### Equation

Posted:

**Fri Feb 21, 2020 9:25 pm**What is the original equation that lnK2/lnK1=-dH/R * (1/T2-1/T1) is derived from?

Created by Dr. Laurence Lavelle

https://lavelle.chem.ucla.edu/forum/

https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=141&t=59867

Page **1** of **1**

Posted: **Fri Feb 21, 2020 9:25 pm**

What is the original equation that lnK2/lnK1=-dH/R * (1/T2-1/T1) is derived from?

Posted: **Fri Feb 21, 2020 9:39 pm**

This equation is derived from a litany of other steps and equations.

First, we know that deltaGknot = -RTlnK, and that deltaGknot = deltaHknot - TdeltaSknot. So, substituting each equation in for deltaGknot, we get that

-RTlnK = deltaHknot - TdeltaSknot.

Next, if we know that deltaHknot is some specific value, we can divide both sides by -RT to get lnK = (deltaHknot/-RT) + (deltaSknot/R).

We know that a given temperature T1, there is a specific equilibrium constant K1, so lnK1 = (deltaHknot/-RT1) + (deltaSknot/R)

Similarly, at a given temperature T2, there is a specific equilibrium constant K2, so lnK2 = (deltaHknot/-RT2) + (deltaSknot/R).

lnK2 - lnK1 = ln(K2/K1), by the properties of logarithms.

ln(K2/K1) = (deltaHknot/-RT2) + (deltaHknot/RT1), assuming that deltaSknot is constant in regards to temperature.

Hence, we can take out a -deltaHknot/R value from the equation via the inverse distributive property to get

ln(K2/K1)=-dH/R * (1/T2-1/T1)

First, we know that deltaGknot = -RTlnK, and that deltaGknot = deltaHknot - TdeltaSknot. So, substituting each equation in for deltaGknot, we get that

-RTlnK = deltaHknot - TdeltaSknot.

Next, if we know that deltaHknot is some specific value, we can divide both sides by -RT to get lnK = (deltaHknot/-RT) + (deltaSknot/R).

We know that a given temperature T1, there is a specific equilibrium constant K1, so lnK1 = (deltaHknot/-RT1) + (deltaSknot/R)

Similarly, at a given temperature T2, there is a specific equilibrium constant K2, so lnK2 = (deltaHknot/-RT2) + (deltaSknot/R).

lnK2 - lnK1 = ln(K2/K1), by the properties of logarithms.

ln(K2/K1) = (deltaHknot/-RT2) + (deltaHknot/RT1), assuming that deltaSknot is constant in regards to temperature.

Hence, we can take out a -deltaHknot/R value from the equation via the inverse distributive property to get

ln(K2/K1)=-dH/R * (1/T2-1/T1)