Imithetho yezinga ehlanganisiwe: ukugxila, isikhathi ne-half-life

Ukuguqula umthetho wezinga ube ukugxila ngokuqhathaniswa nesikhathi

Umthetho wezinga we-differential, njengokuthi izinga = k[A], uxhumanisa izinga nokugxila ngomzuzwana owodwa. Njengoba ukusabela kuqhubeka, u-[A] uyehla, ngakho nezinga liyehla. Ukuhlanganisa umthetho wezinga ngokuhamba kwesikhathi, okuyisinyathelo se-calculus ongadingi ukusiphinda enkingeni ngayinye, kunikeza isibalo sika-[A] nganoma yisiphi isikhathi t ngokokugxila kokuqala [A]₀ ne-rate constant k.

Izibalo ezingezansi zisebenza ekusabeleni izinga lako elincike ekugxileni kwesisabelo esisodwa, A, ezingeni lokushisa elingaguquki. Zisebenza nalapho ezinye izisabelo zikhona ngokweqile okukhulu kangangokuthi ukugxila kwazo kushintsha kancane kakhulu. Leso simo sibizwa ngokuthi ukuziphatha kwe-pseudo-first-order (noma i-pseudo-zero-order).

Izibalo ezintathu okufanele uzigcine eceleni kwesinye

Umthetho wezinga ohlanganisiwe ngamunye ungahlelwa kabusha ube ngefomu lomugqa oqondile, y = mx + b, isikhathi sise-eksisini ye-x. I-half-life, t½, yisikhathi esithathwa ukuthi u-[A] ehle abe yingxenye yenani lakhe lamanje. Ivela ekubekeni u-[A] = ½[A]₀ esibalweni ngasinye.

I-half-life ye-oda lokuqala kuphela engancikile ekugxileni. Lelo qiniso elilodwa lichaza ukuthi kungani izinqubo ze-oda lokuqala, kuhlanganise nokubola kwe-radioactive, zivame kangaka ukuchazwa ngama-half-life azo.

• I-oda le-zero: [A] = [A]₀ − kt. Igrafu ka-[A] ngokuqhathaniswa no-t ingumugqa oqondile onomqansa −k. I-half-life: t½ = [A]₀/(2k), ngakho iba fushane njengoba ukusabela kuqhubeka.
• I-oda lokuqala: ln[A] = ln[A]₀ − kt, noma [A] = [A]₀e⁻ᵏᵗ. Igrafu ka-ln[A] ngokuqhathaniswa no-t ingumugqa oqondile onomqansa −k. I-half-life: t½ = ln 2 / k = 0.693/k, efanayo kuzo zonke izigaba.
• I-oda lesibili: 1/[A] = 1/[A]₀ + kt. Igrafu ka-1/[A] ngokuqhathaniswa no-t ingumugqa oqondile onomqansa +k. I-half-life: t½ = 1/(k[A]₀), ngakho i-half-life ngayinye inde ngokuphindwe kabili kunaleyo eyandulelayo.

Isibonelo esisebenzile: i-cyclobutane, ukubola kwe-oda lokuqala

Isigaba 12.4 se-OpenStax sinikeza i-rate constant ye-oda lokuqala yokubola kwe-cyclobutane, C₄H₈ → 2 C₂H₄, njengo-9.2 × 10⁻³ s⁻¹ ku-500 °C. I-half-life ingu-t½ = 0.693 / (9.2 × 10⁻³ s⁻¹) = 75 s, ezinombolweni ezibalulekile ezimbili ukuze ihambisane no-k.

Kusala ingxenye engakanani ye-cyclobutane ngemva kuka-3.00 min? Guqula isikhathi sibe amasekhondi kuqala, ngoba u-k ungu-s⁻¹: t = 180 s. I-exponent ingu-kt = (9.2 × 10⁻³ s⁻¹)(180 s) = 1.656, ngakho [A]/[A]₀ = e⁻ᵏᵗ = e ephakanyiswe emandleni −1.656 = 0.19, futhi kusala amaphesenti angu-19. Uma ukugxila kokuqala bekungu-0.0800 mol/L, bese [A] = 0.0800 × 0.19 = 0.015 mol/L. Ubungadingi ukugxila kokuqala ukuthola ingxenye. Lokho kuyiqiniso kuphela ekusabeleni kwe-oda lokuqala.

Kuthatha isikhathi esingakanani ukwehla kube ngamaphesenti angu-5.0 enani lokuqala? Hlela kabusha ube t = ln([A]₀/[A]) / k = ln(1/0.050) / (9.2 × 10⁻³ s⁻¹) = 2.996 / (9.2 × 10⁻³ s⁻¹) = 3.3 × 10² s, cishe 5.4 min. Njengokuhlola, lokho kucishe kube ngama-half-life angu-4.3, futhi (½)⁴·³ ≈ 0.05.

Isibonelo esisebenzile: i-dimerization ye-butadiene, ukusabela kwe-oda lesibili

Amamolekyuli e-butadiene ahlangana ngamabhangqa, 2 C₄H₆ → C₈H₁₂, ekusabeleni okuyi-oda lesibili ku-butadiene. Isigaba 12.4 se-OpenStax sicaphuna u-k = 5.76 × 10⁻² L mol⁻¹ min⁻¹ ngaphansi kwezimo esizichazayo. Qala ngo-[C₄H₆]₀ = 0.0500 mol/L.

