Talk:Exponential decay

Merge

• OPPOSE the merge. Decay contributes to the half life. Not the same.

• I OPPOSE the merge to half-life. They are not the same. The half-life is a specific time that characterizes the decay curve. It is used in physics and biology. Engineers usually talk about the time constant. Exponential decay is the process itself. Also, one can have exponential decay to a non-zero value [1]--dependent on the physical system modeled. One doesn't usually talk about half-life in the context of exponential decay to a non-zero steady state value. Nephron 00:21, 19 March 2006 (UTC)

• Oppose. Everything on the half-life article relates specifically to half-lives. I've made sure that that page remains subordinate to this one. --Smack (talk) 03:26, 21 March 2006 (UTC)
  • Mostly oppose. Some elements of Half-life, such as the derivation, should be nixed, but the majority of it should remain. --Smack (talk) 06:44, 24 March 2006 (UTC)

• OPPOSE In the name of all that is holy, for the reasons stated above, please do NOT merge these articles. Barryvalder 08:32, 21 March 2006 (UTC)

• Support, it's fine to argue that half-life and exponential decay are different concept and should remain on separate pages, however the pages as they are currently written have about 80% duplication, and those bits should be consolidated to one page or the other. Perhaps one of you experts could actually read the pages instead of just throwing you uninformed opinions out here? Ewlyahoocom 14:39, 21 March 2006 (UTC)

It is possible to understand half-life without understanding the differential equation involved. So, I think it is easier to keep 'em separate. As for the duplication, it doesn't matter. Wikipedia isn't a paper encyclopedia. Nephron 00:47, 22 March 2006 (UTC)

If you must know, not only have I read both pages, I've written and rewritten much of them. The history of Half-life since last summer shows a repeating pattern of various people adding things and me removing them or trimming them down. However, I've been a little more forgiving than I might have been. --Smack (talk) 06:44, 24 March 2006 (UTC)

• Support. The half-life is just a specific aspect of exponential decay and a commonly used measure within exponential decay. If someone is looking up half-life, they'd likely get the information they need from this article, and I think this one is better organized to explain where half-life comes from. --128.42.162.102 03:13, 1 April 2006 (UTC)

• I OPPOSE the merge with half-life per above. I have also edited this article in such a way that half-life is briefly introduced while suggesting that readers look at the half-life article for more details. Teply 23:10, 10 June 2006 (UTC)
• STRONGLY OPPOSE ANY MERGER. Half-life can be defined and is used (e.g., in pharmacology, physiology, etc.) for non exponential decays. Jclerman 19:20, 17 June 2006 (UTC)

It looks like we have a consensus. I'll remove the merge tag. --Smack (talk) 19:27, 18 June 2006 (UTC)

Decay Constant??

I was redirected here from Decay Constant, but this page doesn't seem to say all that much about the Decay Constant. What is it? Where does it come from? What does it do? (If anything) How do you work it out? I think this needs more explanation. --ayteebee (talk) 14:11, 22 April 2006 (BST)

Can someone put up a definition for decay constant? -- mh 14:32, 4 August 2006 (UTC)

See the very first line of the article. The decay constant is λ, the inverse of the time constant (which is the mean lifetime). The decay constant links the instantaneous absolute rate of decay to the amount of material, according to the differential equation which defines it, also seen at the beginning of the article. Perhaps if you could frame your question more specifically? What is it you'd like to know? The decay constant has units of 1/time. The time-constant and the half-life have units of time.SBHarris 18:40, 4 August 2006 (UTC)

I was taught that the decay constant is the probability of radioactive particles that will decay in a second. Since the activity, $A={\frac {dN}{dt}}=-\lambda N$, $\lambda ={\frac {\frac {dN}{dt}}{-N}}={\frac {-{\frac {dN}{N}}}{dt}}$, which is the probability of rate of decay of particles per unit time. This is what I was taught, but I'm not sure how accurate it is, since no site (that I have searched) seems to back this up.—Dudboi 10:17, 12 November 2006 (UTC)

