}\)) * Number of moles of gaseous reactants (\(n_{\text{g, reactants}}\)) = \(1 + 3 = 4\) (from \(\ce{N2(g)}\) and \(\ce{3H2(g)}\)) Substituting these stoichiometric coefficients: \[\Delta n_g = 2 - (1 + 3) = -2\] Step 2 - State the Relationship Between \(K_p\) and \(K_c\) The thermodynamic relationship between the equilibrium constant in terms of partial pressures (\(K_p\)) and the equilibrium constant in terms of molar concentrations (\(K_c\)) is given by the equation: \[K_p = K_c (RT)^{\Delta n_g}\] Where: * \(R\) is the universal gas constant. * \(T\) is the absolute temperature in Kelvin (\(\text{K}\)). Substituting \(\Delta n_g = -2\) into the relation: \[K_p = K_c (RT)^{-2}\] Step 3 - Convert Variables to Correct Units and Substitute Values We are given the following parameters: * Equilibrium constant in terms of pressure, \(K_p = 1.44 \times 10^{-5}\) * Temperature in Celsius, \(t = 500^\circ\text{C}\) Converting to the absolute Kelvin scale: \[T = t + 273 = 500 + 273 = 773\text{ K}\] * Gas constant in terms of liter-atmosphere units, \(R = 0.082\text{ L atm K}^{-1}\text{ mol}^{-1}\) Rearranging our relation to solve for \(K_c\): \[K_c = \frac{K_p}{(RT)^{-2}}\] Substituting the values with their respective units: \[K_c = \frac{1.44 \times 10^{-5}}{(0.082 \times 773)^{-2}}\] Step 4 - Evaluate the Options * **Option (A) \(\frac{1.44 \times 10^{-5}}{(0.082 \times 500)^{-2}}\)**: Incorrect. This option incorrectly uses the temperature in Celsius (\(500^\circ\text{C}\)) instead of absolute temperature in Kelvin. * **Option (B) \(\frac{1.44 \times 10^{-5}}{(8.314 \times 773)^{-2}}\)**: Incorrect. This option uses the value of the gas constant \(R = 8.314\text{ J K}^{-1}\text{ mol}^{-1}\) in SI units instead of the required liter-atmosphere units (\(0.082\text{ L atm K}^{-1}\text{ mol}^{-1}\)), which is necessary because pressure is measured in atmospheres. * **Option (C) \(\frac{1.44 \times 10^{-5}}{(0.082 \times 773)^{2}}\)**: Incorrect. This uses the incorrect sign in the exponent of \((RT)\), representing \(K_c = K_p / (RT)^2\) which violates the derived relationship. * **Option (D) \(\frac{1.44 \times 10^{-5}}{(0.082 \times 773)^{-2}}\)**: Correct. This option correctly substitutes the absolute temperature (\(773\text{ K}\)), the gas constant in correct units (\(0.082\)), and maintains the correct sign in the exponent in the denominator. \[\boxed{\text{D}}\]