Methods to determine order — initial rates, integrated method, half-life method

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The thing we are trying to find

Everything below is a strategy to extract $n$ from measurable data.

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Method 1 — Initial Rates (Isolation / Differential method)

HOW (derivation from first principles):

Take rate law $r = k[A]^n$. Run two experiments with initial concentrations $[A]_1, [A]_2$ and measure the initial rates $r_1, r_2$ (initial, so no product interferes and concentration is exactly known):

$\frac{r_2}{r_1} = \frac{k[A]_2^{\,n}}{k[A]_1^{\,n}} = \left(\frac{[A]_2}{[A]_1}\right)^n$

Take log of both sides — WHY? To pull $n$ down from the exponent:

$n = \frac{\ln(r_2/r_1)}{\ln([A]_2/[A]_1)}$

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Method 2 — Integrated Rate Law Method (graphical fitting)

Derive the three integrated laws for $A\to$ products, $r=-\frac{d[A]}{dt}=k[A]^n$.

Zero order ($n=0$): $-\frac{d[A]}{dt}=k$. Integrate: $\int_{[A]_0}^{[A]}d[A] = -k\int_0^t dt$ $[A]=[A]_0 - kt \quad\Rightarrow\quad \text{plot }[A]\text{ vs }t\text{ is linear.}$

First order ($n=1$): $-\frac{d[A]}{dt}=k[A]\Rightarrow \int \frac{d[A]}{[A]}=-k\int dt$ $\ln[A]=\ln[A]_0 - kt \quad\Rightarrow\quad \text{plot }\ln[A]\text{ vs }t\text{ is linear.}$

Second order ($n=2$): $-\frac{d[A]}{dt}=k[A]^2\Rightarrow \int \frac{d[A]}{[A]^2}=-k\int dt$ $\frac{1}{[A]}=\frac{1}{[A]_0}+kt \quad\Rightarrow\quad \text{plot }\frac{1}{[A]}\text{ vs }t\text{ is linear.}$

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Method 3 — Half-Life Method

Derive $t_{1/2}$ for each order (set $[A]=[A]_0/2$ in the integrated laws):

Zero order: $\frac{[A]_0}{2}=[A]_0-kt_{1/2}\Rightarrow t_{1/2}=\dfrac{[A]_0}{2k}$ — proportional to $[A]_0$.

First order: $\ln\frac{[A]_0/2}{[A]_0}=-kt_{1/2}\Rightarrow -\ln2=-kt_{1/2}\Rightarrow t_{1/2}=\dfrac{\ln2}{k}=\dfrac{0.693}{k}$ — independent of $[A]_0$.

Second order: $\frac{2}{[A]_0}=\frac{1}{[A]_0}+kt_{1/2}\Rightarrow t_{1/2}=\dfrac{1}{k[A]_0}$ — inversely proportional to $[A]_0$.

General derivation of the exponent rule: For $n\ne 1$, integrated law gives $t_{1/2}\propto [A]_0^{1-n}$. Take log of two experiments: $n = 1 - \frac{\ln(t_{1/2,2}/t_{1/2,1})}{\ln([A]_{0,2}/[A]_{0,1})}$

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Common Mistakes

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Flashcards

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Recall Feynman: explain to a 12-year-old

Imagine a car whose speed depends on how much fuel is in the tank. Order just tells you how strongly the speed depends on fuel.

• Initial rates: fill two tanks differently, see how fast each car starts. If double fuel makes it 4× faster, that's a strong (order-2) dependence.
• Integrated method: watch the fuel drop over time and try three different graph papers — the one that makes a perfectly straight line tells you the rule.
• Half-life: time how long till the tank is half-empty. If that time never changes no matter how full you started (order 1), or shrinks/grows (order 2 or 0), you've spotted the pattern. All three tricks must point to the same answer.

Connections

• Rate Law and Rate Constant — the equation these methods extract.
• Integrated Rate Equations — full derivations reused here.
• Half-Life of Reactions
• Elementary vs Complex Reactions — why order ≠ molecularity.
• Arrhenius Equation — once $k$ is found, temperature dependence.
• Radioactive Decay — the classic first-order constant-half-life case.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, order ko balanced equation dekh ke guess nahi kar sakte — usko sirf experiment se nikalna padta hai. Iske liye teen popular methods hain, aur teeno bas alag-alag "fingerprint" dhoondhte hain.

Initial rates method: Do experiments karo jisme sirf ek reactant ki starting concentration change karo, baaki sab same rakho (isolation). Agar $[A]$ double karne pe rate 4 guna ho gaya, matlab $2^n=4$, toh $n=2$. Formula: $n=\ln(r_2/r_1)/\ln([A]_2/[A]_1)$. Yaad rakho — initial rate hi lo, kyunki baad me product jama ho jaata hai aur reading kharab ho jaati hai.

Integrated method: Concentration vs time ka data lo, aur teen tarah ke graph banao — $[A]$ vs $t$, $\ln[A]$ vs $t$, aur $1/[A]$ vs $t$. Jo bhi seedhi line de, wahi order batata hai. Zero order me $[A]$ straight, first order me $\ln[A]$ straight, second order me $1/[A]$ straight. Trick: "Zero-Amount, First-Log, Second-Reciprocal".

Half-life method: $t_{1/2}$ dekho — jitne time me concentration aadhi ho jaaye. First order me $t_{1/2}=0.693/k$ hamesha constant (chahe kitni bhi starting concentration ho). Zero order me $t_{1/2}$ badhta hai starting concentration ke saath, second order me ghatta hai. General rule: $t_{1/2}\propto [A]_0^{1-n}$. Yeh sab methods same order dena chahiye — agar match nahi ho raha toh kahin galti hai. Isliye yeh important hai: real kinetics problems me aksar yahi tarike lagte hain.