Piv txwv cov lus nug thiab kev sib tham txog Exponential Decay
Kev puas tsuaj exponential yog ib qho xwm txheej ntuj tsim uas pom muaj nyob rau hauv ntau yam kev kawm xws li physics, chemistry, biology, thiab economics. Raws li tus qauv lej, kev puas tsuaj exponential piav qhia txog cov txheej txheem uas ib qho ntau npaum li cas txo qis proportional rau nws cov ntau tam sim no. Hauv lej, kev puas tsuaj exponential ua raws li daim ntawv dav dav:
\[ N(t) = N_0 e^{-\lambda t} \]
Qhov twg:
- \( N(t) \) yog tus nqi seem ntawm lub sijhawm \(t \),
– \( N_0 \) yog tus lej pib,
- \( \lambda \) yog qhov lwj tas mus li (feem ntau hu ua qhov nrawm lwj),
– \( t \) yog lub sijhawm,
- \( e \) yog lub hauv paus ntawm lub logarithm ntuj (kwv yees li 2.718).
Hauv tsab xov xwm no, peb yuav tham txog qee qhov piv txwv ntawm cov teeb meem exponential decay nrog rau lawv cov kev daws teeb meem los pab nkag siab lub tswv yim no tob dua.
Piv txwv lus nug 1: Kev puas tsuaj ntawm cov pa roj radioactive
Lo lus nug:
Ib yam khoom uas muaj radioactive muaj ib nrab ntawm lub neej yog 5 xyoos. Yog tias thaum xub thawj muaj 100 grams ntawm cov khoom no, yuav tshuav pes tsawg tom qab 15 xyoos?
Kev Sib Tham:
Kev puas tsuaj ntawm cov khoom siv hluav taws xob tuaj yeem ua qauv siv cov qauv exponential decay. Lub sijhawm ib nrab (t_{1/2} \)) yog lub sijhawm uas xav tau rau ib nrab ntawm cov khoom siv hluav taws xob kom lwj. Paub tias \(t_{1/2} = 5 \) xyoo.
Ua ntej peb yuav tsum nrhiav qhov lwj tas li \( \lambda \) nrog cov mis:
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
\[ \lambda = \frac{\ln 2}{5} \approx 0.1386 \text{ xyoo}^{-1} \]
Yog li, cov qauv exponential decay yog:
\[ N(t) = N_0 e^{-\lambda t} \]
\[ N(t) = 100 e^{-0.1386 \times 15} \]
Tam sim no, peb xam tus nqi:
\[ N(t) = 100 e^{-2.079} \]
\[ N(t) = 100 \times 0.125 \]
\[ N(t) \approx 12.5 \text{ grams} \]
Yog li, tom qab 15 xyoos, kwv yees li 12.5 grams ntawm cov khoom siv radioactive tseem tshuav.
Piv txwv 2: Kev puas tsuaj ntawm lub capacitor
Lo lus nug:
Ib lub capacitor uas muaj lub zog pib \( Q_0 = 200 \text{ C} \) raug tso cai kom tso tawm hauv lub voj voog. Lub sijhawm tas mus li yog \( \tau = 4 \text{ s} \). Muaj pes tsawg lub zog tseem tshuav tom qab 10 vib nas this?
Kev Sib Tham:
Nyob rau hauv cov ntaub ntawv ntawm capacitor charge decay, tus qauv exponential siv yog:
\[ Q(t) = Q_0 e^{-t/\tau} \]
Muab \( Q_0 = 200 \text{ C} \) thiab \( \tau = 4 \text{ s} \). Peb yuav tsum nrhiav \( Q(10) \):
\[ Q(10) = 200 e^{-10/4} \]
\[ Q(10) = 200 e^{-2.5} \]
Xam cov nqi exponential:
\[ Q(10) = 200 \times 0.0821 \]
\[ Q(10) \kwv yees li 16.42 \text{ C} \]
Yog li, tom qab 10 vib nas this, qhov them seem ntawm lub capacitor yog li 16.42 C.
Piv txwv lus nug 3: Kev lwj ntawm cov tshuaj lom neeg
Lo lus nug:
Ib yam tshuaj muaj qhov lwj tas li ntawm \( \lambda = 0.05 \text{ hnub}^{-1} \). Yuav siv sijhawm ntev npaum li cas rau cov tshuaj kom txo qis mus txog 25% ntawm nws cov nyiaj qub?
Kev Sib Tham:
Peb pib nrog cov qauv dav dav rau kev puas tsuaj exponential:
\[ N(t) = N_0 e^{-\lambda t} \]
Peb xav kom N(t) yog 25% ntawm \( N_0 \), yog li ntawd:
\[ 0.25 N_0 = N_0 e^{-0.05 t} \]
Tshem tawm \(N_0 \) ntawm ob sab:
\[ 0.25 = e^{-0.05 t} \]
Siv cov logarithms ntuj los daws cov teeb meem exponential:
\[ \ln 0.25 = -0.05 t \]
\[ -1.3863 = -0.05 t \]
Kev daws rau \(t\):
\[ t = \frac{1.3863}{0.05} \]
\[ t \kwv yees li 27.726 \text{ hnub} \]
Yog li, lub sijhawm xav tau rau cov tshuaj kom txo qis mus rau 25% ntawm nws cov nyiaj pib yog kwv yees li 27.726 hnub.
Piv txwv lus nug 4: Kev puas tsuaj ntawm cov kab mob
Lo lus nug:
Cov kab mob me me zuj zus sai heev tom qab 3 teev, cov kab mob yuav muaj ib nrab ntawm nws cov lej pib. Yog tias cov lej pib yog 8000 tus kab mob, muaj pes tsawg tus kab mob tseem tshuav tom qab 9 teev?
Kev Sib Tham:
Paub tias lub sijhawm ib nrab ntawm lub neej \( t_{1/2} = 3 \) teev. Ua ntej peb pom qhov lwj tas mus li \( \lambda \):
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
\[ \lambda = \frac{\ln 2}{3} \approx 0.231 \text{ teev}^{-1} \]
Tom qab ntawd, peb siv cov qauv exponential decay:
\[ N(t) = N_0 e^{-\lambda t} \]
\[ N(9) = 8000 e^{-0.231 \times 9} \]
Xam cov nqi exponential:
\[ N(9) = 8000 e^{-2.079} \]
\[ N(9) = 8000 \times 0.125 \]
\[ N(9) \kwv yees li 1000 \]
Yog li ntawd, tom qab 9 teev, yuav muaj li ntawm 1000 tus kab mob nyob.
Xaus
Tus qauv exponential decay muab ib txoj hauv kev zoo los daws cov teeb meem ntsig txog cov txheej txheem decay hauv ntau yam kev siv scientific thiab engineering. Los ntawm kev nkag siab txog cov ntsiab lus yooj yim xws li decay constants, ib nrab-lub neej, thiab kev siv cov qauv exponential, peb tuaj yeem xam qhov kev hloov pauv ntawm qhov ntau dhau sijhawm nrog kev yooj yim. Cov teeb meem kev xyaum uas tau tham saum toj no yuav tsum pab peb nkag siab thiab siv lub tswv yim ntawm exponential decay rau cov xwm txheej nyuaj dua.
