Rate of reaction — average vs instantaneous

2.8.1 · Chemistry › Chemical Kinetics

Overview

Rate of a chemical reaction measure karta hai ki reactants kitni tezi se products mein convert ho rahe hain. Hume do alag definitions ki zarurat isliye hai kyunki reactions constant speed se aage nahi badhte — ye generally slow down ho jaate hain jaise-jaise reactants khatam hote hain. Average rate tumhe ek time interval mein overall speed batata hai, jabki instantaneous rate kisi ek specific moment par exact speed capture karta hai.

Core Concepts

Negative sign [A] ki decrease ko ek positive rate mein convert karta hai (hum rates ko positive numbers ke roop mein report karte hain).

Ek product P ke form hone ke liye: $r_{\text{avg}}=+\frac{\Delta [\text{P}]}{\Delta t}=\frac{[\text{P}{]}_2-[\text{P}{]}_1}{t_2-t_1}$

YE definition kyun? Rate ka matlab hai "concentration per unit time kitna badalta hai." Δ[A]/Δt interval ke upar concentration-time graph ka slope hai. Negative sign ek sign convention hai: chemists chahte hain ki "rate" positive ho, bhale hi [reactant] gire.

KYUN? Jaise-jaise Δt zero ki taraf shrink hota hai, us tiny interval par average rate ek single time point par exact rate ban jaata hai. Mathematically, ye derivative hai — time t par concentration curve ke tangent line ka slope.

$\frac{-1}{a}\frac{d[\text{A}]}{dt}=\frac{-1}{b}\frac{d[\text{B}]}{dt}=\frac{1}{c}\frac{d[\text{C}]}{dt}=\frac{1}{d}\frac{d[\text{D}]}{dt}=r$

Stoichiometry se Derivation:

1. Jab $a$ moles of A disappear hote hain, $c$ moles of C form hote hain (stoichiometry).
2. Toh agar A ki concentration $\Delta t$ time mein $a$ mol/L gir jaati hai, C ki concentration $c$ mol/L badhti hai.
3. A ke per mole rate = $-\frac{1}{a}\frac{d[\text{A}]}{dt}$; C ke per mole rate = $\frac{1}{c}\frac{d[\text{C}]}{dt}$.
4. Ye equal hone chahiye kyunki ye same reaction event ko same speed par describe karte hain.

Stoichiometric coefficients se kyun divide karte hain? Ek balanced equation 1 mole of B ke liye 2 moles of A consume kar sakta hai. Tab [A] [B] se twice as fast gir jaata hai, lekin reaction ek rate par proceed karta hai. Coefficients se divide karna "reaction rate" tak normalize karta hai.

Worked Examples

Solution:

Step 1: N₂O₅ consumption ka average rate. $-\frac{\Delta [{\text{N}}_2{\text{O}}_5]}{\Delta t}=-\frac{0.0250-0.0400}{100-0}=-\frac{-0.0150}{100}=1.50\times 10^{-4}\text{ M/s}$ Ye step kyun? Hum directly definition apply karte hain. Negative ka negative positive rate deta hai.

Step 2: Stoichiometry use karke reaction rate mein convert karo. $r=\frac{-1}{2}\frac{\Delta [{\text{N}}_2{\text{O}}_5]}{\Delta t}=\frac{1}{2}\times 1.50\times 10^{-4}=7.50\times 10^{-5}\text{ M/s}$ 2 se kyun divide? N₂O₅ ka coefficient 2 hai. Reaction utni "speed" par "hoti" hai jitni speed se N₂O₅ molecules disappear hoti hain, uski aadhi.

Step 3: NO₂ formation ka rate. $\frac{1}{4}\frac{d[{\text{NO}}_2]}{dt}=r⟹\frac{d[{\text{NO}}_2]}{dt}=4r=4\times 7.50\times 10^{-5}=3.00\times 10^{-4}\text{ M/s}$ 4 se kyun multiply? Har "reaction event" mein 4 NO₂ molecules form hote hain (stoichiometry). Toh NO₂ normalized reaction rate se 4 times faster appear hota hai.

Answer: Reaction rate = $7.50\times 10^{-5}$ M/s; NO₂ formation rate = $3.00\times 10^{-4}$ M/s.

Solution:

Step 1: Tangent slope HI derivative $\frac{d[\text{A}]}{dt}$ hai t=50 s par. $\frac{d[\text{A}]}{dt}=-0.0030\text{ M/s}$ Kyun? Derivative ki definition: tangent ka slope.

Step 2: Positive rate paane ke liye negative sign lagao. $r_{\text{inst}}=-\frac{d[\text{A}]}{dt}=-(-0.0030)=0.0030\text{ M/s}$ Kyun? Sign convention: [A] decrease hone ke bawajood rate positive hoti hai.

