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which means that we have lost (1 โ 0.37 = 0.63) 63% of A, with only 37% left. With this, we now know that if we have 1 lifetime passed, we have gone through 1 "e-folding". What would 2 "e-foldings" look like? After two lifetimes, we would have
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are the final and initial concentrations of A. Upon comparing the ratio on the left hand side to the equation on the right hand side, we conclude that the ratio between the final and initial concentrations follows an exponential function, of which
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are inverse time. If we were to take the reciprocal of this, we would be left with units of time. For this reason, we often state that the lifetime of a species that undergoes first order decay is equal to the reciprocal of
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1146:{\displaystyle {\begin{aligned}&\int _{_{i}}^{_{f}}{\frac {d}{}}=\int _{t=0}^{t}-k\,dt\\&\ln_{f}-\ln_{i}=-kt-k(0)\\&\ln {\frac {_{f}}{_{i}}}=-kt\\&{\frac {_{f}}{_{i}}}=e^{-kt}\end{aligned}}}
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Let us assume that this reaction follows first order kinetics, meaning that the conversion of A into B depends only on the concentration of A, and the
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1345:), the ratio of final to initial concentrations is equal to about 0.37. Stated another way, after one lifetime, we have
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519:. Consider a chemical species A, which decays into another chemical species, B. We could depict this as an equation:
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1603:{\displaystyle {\frac {_{f}}{_{i}}}=e^{-k\left({\frac {2}{k}}\right)}=e^{-2}={\frac {1}{e^{2}}}\approx 0.14=14\%}
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1617:-folding lends us an easy way to describe the number of lifetimes that have passed. After 1 lifetime, we have 1/
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1331:{\displaystyle {\frac {_{f}}{_{i}}}=e^{-k\left({\frac {1}{k}}\right)}=e^{-1}={\frac {1}{e}}\approx 0.37}
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148:
431:{\displaystyle T_{e}={\frac {T_{d}}{\ln(2)}}={\frac {t}{\ln({N(t)}/{N(0)})}}={\frac {t}{\ln(1+r/100)}}}
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would need to allow for these to be replicated on the right hand side. For this reason, the units of
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would be. On the left hand side, we have a concentration divided by a unit of time. The units for
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247:, the initial difference between the initial value of the observable and the equilibrium value, ฮ
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The process of evolving to equilibrium is often characterized by a time scale called the
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567:. We could write the following reaction to describe this first order kinetic process:
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differential equation, we may separate the terms such that the equation becomes:
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states that a change (in this case the disappearance) of the concentration of A,
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both sides of this equation, which results in the inclusion of the constant
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1629:-folding time, which is the most descriptive way of stating the decay.
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which says that only about 14% of A remains. It is in this manner that
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1428:{\displaystyle {\frac {_{f}}{_{i}}}\approx {\frac {37}{100}}=37\%}
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multiplied by the concentration of A. Consider what the units of
1185:. Consider, now, what would happen if we were to set the time,
147:. This term is often used in many areas of science, such as in
212:-folding time is also sometimes used similarly in the case of
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is investigated. Physicists and chemists often talk about the
15:
216:, to refer to the timescale for a quantity to decrease to 1/
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which dictates the velocity at which this happens,
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549:{\displaystyle {\rm {A}}\rightarrow {\rm {B}}}
515:-folding time may be used in the analysis of
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106:Learn how and when to remove this message
1621:remaining. After 2 lifetimes, we have 1/
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1341:This states that after one lifetime (1/
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751:{\displaystyle {\frac {d}{}}=-k\,dt}
44:adding citations to reliable sources
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243:, for example) then after a time,
174:in which the length of a patch of
133:quantity increases by a factor of
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129:is the time interval in which an
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1659:"E-folding explaination needed"
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195:continuously compounded return
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495:% growth rate in time
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1695:Temporal exponentials
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131:exponentially growing
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227:-folding time,
40:improve this article
1201:. This would yield
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168:-folding time scale
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673:Because this is a
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463:) amount at time
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269:where the number
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1639:Doubling time
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561:rate constant
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507:-folding time
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145:doubling time
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96:December 2023
88:
85:
81:
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74:
71:
67:
64:
60:
57: โ
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
1667:. Retrieved
1665:. 2013-02-27
1662:
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766:
761:We may then
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93:
83:
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62:
50:
38:Please help
33:verification
30:
679:homogeneous
237:temperature
172:proper time
55:"E-folding"
1684:Categories
1669:2023-12-19
1645:References
143:analog of
66:newspapers
1598:%
1586:≈
1558:−
1524:−
1423:%
1404:≈
1323:≈
1302:−
1268:−
1129:−
1059:−
1005:
982:−
973:−
947:
941:−
918:
897:−
877:∫
788:∫
763:integrate
736:−
683:separable
611:−
537:→
400:
343:
316:
274:โ 2.71828
208:The term
180:spacetime
1633:See also
517:kinetics
265: /
153:medicine
127:-folding
1193:, i.e.
