Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o ka Exponential Decay
He hanana kūlohelohe ka palaho exponential i loaʻa i nā ʻano aʻo like ʻole e like me ke kino, ke kemika, ka biology, a me ka hoʻokele waiwai. Ma ke ʻano he kumu hoʻohālike makemakika, wehewehe ka palaho exponential i ke kaʻina hana e emi ai kahi nui i hāʻawi ʻia me ka like me kona nui o kēia manawa. I ka makemakika, hahai ka palaho exponential i ke ʻano maʻamau:
\[ N(t) = N_0 e^{-\lambda t} \]
Ma hea:
– ʻO \( N(t) \) ka nui i koe i ka manawa \(t \),
– ʻO \( N_0 \) ka helu mua,
– ʻO \( \lambda \) ke kūpaʻa palaho (i kapa pinepine ʻia ʻo ka helu palaho),
– ʻo \( t \) ka manawa,
– ʻO \( e \) ke kumu o ka logarithm kūlohelohe (ma kahi o 2.718).
Ma kēia ʻatikala, e kūkākūkā mākou i kekahi mau laʻana o nā pilikia palaho exponential me kā lākou mau hoʻonā e kōkua i ka hoʻomaopopo hohonu ʻana i kēia manaʻo.
Laʻana Nīnau 1: Palaho Radioactive
Nīnau:
He 5 mau makahiki ka hapalua ola o kahi mea radioactive. Inā he 100 grams o ia mea i ka wā mua, ehia ka nui e koe ma hope o 15 mau makahiki?
Kūkākūkā:
Hiki ke hoʻohālikelike ʻia ka palaho radioactive me ka hoʻohana ʻana i ke ʻano palaho exponential. ʻO ka hapalua ola (\( t_{1/2} \)) ka manawa e pono ai no ka hapalua o ka nui o nā mea radioactive e palaho. Ua ʻike ʻia ʻo \( t_{1/2} = 5 \) mau makahiki.
ʻO ka mea mua, pono mākou e ʻimi i ke kūpaʻa palaho \( \lambda \) me ke ʻano hana:
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
\[ \lambda = \frac{\ln 2}{5} \approx 0.1386 \text{ makahiki}^{-1} \]
No laila, ʻo ke ʻano hoʻohaʻahaʻa exponential:
\[ N(t) = N_0 e^{-\lambda t} \]
\[ N(t) = 100 e^{-0.1386 \times 15} \]
I kēia manawa, ke helu nei mākou i ka waiwai:
\[ N(t) = 100 e^{-2.079} \]
N(t) = 100 manawa 0.125
\[ N(t) \approx 12.5 \text{ grams} \]
No laila, ma hope o 15 mau makahiki, aia ma kahi o 12.5 grams o ka mea radioactive i koe.
Laʻana 2: Ka palaho o ka Capacitor
Nīnau:
Ua ʻae ʻia kahi capacitor me ka hoʻopiʻi mua \( Q_0 = 200 \text{ C} \) e hoʻokuʻu i loko o kahi kaapuni. ʻO ke kūpaʻa manawa ʻo \( \tau = 4 \text{ s} \). Ehia ka nui o ka hoʻopiʻi i koe ma hope o 10 kekona?
Kūkākūkā:
I ke ʻano o ka palaho o ka hoʻouka capacitor, ʻo ke kumu hoʻohālike exponential i hoʻohana ʻia:
\[ Q(t) = Q_0 e^{-t/\tau} \]
Hāʻawi ʻia \( Q_0 = 200 \text{ C} \) a me \( \tau = 4 \text{ s} \). Pono mākou e ʻimi \( Q(10) \):
\[ Q(10) = 200 e^{-10/4} \]
\[ Q(10) = 200 e^{-2.5} \]
Ke helu nei i nā waiwai exponential:
\[ Q(10) = 200 \times 0.0821 \]
\[ Q(10) \approx 16.42 \text{ C} \]
No laila, ma hope o 10 kekona, ʻo ke koena o ka uku ma ka capacitor ma kahi o 16.42 C.
Laʻana Nīnau 3: Palaho Kemika
Nīnau:
He kūpaʻa palaho ko kahi kemika o \( \lambda = 0.05 \text{ lā}^{-1} \). Pehea ka lōʻihi e emi ai ke kemika i 25% o kona nui mua?
Kūkākūkā:
Hoʻomaka mākou me ke ʻano maʻamau no ka palaho exponential:
\[ N(t) = N_0 e^{-\lambda t} \]
Makemake mākou iā N(t) e lilo i 25% o \( N_0 \), no laila:
\[ 0.25 N_0 = N_0 e^{-0.05 t} \]
Ke kāpae nei iā \( N_0 \) mai nā ʻaoʻao ʻelua:
\[ 0.25 = e^{-0.05 t} \]
Ke hoʻohana nei i nā logarithms kūlohelohe e hoʻoponopono i nā hihia exponential:
\[ \ln 0.25 = -0.05 t \]
\[ -1.3863 = -0.05 t \]
Ke hoʻoholo nei no \( t \):
\[ t = \frac{1.3863}{0.05} \]
\[ t \approx 27.726 \text{ lā} \]
No laila, ʻo ka manawa e pono ai no ka kemika e hōʻemi i 25% o kona nui mua ma kahi o 27.726 mau lā.
Laʻana Nīnau 4: Ka Palaho ʻana o ka Helu Bacteria
Nīnau:
Ua emi ka heluna bacteria ma ka wikiwiki exponential i hiki ai ma hope o 3 mau hola, ua hapalua ka heluna o kona helu mua. Inā he 8000 bacteria ka heluna mua, ehia mau bacteria i koe ma hope o 9 mau hola?
Kūkākūkā:
Ua ʻike ʻia ʻo ka hapalua ola \( t_{1/2} = 3 \) hola. ʻO ka mea mua, loaʻa iā mākou ke kūpaʻa palaho \( \lambda \):
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
\[ \lambda = \frac{\ln 2}{3} \approx 0.231 \text{ hola}^{-1} \]
Ma hope o kēlā, hoʻohana mākou i ke ʻano hana hoʻoheheʻe exponential:
\[ N(t) = N_0 e^{-\lambda t} \]
\[ N(9) = 8000 e^{-0.231 \times 9} \]
Ke helu nei i nā waiwai exponential:
\[ N(9) = 8000 e^{-2.079} \]
\[ N(9) = 8000 \times 0.125 \]
\[ N(9) \approx 1000 \]
No laila, ma hope o 9 mau hola, e koe ana ma kahi o 1000 mau bacteria.
Ka hopena
Hāʻawi ke kumu hoʻohālike decay exponential i kahi ala kūpono no ka hoʻoponopono ʻana i nā pilikia e pili ana i nā kaʻina hana decay i nā ʻano noi ʻepekema a me nā ʻenekinia. Ma ka hoʻomaopopo ʻana i nā manaʻo kumu e like me nā constants decay, ka hapalua ola, a me ka hoʻohana ʻana i nā formula exponential, hiki iā mākou ke helu i ka loli o kahi nui i ka hala ʻana o ka manawa me ka maʻalahi. Pono nā pilikia hoʻomaʻamaʻa i kūkākūkā ʻia ma luna e kōkua iā mākou e hoʻomaopopo a hoʻopili i ke kumumanaʻo o ka decay exponential i nā kūlana paʻakikī.
