2.8.3
Differential rate equations ko samajhna matlab yeh connect karna ki ek reaction kitni tezi se hoti hai (rate) aur reactants ki abhi ki concentration kya hai. Order of reaction ka har ek case ek alag mathematical relationship deta hai, aur inhe scratch se derive karne par pata chalta hai ki kyun concentration rate ko uss tarah affect karti hai jaise karti hai.
Intuition Alag-Alag Orders Kyun Hote HainChemical reactions ko highway pe chalti cars ki tarah socho. Zero-order reaction mein, highway pe ek narrow toll booth hai—chahe kitni bhi cars pile up ho jaayein (concentration), uss se same fixed number hi per minute nikal sakti hain (rate constant rehti hai). First-order mein, har car independently chalti hai—cars double karo, throughput double ho jaata hai. Second-order mein, cars ko merge karne ke liye collide karna padta hai—agar cars double karo, to collisions quadruple ho jaati hain kyunki har car ke paas double partners hote hain jisse hit ho sake.
Definition Rate Law (General Form)Ek reaction aA → products differential rate equation yeh hai:
Rate = − a 1 d t d [ A ] = k [ A ] n
k rate constant (units n n order of reaction (experimentally determine hota hai)Negative sign isliye: concentration time ke saath kamat hai
Starting point : Rate, concentration ko power zero tak raise karne pe depend karti hai.
− d t d [ A ] = k [ A ] 0 = k
YEH STEP KYUN? Zero power ka matlab hai concentration matter nahi karti—rate hamesha k
Variables separate karo :
d [ A ] = − k d t
Dono sides ko integrate karo t = 0 [ A ] 0 t [ A ] t
∫ [ A ] 0 [ A ] t d [ A ] = − k ∫ 0 t d t
[ A ] t − [ A ] 0 = − k t
Worked example Worked Example: Surface-Catalyzed DecompositionAmmonia platinum surface pe decompose hoti hai: 2NH 3 (g) → N 2 (g) + 3H 2 (g)
Diya hai: [ NH 3 ] 0 = 0.10 M k = 2.5 × 1 0 − 4 M s − 1
Dhundho : 200 s ke baad concentration
Solution :
[ NH 3 ] t = 0.10 − ( 2.5 × 1 0 − 4 ) ( 200 ) = 0.10 − 0.05 = 0.05 M Yeh step kyun? [ A ] t = [ A ] 0 − k t NH 3
Starting point : Rate, concentration ke proportional hoti hai.
− d t d [ A ] = k [ A ]
Variables separate karo :
[ A ] d [ A ] = − k d t
YEH STEP KYUN? Hume [ A ] t
Integrate karo :
∫ [ A ] 0 [ A ] t [ A ] d [ A ] = − k ∫ 0 t d t
ln [ A ] t − ln [ A ] 0 = − k t
ln [ A ] 0 [ A ] t = − k t
Worked example Worked Example: Radioactive Decay14 C t 1/2 = 5730 years
Dhundho : k
Solution :
Step 1 : k k = 5730 0.693 = 1.21 × 1 0 − 4 yr − 1
Yeh step kyun? Half-life formula ko rearrange kiya.
Step 2 : Integrated law use karo
ln [ A ] 0 [ A ] t = − ( 1.21 × 1 0 − 4 ) ( 10000 ) = − 1.21
[ A ] 0 [ A ] t = e − 1.21 = 0.298
Yeh step kyun? Dono sides ko exponentiate karne se natural log invert ho jaata hai. 10,000 saal baad lagbhag 30% bachta hai (note: t = 10 , 000 k t = 1.21 × 1 0 − 4 × 10 , 000 = 1.21
Starting point : Rate, concentration squared ke proportional hoti hai.
− d t d [ A ] = k [ A ] 2
SQUARED KYUN? Iska matlab aksar yeh hota hai ki A ke do molecules collide karne chahiye. Collision ki probability [ A ] × [ A ]
Variables separate karo :
[ A ] 2 d [ A ] = − k d t
Left side ko rewrite karo :
[ A ] − 2 d [ A ] = − k d t
Integrate karo :
∫ [ A ] 0 [ A ] t [ A ] − 2 d [ A ] = − k ∫ 0 t d t
[ − [ A ] − 1 ] [ A ] 0 [ A ] t = − k t
− [ A ] t 1 + [ A ] 0 1 = − k t
[ A ] t 1 − [ A ] 0 1 = k t
Worked example Worked Example: Dimerization2 NO 2 ( g ) → N 2 O 4 ( g ) NO 2
Diya hai: [ NO 2 ] 0 = 0.0100 M k = 0.543 M − 1 s − 1 t = 300 s
Dhundho : [ NO 2 ] t
Solution :
[ NO 2 ] t 1 = 0.0100 1 + ( 0.543 ) ( 300 ) Yeh step kyun? Second-order law mein direct substitution.
