Question

at - 15.0 ^ { circ } mathrm { c } the pressure equilibrium constant k _ { p } = 7.0 \times 10 ^ { - 5 } for a certain reaction.
here are some facts about the reaction:

• the constant pressure molar heat capacity c _ { p } = 2.05 mathrm { j } cdot mathrm { mol } ^ { - 1 } cdot mathrm { k } ^ { - 1 }
• the initial rate of the reaction is 8.8 mathrm { mol } cdot mathrm { l } ^ { - 1 } cdot mathrm { s } ^ { - 1 }
• if the reaction is run at constant pressure, 147. mathrm { kj } / mathrm { mol } of heat are absorbed.

using these facts, can you calculate k _ { p } at - 30. ^ { circ } mathrm { c }?

if you said yes, then enter your answer at right. round it to 2 significant digits.

if you said no, can you at least decide whether k _ { p } at - 30. ^ { circ } mathrm { c } will be bigger or smaller than k _ { p } at - 15.0 ^ { circ } mathrm { c }?

Explanation:

Step1: Recall the van't Hoff equation

The van't Hoff equation is \(\ln\frac{K_{p2}}{K_{p1}}=\frac{\Delta H^{\circ}}{R}(\frac{1}{T_1}-\frac{1}{T_2})\) when \(\Delta C_p = 0\). Here, we are given \(\Delta C_p=2.05\ J\cdot mol^{-1}\cdot K^{-1}\). The more general form of the van't Hoff equation considering \(\Delta C_p\) is \(\ln\frac{K_{p2}}{K_{p1}}=\frac{\Delta H^{\circ}}{R}(\frac{1}{T_1}-\frac{1}{T_2})+\frac{\Delta C_p}{R}\ln\frac{T_2}{T_1}\). But we are not given \(\Delta H^{\circ}\) at a reference temperature (usually \(298\ K\)). Also, the initial rate of the reaction (\(8.8\ mol\cdot L^{-1}\cdot s^{-1}\)) is kinetic data and not relevant for equilibrium constant calculations.

Step2: Analyze the information sufficiency

To calculate \(K_p\) at a new temperature (\(T_2=- 30.0^{\circ}C=(273 - 30)K = 243\ K\), \(T_1=-15.0^{\circ}C=(273-15)K = 258\ K\)), we need \(\Delta H^{\circ}\) at a reference temperature (e.g., \(298\ K\)) to use the extended van't Hoff equation (because \(\Delta C_p
eq0\)). Since we are not given \(\Delta H^{\circ}\) at a reference temperature, we cannot calculate the exact value of \(K_p\) at \(-30.0^{\circ}C\).

Step3: Analyze the endothermic nature and temperature - equilibrium constant relation

The reaction is endothermic (\(q = \Delta H=147\ kJ/mol>0\)). According to Le - Chatelier's principle, for an endothermic reaction (\(\Delta H>0\)), when the temperature decreases (\(T_2=-30^{\circ}C

Answer:

No. Yes, and \(K_p\) will be smaller.