Lipotso tsa Mehlala le Puisano ea ho Bola ha Exponential
Ho bola ha exponential ke ketsahalo ea tlhaho e fumanoang mafapheng a fapaneng a kang fisiks, k'hemistri, baeloji le moruo. Jwalo ka mohlala oa lipalo, ho bola ha exponential ho hlalosa mokhoa oo bongata bo fanoeng bo fokotsehang ka oona ka ho lekana ho ea ka bongata ba hona joale. Lipalong, ho bola ha exponential ho latela sebopeho se akaretsang:
\[ N(t) = N_0 e^{-\lambda t} \]
Moo:
– \( N(t) \) ke tjhelete e setseng ka nako \(t \),
– \( N_0 \) ke nomoro ya pele,
– \( \lambda \) ke ntho e sa fetoheng ya ho bola (hangata e bitswang sekgahla sa ho bola),
– \(t \) ke nako,
– \( e \) ke motheo oa logarithm ea tlhaho (hoo e ka bang 2.718).
Sehloohong sena, re tla tšohla mehlala e meng ea mathata a ho bola ha exponential hammoho le litharollo tsa ona ho thusa ho utloisisa mohopolo ona ka botebo.
Mohlala oa Potso ea 1: Ho Bola ha Mahlaseli a Leseli
Potso:
Sehlahisoa se nang le mahlaseli a kotsi se na le nako ea bophelo ba halofo ea lilemo tse 5. Haeba qalong ho ne ho e-na le ligrama tse 100 tsa ntho eo, ho ne ho tla sala bokae ka mor'a lilemo tse 15?
Puisano:
Ho bola ha mahlaseli a kotsi ho ka etswa mohlala ho sebediswa foromo ya ho bola ha maqhubu a kotsi. Half-life (\(t_{1/2} \)) ke nako e hlokahalang hore halofo ya bongata ba thepa ya mahlaseli a kotsi e bole. Ho a tsebahala hore \(t_{1/2} = 5 \) dilemo.
Taba ea pele re hloka ho fumana constant ea decay \( \lambda \) ka foromo:
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
\[ \lambda = \frac{\ln 2}{5} \hoo e ka bang 0.1386 \mongolo{selemo}^{-1} \]
Ka hona, foromo ea ho bola ha exponential ke:
\[ N(t) = N_0 e^{-\lambda t} \]
\[ N(t) = 100 e^{-0.1386 \makgetlo a 15} \]
Jwale, re bala boleng:
\[ N(t) = 100 e^{-2.079} \]
\[ N(t) = 100 \makgetlo a 0.125 \]
\[ N(t) \hoo e ka bang 12.5 \mongolo{ digrama} \]
Kahoo, kamora dilemo tse 15, ho setse digrama tse ka bang 12.5 tsa dintho tse nang le mahlasedi.
Mohlala oa 2: Ho bola ha Capacitor
Potso:
Capacitor e nang le tjhaja ya pele \( Q_0 = 200 \text{C} \) e dumellwa ho ntsha motlakase ka hara potoloho. Nako e sa fetoheng ke \( \tau = 4 \text{s} \). Ho sala tjhaja e kae kamora metsotsoana e 10?
Puisano:
Tabeng ea ho bola ha tjhaja ea capacitor, mohlala oa exponential o sebelisitsoeng ke:
\[ Q(t) = Q_0 e^{-t/\tau} \]
Ho fanoe \( Q_0 = 200 \text{ C} \) le \( \tau = 4 \text{ s} \). Re hloka ho fumana \( Q(10) \):
\[ Q(10) = 200 e^{-10/4} \]
\[ Q(10) = 200 e^{-2.5} \]
Ho bala boleng ba exponential:
\[ Q(10) = 200 \makgetlo 0.0821 \]
\[ Q(10) \hoo e ka bang 16.42 \mongolo{ C} \]
Kahoo, kamora metsotsoana e 10, tjhaja e setseng ho capacitor ke hoo e ka bang 16.42 C.
Mohlala oa Potso ea 3: Ho Bola ha Lik'hemik'hale
Potso:
Khemikhale e na le botsitso ba ho bola ba \( \lambda = 0.05 \text{ days}^{-1} \). Ho tla nka nako e kae hore k'hemik'hale e fokotsehe ho fihlela ho 25% ea bongata ba eona ba pele?
Puisano:
Re qala ka foromo e akaretsang ea ho bola ha exponential:
\[ N(t) = N_0 e^{-\lambda t} \]
Re batla hore N(t) e be 25% ea \( N_0 \), e le hore:
\[ 0.25 N_0 = N_0 e^{-0.05 t} \]
Ho tlosa \( N_0 \) mahlakoreng ka bobeli:
\[ 0.25 = e^{-0.05 t} \]
Ho sebelisa li-logarithm tsa tlhaho ho rarolla linyeoe tsa exponential:
\[ \ln 0.25 = -0.05 t \]
\[ -1.3863 = -0.05 t \]
Ho rarolla \(t \):
\[t = \frac{1.3863}{0.05} \]
\[t \hoo e ka bang 27.726 \mongolo{matsatsi} \]
Kahoo, nako e hlokahalang hore k'hemik'hale e fokotsehe ho fihlela ho 25% ea bongata ba eona ba pele ke matsatsi a ka bang 27.726.
Mohlala oa Potso ea 4: Ho Bola ha Palo ea Baktheria
Potso:
Palo ea libaktheria e fokotseha ka sekhahla se seholo hoo ka mor'a lihora tse 3, palo ea baahi e leng halofo ea palo ea eona ea pele. Haeba palo ea pele e ne e le libaktheria tse 8000, ho sala libaktheria tse kae ka mor'a lihora tse 9?
Puisano:
Hoa tsebahala hore halofo ea bophelo \(t_{1/2} = 3 \) lihora. Pele re fumana ho bola ho sa fetoheng \( \lambda \):
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
\[ \lambda = \frac{\ln 2}{3} \hoo e ka bang 0.231 \text{ hour}^{-1} \]
Kamora moo, re sebelisa foromo ea ho bola ha exponential:
\[ N(t) = N_0 e^{-\lambda t} \]
\[ N(9) = 8000 e^{-0.231 \makgetlo a 9} \]
Ho bala boleng ba exponential:
\[ N(9) = 8000 e^{-2.079} \]
\[ N(9) = 8000 \makgetlo a 0.125 \]
\[ N(9) \hoo e ka bang 1000 \]
Kahoo, ka mora dihora tse 9, dibaktheria tse ka bang 1000 di tla sala.
Qetello
Mohlala oa ho bola ha exponential o fana ka mokhoa o sebetsang oa ho rarolla mathata a amanang le lits'ebetso tsa ho bola lits'ebetsong tse fapaneng tsa saense le tsa boenjiniere. Ka ho utloisisa likhopolo tsa motheo tse kang li-constant tsa ho bola, halofo ea bophelo, le ts'ebeliso ea liforomo tsa exponential, re ka bala phetoho ea bongata ha nako e ntse e ea habonolo. Mathata a mokhoa a tšohliloeng ka holimo a lokela ho re thusa ho utloisisa le ho sebelisa mohopolo oa ho bola ha exponential maemong a rarahaneng haholoanyane.