The equations you give are re-arrangements of those which appear in the article. And yes, the decay constant is the differential probability of decay with regard to time (in whatever units of time you're using). That is also the reciprocal of the particle mean lifetime (which is not the same as half-life). A particle which has a 0.01 chance of decay in any given time (say a second) will have a mean lifetime of 100 seconds (a collection will show this). SBHarris 22:20, 12 November 2006 (UTC)

Indeed it is, so shouldn't we mention that it is the probability, rather than the inverse of the mean lifetime? After all, saying it is the inverse of the mean lifetime doesn't really explain what it's about.—Dudboi 01:24, 13 November 2006 (UTC)

Right heres a problem to put all this into context. The half life of bismuth is 19*10^18 years. Its molecular mass is 208.98040g·mol−1. Accordingly there are about 2.87108E+27 atoms in a ton of bismuth. How long would I have to observe the one ton mass in order to be 95% sure that I would observe a single disintigration (assuming there were no technical problems to observation of every event)?--ManInStone 15:57, 17 May 2007 (UTC)

Well, the equation is dN/dt = γN = [ln2/t(1/2)]N. If t(1/2) is 1.9e19 years = 6e26 sec and N is 2.9e27, then dN/dt is 3.35 disintigrations per sec. From here, you have to know that you need to calculate from the Poisson distribution the 0.05 probability time for 0 events, when gamma (expected event rate) is 3.35/sec, is exp-(γ*t), where the lamba here is the total expected event rate for our entire mass, not the one for each atom. So 0.05 = exp-(3.35*t). Solve for t and you get a time about 0.893 sec. You have to wait that long for a 95% chance to see a single disintigration with an expected rate of 3.35/sec. SBHarris 02:06, 28 July 2007 (UTC)

Excuse me if I'm missing the obvious, but what is the name of character used to represent the constant? That is, λ. Phyte 15:12, 27 July 2007 (UTC)

Okay, sorry about that, it's lambda isn't it. Forgot google could search symbols Phyte 22:53, 27 July 2007 (UTC)

One thing that cropped into my head. Say we have a substance with half-life 0.5 seconds. Then lambda = ln2/0.5, which is greater than 1. But clearly the probability that a particle will decay in the next second is not greater than 1, and is actually 3/4. What gives? Andymc 17:38, 4 October 2007 (UTC)

Figured it. Basically λ is a number such that the probability of decay in the next Δt seconds is Δλ. When Δt is small compared to τ, it's fine to use it as "chance of decay per second", but when it isn't small, there can be a big difference.Andymc 18:27, 7 October 2007 (UTC)

"exponential decay if it decreases at a rate proportional to its value." The term rate is inexact here. The values at any time decrease at a constant "rate" (the decay constant), which means that the actual value of decay decreases also, the "rate", however, remains constant, or not, just whether you understand by "rate" the decay constant or the actual proportion. HJJHolm (talk) 10:40, 5 March 2012 (UTC)

Errors/Standard Deviations

Does anyone know if there's a variance or stdev associated with exponential decay?? y=Cexp(-x/λ)

for exponential prob. distribution y=λexp(-λx), the mean value is 1/λ and the variance is 1/λ² [ish!], but this distribution has only one parameter (λ), whilst exponential decay allows for 2 (C and λ).

insight greatly appreciated! Izzie— Preceding unsigned comment added by 131.111.85.79 (talk) 10:35, 1 June 2006 (UTC)

Deleted statements

These unsourced (subjective) statements have been deleted from the article, as per discusion of the terminology

The concept of half-life is generally more traditional, particularly in the study of radioactive decay, where exponential decay first received the most study. Although understanding the concept of half-life is generally easy, the additional logarithmic factor is now usually considered unattractive. Most scientists and current textbooks are now opting for the mean lifetime over the half-life. See the mean lifetime and half-life articles for more about their derivation and uses.

