### Van't Hoff Equation

Posted:

**Fri Feb 02, 2018 12:02 pm**Could someone explain the steps involved in deriving lnK=(-deltaH/RT)+(deltaS/R) from deltaH-TdeltaS=-RTlnK?

Created by Dr. Laurence Lavelle

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

https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=135&t=26737

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Posted: **Fri Feb 02, 2018 12:02 pm**

Could someone explain the steps involved in deriving lnK=(-deltaH/RT)+(deltaS/R) from deltaH-TdeltaS=-RTlnK?

Posted: **Fri Feb 02, 2018 1:25 pm**

You start with deltaH - TdeltaS = -RTlnK. Then divide both sides by -RT to get (-deltaH/RT) + (deltaS/R) = lnK , or flip it around to get lnK = (-deltaH/RT) + (deltaS/R).

Posted: **Fri Feb 02, 2018 1:25 pm**

From ΔH - TΔS = -RTln(K), you simply divide by -RT to get ln K = -(ΔH/RT) + (ΔS/R).

However, this form of the Van't Hoff equation is not that commonly used. To derive the more widespread version of the Van't Hoff equation, you add two Van't Hoff equations in order to calculate the equilibrium constant K at different temperatures (as long as ΔH is known).

1. Van't Hoff equations for two different temperatures

ln K1 = -(ΔH/RT1) + (ΔS/R)

ln K2 = -(ΔH/RT2) + (ΔS/R)

2. subtract ln K1 from ln K2

ln K2 - ln K1 = ln (K2/K1) = -(ΔH/RT2) + (ΔS/R) -[-(ΔH/RT1) + (ΔS/R)]

The two (ΔS/R) factors cancel out, and the resulting equation is:

ln (K2/K1) = -(ΔH/RT2) + (ΔH/RT1), which can also be written as:

ln (K2/K1) = (-ΔH/R)[(1/T2)-(1/T1)]

These derivations are under the assumptions that ΔS and ΔH are constant.

However, this form of the Van't Hoff equation is not that commonly used. To derive the more widespread version of the Van't Hoff equation, you add two Van't Hoff equations in order to calculate the equilibrium constant K at different temperatures (as long as ΔH is known).

1. Van't Hoff equations for two different temperatures

ln K1 = -(ΔH/RT1) + (ΔS/R)

ln K2 = -(ΔH/RT2) + (ΔS/R)

2. subtract ln K1 from ln K2

ln K2 - ln K1 = ln (K2/K1) = -(ΔH/RT2) + (ΔS/R) -[-(ΔH/RT1) + (ΔS/R)]

The two (ΔS/R) factors cancel out, and the resulting equation is:

ln (K2/K1) = -(ΔH/RT2) + (ΔH/RT1), which can also be written as:

ln (K2/K1) = (-ΔH/R)[(1/T2)-(1/T1)]

These derivations are under the assumptions that ΔS and ΔH are constant.