CHM-26

Second-order 1/c law

1/c = 1/c0 + k t. Remaining concentration of a 2A → products step.

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KineticsSecond-order kinetics

Governing equation

\dfrac1c=\dfrac1{c_0}+kt

$t \in \left[0, 500\right]$

where

c_0: Initial concentration (mol/L)
k: Rate constant (L/(mol·s))
t: Time (s)
c: Remaining c (mol/L)
1/c: Reciprocal c (L/mol)

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

\dfrac1c=\dfrac1{c_0}+kt

$t \in \left[0, 500\right]$

$t = 0$ $c = 0.5$

Along the curve

②

Domain of the variable — blue

t_{\min} = 0,\quad t_{\max} = 500

$t \in \left[0, 500\right]$

$t = 0$ $t = 0$

Along the curve

③

Response curve — green

\dfrac1c=\dfrac1{c_0}+kt

$t \in \left[0, 500\right]$

$t = 0$ $c = 0.5$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}c}{\mathrm{d}t}

$t \in \left[0, 500\right]$

$t = 3.145$ $c^{\prime} = -0.005$

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-26 — Second-order 1/c law) is the form associated with Second-order kinetics. Working symbols: $c_0$ , $k$ , $t$ $\rightarrow$ $c$ , $1/c$ . Integrate dc/dt = −k c². A plot of 1/c vs t is linear with slope k.

Purpose

Purpose: compute

c

1/c

from

c_0

k

t

in Physical chemistry via

\dfrac1c=\dfrac1{c_0}+kt

1/c = 1/c0 + k t. Remaining concentration of a 2A → products step. 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

c_0 = 0.500\,\mathrm{mol/L}

k = 0.020\,\mathrm{L/(mol·s)}

t = 30.000\,\mathrm{s}

, the governing relation

\dfrac1c=\dfrac1{c_0}+kt

yields

c = 0.3846\,\mathrm{mol/L}

1/c = 2.6000\,\mathrm{L/mol}

. Equal-concentration second order, constant volume. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Initial concentration c_00.5 mol/LRate constant k0.02 L/(mol·s)Time t30 s

Outputs

Remaining c c0.3846 mol/L
Reciprocal c 1/c2.6000 L/mol

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Narration of this film

Equal-concentration second order, constant volume.

Integrate dc/dt = −k c². A plot of 1/c vs t is linear with slope k.

Reading speed

Watch on YouTube