Question

b) General chemistry textbooks often show this simplified derivation of the van't Hoff equation A, G In Kp - dln K Then, under the assumption that Д.Ho and Δ,So are independent of temperature dln K, However, the Gibbs-Helmholtz equation and, correspondingly, the van't Hoff equation, is a general re- lationship that gives the proper expression for the derivative of Kp at a specific temperature even when ΔΤΗ"and Δ,So are functions of temperature, i.e., when Δ7"Ho Δ,Ho(T) and Δ,So-Δ7S®(T) Show, directly (i.e., without using the Gibbs-Helmholtz equation), that: dln K, RT2 Hint: Since ΔΤ.H"(T) and ΔΤ'S®(T) are both temperature-dependent, you're going to need to use the prod- uct rule when you take the derivative. And you're then going to need to show that these two temperature- dependencies somehow cancel out

Answer 1

b) Given ln Kp = -Delta rG/RT = -Delta_r H degree/RT + Delta_r S degree/R Hence d/dT (ln Kp) = d/dT (-Delta_r H degree/RT + Delta_r S degree/R) = Delta_r H degree/RT^2 - Delta_r d/dT (H degree/RT) + Delta_r d/dT (S degree/R) (1) Now Delta_r (S_0) = Delta (H degree)/T Hence Delta_r d/dT (s degree) = Delta_r d/dT (H degree/T) Hence - Delta_r d/dT (H degree/RT) + Delta_r d/dT (S degree/R) = 0 rightarrow (2) From eqn (1) and (2) is \r\n d ln Kp/dT = d/dT (-Delta_r H degree/RT + Delta_r S degree/R) = Delta_r H degree/RT^2 - 0 Hence d ln Kp/dT = d/dT (-Delta_r H degree/RT + Delta_r S degree/R) = Delta_r H degree/RT^2