= 100 + 162.9 = 262.9 M − 1
[ NO 2 ] t = 262.9 1 = 3.80 × 1 0 − 3 M
Yeh step kyun? Reciprocal lene se concentration milti hai. Concentration initial ki ~38% tak drop ho gayi kyunki do NO 2
Order
Differential
Integrated
Linear Plot
Slope
Units of k
Half-life
0
− d t d [ A ] = k [ A ] t = [ A ] 0 − k t [ A ] t − k M s − 1 2 k [ A ] 0
1
− d t d [ A ] = k [ A ] ln [ A ] t = ln [ A ] 0 − k t ln [ A ] t − k s − 1 k 0.693
2
− d t d [ A ] = k [ A ] 2 [ A ] t 1 = [ A ] 0 1 + k t [ A ] 1 t k M − 1 s − 1 k [ A ] 0 1
Common mistake Common Error: Kaunsa Graph Use Karna Hai Yeh Confuse KarnaGalat soch : "Main har reaction ke liye [ A ] t
Kyun sahi lagta hai : Sabse simple plot hai, aur zero-order ke liye kaam karta hai.
Fix : Har order ek alag y-axis transformation se linearize hota hai:
Zero-order: [ A ] t
First-order: ln [ A ] t
Second-order: [ A ] 1 t Yaad kaise rakhein : Integrated form batata hai kya plot karna hai—left side hi tumhara y-axis hai!
Common mistake Common Error:k Galat soch : "k
Kyun sahi lagta hai : First-order sabse common hai, aur usme k s − 1
Fix : Units aisi honi chahiye ki rate equation dimensionally consistent rahe. Rate ki units hamesha M s − 1 k [ A ] n must equal M s − 1 k = [ M ] n M s − 1 = M 1 − n s − 1
Recall Feynman Explain-to-a-12-Year-OldImagine karo tum ek video game khel rahe ho jisme enemies spawn hoti hain aur tumhe unhe shoot karna hai.
Zero-order : Teri gun exactly 10 bullets per second fire karti hai chahe screen pe kitni bhi enemies ho. Agar 5 enemies aayein, tum 10 baar shoot karte ho. Agar 500 aayein, phir bhi 10 baar per second. "Rate" ko koi fark nahi padata ki kitne targets hain.
First-order : Ab teri gun automatic hai aur enemies ko track karti hai—agar 10 enemies hain, to 10 shots per second fire hoti hain. Agar 20 enemies hain, to 20 shots. Rate seedha proportional hai ki kitne hain.
Second-order : Tumhe ek special laser se do enemies ko ek saath hit karna hota hai jo tabhi fire hoti hai jab enemies align hoti hain. Agar 10 enemies hain, to shayad 45 possible pairs ho sakte hain. Agar 20 enemies hain, to 190 pairs hain! Successful hits ki number bahut tezi se badhti hai kyunki har enemy baaki har enemy ke saath pair ho sakti hai. Isliye rate, concentration squared pe depend karti hai.
Mnemonic Order Memory Aid"Zero, One, Two = Flat, Ln, Divide"
Zero : Plot time ke saath flat rehti hai (constant rate), [ A ] One : Concentration ka natural Ln use karoTwo : Divide karo (reciprocal lo [ A ] 1
Half-life mnemonic : "Zero Depends, One Never, Two Inverts"
Zero-order t 1/2 [ A ] 0 depend karta hai
First-order kabhi [ A ] 0
Second-order inverted hai (bada [ A ] 0 t 1/2
#flashcards/chemistry
Zero-order reaction ke liye integrated rate law kya hai? [ A ] t = [ A ] 0 − k t [ A ] t − k
First-order reaction mein half-life, initial concentration pe kaise depend karta hai? Karta hi nahi. t 1/2 = k 0.693 [ A ] 0
Second-order reaction ke liye rate constant ki units kya hain? M − 1 s − 1 L mol − 1 s − 1
Graphically reaction order kaise determine karte hain? [ A ] t ln [ A ] t [ A ] 1 t
− d t d [ A ] = k [ A ] Variables separate karo: [ A ] d [ A ] = − k d t ∫ [ A ] d [ A ] = − k ∫ d t ln [ A ] t − ln [ A ] 0 = − k t ln [ A ] t = ln [ A ] 0 − k t
Second-order reaction mein rate, [ A ] 2 Kyunki A ke do molecules collide karne chahiye. Collision frequency unki concentrations ke product ke proportional hoti hai: [ A ] × [ A ] = [ A ] 2
Zero-order reaction mein agar initial concentration double kar do to half-life mein kya change aata hai? Yeh bhi double ho jaata hai. t 1/2 = 2 k [ A ] 0 [ A ] 0 t 1/2
Rate constant ki units aur reaction order n k M 1 − n s − 1 n = 0 M s − 1 n = 1 s − 1 n = 2 M − 1 s − 1
General Rate Law k A to n
1 over A = 1 over A0 + kt
