CHM-03

van 't Hoff isochore

ln(K2/K1) = −(ΔH/R)(1/T2 − 1/T1). How K moves with T.

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Equilibriumvan 't Hoff 1884

Governing equation

\ln\dfrac{K_2}{K_1}=-\dfrac{\Delta H}{R}\left(\dfrac1{T_2}-\dfrac1{T_1}\right)

$T_2 \in \left[200, 900\right]$

where

K_1: K at T1 (—)
T_1: T1 (K)
T_2: T2 (K)
\Delta H: Enthalpy of reaction (kJ/mol)
K_2: K at T2 (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\ln\dfrac{K_2}{K_1}=-\dfrac{\Delta H}{R}\left(\dfrac1{T_2}-\dfrac1{T_1}\right)

$T_2 \in \left[200, 900\right]$

$T_2 = 200$ $K_2 = 5.08\times 10^{-5}$

Along the curve

②

Domain of the variable — blue

T_2_{\min} = 200,\quad T_2_{\max} = 900

$T_2 \in \left[200, 900\right]$

$T_2 = 200$ $T_2 = 200$

Along the curve

③

Response curve — green

\ln\dfrac{K_2}{K_1}=-\dfrac{\Delta H}{R}\left(\dfrac1{T_2}-\dfrac1{T_1}\right)

$T_2 \in \left[200, 900\right]$

$T_2 = 200$ $K_2 = 5.08\times 10^{-5}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}K_2}{\mathrm{d}T_2}

$T_2 \in \left[200, 900\right]$

$T_2 = 204.4$ $K_2^{\prime} = 1.47\times 10^{-5}$

Along the curve

Lecture brief

Historical brief

Ideal-gas law, van ’t Hoff, Nernst, Michaelis–Menten and Clausius–Clapeyron are physical chemistry’s working equations of equilibrium and rate. The lab is pressure, potential and kinetics. This sheet (CHM-03 — van 't Hoff isochore) is the form associated with van 't Hoff 1884. Working symbols: $K_1$ , $T_1$ , $T_2$ , $\Delta H$ $\rightarrow$ $K_2$ . From d ln K / dT = ΔH / RT² (constant ΔH). Endothermic equilibria heat-shift to the right.

Purpose

Purpose: compute

K_2

from

K_1

T_1

T_2

\Delta H

in Physical chemistry via

\ln\dfrac{K_2}{K_1}=-\dfrac{\Delta H}{R}\left(\dfrac1{T_2}-\dfrac1{T_1}\right)

ln(K2/K1) = −(ΔH/R)(1/T2 − 1/T1). How K moves with T. Use it when a real physical chemistry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

K_1 = 1.000\,\mathrm{—}

T_1 = 298.000\,\mathrm{K}

T_2 = 350.000\,\mathrm{K}

\Delta H = 50.000\,\mathrm{kJ/mol}

, the governing relation

\ln\dfrac{K_2}{K_1}=-\dfrac{\Delta H}{R}\left(\dfrac1{T_2}-\dfrac1{T_1}\right)

yields

K_2 = 20.0486\,\mathrm{—}

. Constant ΔH between T1 and T2. K1 at T1. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

K at T1 K_11 —T1 T_1298 KT2 T_2350 KEnthalpy of reaction \Delta H50 kJ/mol

Outputs

K at T2 K_220.0486 —

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Narration of this film

Constant ΔH between T1 and T2. K1 at T1.

From d ln K / dT = ΔH / RT² (constant ΔH). Endothermic equilibria heat-shift to the right.

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