Answer: t=50 s par 0.0030 M/s.

Solution:

Step 1: Average rate (H₂O₂ ka consumption). $-\frac{\Delta [{\text{H}}_2{\text{O}}_2]}{\Delta t}=-\frac{0.50-1.00}{1000-0}=5.0\times 10^{-4}\text{ M/s}$ Kyun? Average rate ki definition.

Step 2: Reaction rate (stoichiometry account karo). $r_{\text{avg}}=\frac{1}{2}\times 5.0\times 10^{-4}=2.5\times 10^{-4}\text{ M/s}$ 2 se kyun divide? Coefficient 2 hai.

Step 3: Initial instantaneous rate (directly diya gaya hai). $r_{\text{inst}}(t=0)=\frac{1}{2}\times 1.5\times 10^{-3}=7.5\times 10^{-4}\text{ M/s}$ Ruko, kyun divide? Problem mein "instantaneous rate" = 1.5×10⁻³ M/s diya gaya hai. Agar ye H₂O₂ ki consumption rate hai, toh reaction rate ke liye 2 se divide karte hain. (Real problems mein units clarify karo!)

Comparison: $r_{\text{inst}}(t=0)=7.5\times 10^{-4}$ M/s, $r_{\text{avg}}=2.5\times 10^{-4}$ M/s se 3 times bada hai.

KYUN? Reactions typically slow down hoti hain kyunki [reactant] decrease hota hai. Initial rate (jab [H₂O₂] sabse zyada hai) baad ke average se zyada fast hai (jo 1000 s include karta hai, jisme baad ke slow periods bhi hain).

Ye sahi kyun lagta hai: Tumne directly slope calculate kiya aur negative number mila.

Fix: Rates defined hoti hain positive quantities ke roop mein. Ek disappearing reactant ke liye, tumhe ZARUR negative sign include karna hai: $r=-\frac{\Delta [\text{A}]}{\Delta t}=-(-0.03)=+0.03$ M/s. Definition mein negative sign, negative slope ko compensate karta hai.

Steel-man: Confusion isliye hoti hai kyunki math mein slopes negative ho sakte hain. Lekin chemistry mein, "rate" ek speed concept hai (hamesha positive), aur hum definition mein sign isliye choose karte hain taaki ye ensure ho sake.

Ye sahi kyun lagta hai: Tumne measure kiya ki A kitna fast disappear hota hai, that's the rate, right?

Fix: Reaction rate per reaction event define hoti hai, per molecule nahi. $r=\frac{-1}{2}\frac{d[\text{A}]}{dt}=\frac{0.04}{2}=0.02\text{ M/s}$ Kyun? Har reaction mein 2 A molecules vanish hote hain, isliye reaction utni baar hoti hai jitni baar A molecules disappear hote hain, uski aadhi.

Steel-man: Confusion everyday language se aati hai: "the rate A disappears" sunta hai "the rate of reaction" jaisa. Lekin chemists ko ek single number chahiye jo same ho chahe tum koi bhi species dekho. Isliye hum stoichiometry se normalize karte hain.

Ye sahi kyun lagta hai: 50 s interval ke beech mein hai, toh ye ek acha estimate lagta hai.

Fix: Δ[A]/Δt, 0-100 s ka average hai. t=50 s par instantaneous rate ke liye exactly t=50 s par tangent slope chahiye, jo bahut alag ho sakta hai agar reaction speed up ya slow down ho rahi ho.

Approximation kab theek hai? Agar reaction rate almost constant ho (rare!), ya agar Δt bahut chhota ho, toh average ≈ instantaneous.

Memory Aids

Active Recall Practice

#flashcards/chemistry

Reactant A ke terms mein average rate of reaction ka formula kya hai? :: $r_{\text{avg}}=-\frac{\Delta [\text{A}]}{\Delta t}=-\frac{[\text{A}{]}_2-[\text{A}{]}_1}{t_2-t_1}$ (negative sign decrease ko positive rate mein convert karta hai)

Connections

• Collision theory — explain karta hai ki rates kyun decrease hoti hain (kam reactant molecules → kam collisions)
• Rate law and order of reaction — dikhata hai ki instantaneous rate [reactant] par kaise depend karti hai
• Integrated rate laws — instantaneous rate expressions ko [A] vs. t equations mein convert karta hai
• Half-life of reactions — characteristic time define karne ke liye instantaneous rate concept use karta hai
• Reaction mechanisms — multi-step reactions mein rate-determining step hota hai (instantaneous rate concept)
• Catalysts — stoichiometry mein appear hue bina instantaneous rate change karte hain
• Arrhenius equation — rate constant ki temperature dependence (instantaneous rate affect karti hai)

Concept Map