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Index/Physical chemistry/CHM-25

CHM-25

First-order half-life

t½ = ln 2 / k. Independent of the starting concentration.

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KineticsFirst-order kinetics

Governing equation

t1/2=ln⁡2/kt_{1/2}=\ln 2/k

k∈[1×10−6,2]k \in \left[1\times 10^{-6}, 2\right]

where

k
Rate constant (1/s)
t_{1/2}
Half-life (s)

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CHM-25 · decay
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①

Governing law — red

t1/2=ln⁡2/kt_{1/2}=\ln 2/k

k∈[1×10−6,2]k \in \left[1\times 10^{-6}, 2\right]

k=1×10−6k = 1\times 10^{-6}t1/2=6.93×105t_{1/2} = 6.93\times 10^{5}

②

Domain of the variable — blue

kmin⁡=1×10−6,kmax⁡=2k_{\min} = 1\times 10^{-6},\quad k_{\max} = 2

k∈[1×10−6,2]k \in \left[1\times 10^{-6}, 2\right]

k=1×10−6k = 1\times 10^{-6}k=1×10−6k = 1\times 10^{-6}

③

Response curve — green

t1/2=ln⁡2/kt_{1/2}=\ln 2/k

k∈[1×10−6,2]k \in \left[1\times 10^{-6}, 2\right]

k=1×10−6k = 1\times 10^{-6}t1/2=6.93×105t_{1/2} = 6.93\times 10^{5}

④

Rate of change — amber

dt1/2dk\frac{\mathrm{d}t_{1/2}}{\mathrm{d}k}

k∈[1×10−6,2]k \in \left[1\times 10^{-6}, 2\right]

k=0.013k = 0.013t1/2′=−2.76×107t_{1/2}^{\prime} = -2.76\times 10^{7}

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-25 — First-order half-life) is the form associated with First-order kinetics. Working symbols: kk →\rightarrow t1/2t_{1/2}. From c = c0 e^{−kt}. Radioactive decay is the textbook first-order process.

Purpose

Purpose: compute t1/2t_{1/2} from kk in Physical chemistry via t1/2=ln⁡2/kt_{1/2}=\ln 2/k t½ = ln 2 / k. Independent of the starting concentration. 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=0.010 1/sk = 0.010\,\mathrm{1/s}, the governing relation t1/2=ln⁡2/kt_{1/2}=\ln 2/k yields t1/2=69.315 st_{1/2} = 69.315\,\mathrm{s}. Irreversible first-order, constant k. Move a slider: the numbers are this situation, not a canned story.

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Outputs

  • Half-life t_{1/2}69.315 s
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CHM-25 · decay
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Narration of this film

Irreversible first-order, constant k.

From c = c0 e^{−kt}. Radioactive decay is the textbook first-order process.

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