241:density
191:finance
120:science
80:scholar
1156:where
675:linear
82:
75:
68:
61:
53:
642:This
176:space
87:JSTOR
73:books
1589:0.14
1452:= 2/
1448:+ 1/
1444:= 1/
1326:0.37
1162:and
681:and
155:and
59:news
1412:100
420:100
239:or
189:In
178:or
118:In
42:by
1686::
1661:.
1595:14
1420:37
1409:37
1002:ln
944:ln
915:ln
769::
677:,
652:dt
397:ln
340:ln
313:ln
276:.
151:,
122:,
1672:.
1627:e
1623:e
1619:e
1615:e
1592:=
1579:2
1575:e
1571:1
1566:=
1561:2
1554:e
1550:=
1544:)
1539:k
1536:2
1531:(
1527:k
1520:e
1516:=
1508:i
1504:]
1498:A
1493:[
1486:f
1482:]
1476:A
1471:[
1454:k
1450:k
1446:k
1442:t
1417:=
1396:i
1392:]
1386:A
1381:[
1374:f
1370:]
1364:A
1359:[
1343:k
1318:e
1315:1
1310:=
1305:1
1298:e
1294:=
1288:)
1283:k
1280:1
1275:(
1271:k
1264:e
1260:=
1252:i
1248:]
1242:A
1237:[
1230:f
1226:]
1220:A
1215:[
1199:k
1195:t
1191:k
1187:t
1183:k
1178:k
1171:e
1165:i
1159:f
1135:t
1132:k
1125:e
1121:=
1113:i
1109:]
1103:A
1098:[
1091:f
1087:]
1081:A
1076:[
1065:t
1062:k
1056:=
1048:i
1044:]
1038:A
1033:[
1026:f
1022:]
1016:A
1011:[
994:)
991:0
988:(
985:k
979:t
976:k
970:=
965:i
961:]
955:A
950:[
936:f
932:]
926:A
921:[
907:t
904:d
900:k
892:t
887:0
884:=
881:t
873:=
867:]
862:A
857:[
852:]
847:A
842:[
839:d
829:f
825:]
819:A
814:[
807:i
803:]
797:A
792:[
767:e
746:t
743:d
739:k
733:=
727:]
722:A
717:[
712:]
707:A
702:[
699:d
668:k
664:k
660:k
656:k
650:/
648:d
627:]
622:A
617:[
614:k
608:=
602:t
599:d
594:]
589:A
584:[
581:d
565:k
542:B
532:A
513:e
505:e
497:t
493:r
480:d
476:T
470:N
465:t
461:t
459:(
457:N
451:e
447:e
443:T
423:)
416:/
412:r
409:+
406:1
403:(
393:t
388:=
382:)
378:)
375:0
372:(
369:N
364:/
359:)
356:t
353:(
350:N
346:(
336:t
331:=
325:)
322:2
319:(
307:d
303:T
297:=
292:e
288:T
272:e
267:e
262:i
258:X
253:i
249:X
245:ฯ
233:X
229:ฯ
225:e
218:e
210:e
203:e
184:e
166:e
141:e
136:e
125:e
109:)
103:(
98:)
94:(
84:ยท
77:ยท
70:ยท
63:ยท
36:.
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