Foundations — Differential rate equations for 0, 1st, 2nd order — derivations

2.8.3 · D1 · Chemistry › Chemical Kinetics › Differential rate equations for 0, 1st, 2nd order — derivati

Pehle aapko parent note mein ek bhi derivation padhne se pehle, usme use kiya gaya har squiggle obvious lagni chahiye. Yeh page unhe ek-ek karke zero se build karta hai — koi prior chemistry nahi, koi prior calculus nahi assumed.

1. Concentration aur square-bracket notation $[\text{A}]$

Ek fixed 1-litre jar socho. Substance A us mein floating coloured dots hain. $[\text{A}]$ simply dots kitne crowded hain — yahi hai.

Subscripts jo aap dekhenge:

• $[\text{A}{]}_0$ — shuru mein concentration (chhota $0$ matlab "time zero").
• $[\text{A}{]}_t$ — kisi baad ke time $t$ par concentration.

2. Time $t$, aur change $\Delta$

$t$ bas reaction shuru hone ke baad ka elapsed time hai (seconds, ya dheeми reactions ke liye saal).

Greek capital delta $\Delta$ ka matlab hai "change in" — baad ki value minus pehle ki value. Toh kisi bhi quantity ke liye, $\Delta (\text{quantity})=(\text{baad ki value})-(\text{pehle ki value})$.

Kyunki reactant khatam hota jaata hai, $[\text{A}{]}_t$, $[\text{A}{]}_0$ se chhhota hai, isliye $\Delta [\text{A}]$ negative hai. Yeh minus sign yaad rakho — yehi wajah hai ki aage minus everywhere dikhta hai.

3. Rate = concentration curve ki steepness

Time $t$ ko horizontal ($x$) axis par aur concentration $[\text{A}]$ ko vertical ($y$) axis par rakho. Jaise time right jaata hai, curve neeche slide karta hai kyunki reactant consume ho raha hai. Rate yahi hai ki woh curve kitna steeply girta hai.

Woh "ek instant par slope" exactly wohi hai jo derivative aage deta hai.

4. Derivative $\frac{d[\text{A}]}{dt}$

Ise zor se padho "dee A dee tee" — yeh $\Delta [\text{A}]/\Delta t$ hai jab aap run $\Delta t$ ko zero ki taraf shrink karte ho, toh do points ek mein collapse ho jaate hain aur straight line tangent ban jaati hai.

Kyunki concentration girta hai, yeh slope negative hai — isliye topic rate ko leading minus ke saath likhta hai taaki rate khud positive rahe:

$\text{Rate}=-\frac{d[\text{A}]}{dt}(\text{ek positive number})$

5. Rate constant $k$ aur order $n$

Saath mein, poora topic ek sentence par tikaa hai:

Teen cases jo parent derive karta hai wo bas $n$ ki teen choices hain:

• $n=0$: crowding ko power $0$ raise karo toh $1$ milta hai, isliye speed $=k$, flat — crowding ignore hoti hai.
• $n=1$: speed $=k[\text{A}]$, origin se straight — crowding double, speed double.
• $n=2$: speed $=k[\text{A}{]}^2$, upar curve karta hai — crowding double, speed chaar guna.

$n$ kaise measure hota hai yeh dekhne ke liye Rate Laws and Reaction Order dekho, aur $n=2$ ka matlab "do molecules milne chahiye" kyun hai yeh Collision Theory mein hai.

6. Powers — general idea, phir teen jo actually chahiye

Exponent bas "base ko khud se kitni baar multiply karo" hai — $[\text{A}{]}^3=[\text{A}]\cdot [\text{A}]\cdot [\text{A}]$. Yeh picture smoothly kisi bhi number exponent tak extend hoti hai:

Reaction orders fractional ho sakte hain (kuch real reactions order $\frac{1}{2}$ ya $\frac{3}{2}$ ki hoti hain), isliye general rule matter karta hai. Lekin parent mein teen derivations sirf whole-number aur $-1$ cases use karte hain neeche — isliye yeh teen har worked example ke liye kaafi hain:

• Kisi bhi cheez ko power $0$ = $1$ — isliye zero-order rate bas $k$ mein collapse hoti hai.
• Power $-1$ matlab reciprocal (one-over) — isliye second-order answer $\frac{1}{[\text{A}]}$ ke roop mein aata hai.

7. Integration: saare chhote changes jodna

Derivative ne curve liya aur slope diya. Integration ulta karta hai — yeh slope rule leta hai aur curve rebuild karta hai. Lamba "S" symbol $\int$ matlab "continuously sum karo."

Teen specific integrals parent mein aate hain, ek per order. Inhe yahan derive karne ki zarurat nahi — bas inhe pehchano:

Beech ka integral ek brand-new function introduce karta hai, $\ln$ — aage milte hain.

8. Natural logarithm $\ln$ aur iska partner $e^x$

$\ln$ first-order mein milta hai kyunki $\frac{1}{[\text{A}]}$ integrate karne par exactly $\ln [\text{A}]$ milta hai — koi aur elementary function nahi hai jiska slope $\frac{1}{x}$ ho.

$e^x$ aur $\ln x$ ek doosre ko undo karte hain, jaise $+$ aur $-$:

Yahi move parent ke carbon-14 (${}^{14}\mathrm{C}$) example mein use hota hai: jab equation $\ln (\text{ratio})=-1.21$ padhta hai, aap $\ln$ ko peel off karne ke liye dono sides par $e^{(\cdots )}$ apply karte ho aur ratio khud milta hai, $e^{-1.21}=0.298$.

9. Straight-line graph padhna: slope aur intercept

Har integrated law ko rearrange kiya jaata hai taaki plot straight nikle. Ek straight line obey karta hai:

• slope = $x$ ke har step par $y$ kitna tezi se badhta hai (steepness).
• intercept = line $y$-axis se kahan milti hai ($x=0$ par).

Prerequisite map

Neeche ka diagram dikhata hai ki is page par har foundation agale mein kaise feed hota hai, rate equations tak pohonchte hue. Ise upar se neeche padho: concentration aur time aapko rate as a slope dete hain; slope idea derivative mein sharpen hota hai; derivative plus power rules rate law build karte hain; integration (aur uske offshoots $\ln$/$e$ aur straight-line reading) phir us rate law ko usable formulas mein badalta hai. Agar diagram aapke device par render nahi hota, toh wohi chain is paragraph mein words mein spelled out hai.

Aage, yeh Half-Life Calculations, Arrhenius Equation, aur Steady-State Approximation mein feed karte hain.

Equipment checklist

Khud test karo — sirf jawab dene ke baad reveal karo.