I-half-life yokuqala ingu-t½ = 1/(k[A]₀) = 1 / (5.76 × 10⁻² L mol⁻¹ min⁻¹ × 0.0500 mol/L) = 347 min. I-half-life yesibili iqala ku-0.0250 mol/L futhi ithatha u-1/(5.76 × 10⁻² × 0.0250) = 694 min, okuphindwe kabili ubude. I-half-life ekhula njengoba ukugxila kwehla iwuphawu lwe-kinetics ye-oda lesibili.

Ukuthola ukugxila ngemva kuka-2.00 h, sebenzisa amaminithi ukuze amayunithi ahambisane no-k: t = 120 min. 1/[A] = 1/0.0500 + (5.76 × 10⁻²)(120) = 20.0 + 6.912 = 26.91 L/mol, ngakho [A] = 0.0372 mol/L. Impendulo incane kunenani lokuqala futhi inkulu kunengxenye yalo, njengoba kufanele kube njalo esikhathini esifushane kune-half-life yokuqala.

Ukubala ama-half-life ezinkingeni ze-oda lokuqala

Ekusabeleni kwe-oda lokuqala, ingxenye esele ngemva kwama-half-life angu-n ingu-(½)ⁿ. Ngemva kwe-half-life eyodwa, kusala amaphesenti angu-50; ngemva kwamabili, angu-25; ngemva kwamathathu, angu-12.5. Lapho ukugxila okuhlosiwe kukude ngenani eliphelele lokuhlukaniswa kabili, ungabala esikhundleni sokusebenzisa ama-logarithm.

Isibonelo esisebenzile. Isigaba 12.4 se-OpenStax sibika ukuthi i-hydrogen peroxide ibola nge-kinetics ye-oda lokuqala ku-40 °C ne-half-life engu-2.16 × 10⁴ s, okungu-6.00 h. I-rate constant ingu-k = 0.693 / (2.16 × 10⁴ s) = 3.21 × 10⁻⁵ s⁻¹. Ukusuka ku-0.800 mol/L kuya ku-0.100 mol/L kungukuhlukaniswa kabili kathathu (0.800 → 0.400 → 0.200 → 0.100), ngakho kuthatha u-3 × 6.00 h = 18.0 h.

Ku-0.800 mol/L kuya ku-0.150 mol/L inani aliyona inombolo ephelele, ngakho sebenzisa isibalo: t = ln(0.800/0.150) / k = 1.674 / (3.21 × 10⁻⁵ s⁻¹) = 5.22 × 10⁴ s, noma 14.5 h. Lokho kuwela phakathi kwama-half-life amabili (12.0 h) namathathu (18.0 h), njengoba bekulindelekile.

Ukukhetha i-oda kusuka etafuleni lokugxila nesikhathi

Uma unokugxila okulinganiswe ezikhathini eziningana, hlola ifomu ngalinye le-linear. Bala u-[A], ln[A] no-1/[A] kuwo wonke amaphuzu bese ubona ukuthi yikuphi okushintsha ngenani elifanayo esinyathelweni ngasinye sesikhathi esilinganayo. Ngedatha yangempela enomsindo, bonisa kokuthathu kugrafu bese ukhetha umugqa oqonde kakhulu, noma uqhathanise i-correlation yokufanisa kwe-linear.

Isibonelo esisebenzile. Idatha elandelayo yakhelwe ukuzijwayeza. Ku-t = 0, 50, 100, 150 no-200 s, [A] = 0.1000, 0.0667, 0.0500, 0.0400 no-0.0333 mol/L. Ukwehla kuka-[A] esinyathelweni ngasinye sika-50 s kungu-0.0333, 0.0167, 0.0100 no-0.0067 mol/L, okungaguquki, ngakho ukusabela akuyona i-oda le-zero. Ushintsho ku-ln[A] lungu-−0.405, −0.288, −0.223 no-−0.183, nalo olungaguquki, ngakho akuyona i-oda lokuqala. Amanani ka-1/[A] angu-10.0, 15.0, 20.0, 25.0 no-30.0 L/mol, akhuphuka ngo-5.0 L/mol njalo ngemva kuka-50 s. Ukusabela kuyi-oda lesibili, futhi umqansa unikeza k = 5.0 / 50 s = 0.100 L mol⁻¹ s⁻¹.

Ama-half-life ayaqinisekisa i-oda. U-[A] wehla esuka ku-0.100 eya ku-0.0500 mol/L ku-100 s, futhi ifomula ye-oda lesibili ibikezela ukuthi ukuhlukaniswa kabili okulandelayo, kuya ku-0.0250 mol/L, kuzothatha u-1/(0.100 × 0.0500) = 200 s.

Lapho izimpendulo zomthetho wezinga ohlanganisiwe zihamba kabi

Amaphutha amaningi avela ekuxubeni izibalo noma amayunithi. Ukuhlola okusheshayo ukuthi konke ukugxila okubalayo kufanele kube kuncane kuno-[A]₀ futhi kube kukhulu kune-zero.

• Ukusebenzisa isibalo se-oda lokuqala ekusabeleni i-oda lako ongakaliqinisekisi.
• Ukuxuba amayunithi esikhathi: u-k ngo-s⁻¹ no-t ngamaminithi kunikeza inani lika-kt eliphambuke ngesici esingu-60.
• Ukubhala u-ln([A]/[A]₀) = kt futhi ulahle uphawu lokususa, okwenza ukugxila kukhule ngokuhamba kwesikhathi.
• Ukusebenzisa u-log (isisekelo 10) lapho isibalo sidinga u-ln (isisekelo e).
• Ukusebenzisa u-t½ = 0.693/k ekusabeleni kwe-oda lesibili, i-half-life yako encike ekugxileni.
• Ukuthatha umqansa wegrafu ka-1/[A] njengo-−k. Ku-oda lesibili umqansa ungu-+k.