For the discussion of the terminology see [2]

Jclerman 19:16, 17 June 2006 (UTC)

Proposed merge

At present, we have four articles on what is basically the same subject: exponential decay and half-life as the most common useful measure. That strikes me as too many: much of the material is common to three of the articles, and Elimination half-life is very short. I propose to figure out how to consolidate them. Robert A.West (Talk) 00:18, 26 June 2006 (UTC)

Partially deja-vu. As recently as one week ago a merge tag was removed. See [3]. 69.9.26.206 01:02, 26 June 2006 (UTC)

OIC. I was coming at this from the Half-life direction and didn't want to discuss any of the four without the others. A merge of the three half-life articles without a merge of this one would be a start, but I still don't see what the point is of having separate articles for half-life. It smacks of content-forking. As for the assertion that there can be such a concept as a half-life apart from an exponential decay strikes me as very odd indeed, since the two are logically equivalent. Robert A.West (Talk) 01:09, 26 June 2006 (UTC)

Both concepts are related, if not equivalent, only for a restricted type of processes. Jclerman 02:16, 26 June 2006 (UTC)

don't think that is really true. If there is a well-defined half-life (i.e. the same for all populations/concentrations), the distribution has to be exponential. That's by definition of what an exponential is. As I understand, the term is used outside the exponential range for a biological process because the process is exponential over part of its range, so half-life is already in use. In any event, there may be sufficient reason for two articles, but four just looks like a content-fork between the various groups. I think it is silly to consider one pair in isolation. Robert A.West (Talk) 11:40, 26 June 2006 (UTC)

The above is only partially correct. It is valid only for one of several types of processes. In fact, the definition of half-life does not assume any one of possible processes of time-decrease of a quantity. For the comparison of cases in which t(1/2) can be related to rate-law constants, see the bottom row of the table [4] 69.9.26.206 15:14, 26 June 2006 (UTC)

It looks like somehow you reverted the talk page when you edited. [5] Strange and wondrous are the ways of Wikimedia software. In any case, I have reverted my comments back to the version that was allegedly current when you replied. My understanding has always been that the use of half-life for any non-exponential decay is derivative and artificial. I'll look more carefully at your point. Robert A.West (Talk) 21:24, 26 June 2006 (UTC)

Half-life doesn't have to be related to an exponential decay. --The definition does not state this is always so.[6][7] While clearance is typically 1st order, it can also be higher order or zeroth order.[8] Also see Jclerman's comments above in the discussion about the proposed merge with half-life. Nephron  T|C 00:31, 29 June 2006 (UTC)

It's fairly easy to prove than unless the decay IS exponential, the "half life" won't be a constant over time. It will change with time. Which brings us to ask: what's the use of talking about a changing "half-life"? Why even bother to use the word, now with implications that are wrong? A major advantage of the concept of half-life is that it's usually understood to be a constant, which means by definition, first order process. In any case, if there are cases where people insist on a non-constant half-life when talking about other than first-order (exponential) systems, then this needs to be so-noted as a bastardization of the idea of half-life, where it's being used in a nonstandard way, and where the normal assumptions about it do not hold.

In a simple exponential system, note that all "lives" are constants. So the "half-life" is a constant, the "third-life" is another constant, and the "1/eth" life is a constant = 1/γ. We just happened to have picked these two "lives" as times of interest, but we really could have picked "1/10th life" and this would work just as well. All these things are constant and related to each other by simple constants, so long as we stay with first-order (exponential) processe.SBHarris 19:00, 16 July 2006 (UTC)

See MeSH H01.862.405[9], which is a more general definition, from medicine to physics. Those based on exponentials are particular cases of the general definition. --69.9.27.14 19:17, 16 July 2006 (UTC)

Well, I dislike this definition intensely (since it applies only to the first 50% decrease, and is not in general a constant number describing a process at all times), but if it's used that way, out there, I suppose there's nothing I can do about it. This argues for a "half-life" article with exponential decay as a special case, but "half-life" in the general sense needs to be (re)defined so that it's clear that it means various things in various circumstances. In a zero order reaction, after two "half-lives" there's nothing left, for example. That's not the usual way we think about half-lives. In a second order, after two half lives there's MORE than a quarter left, etc. SBHarris 20:27, 16 July 2006 (UTC)

For the comparison of cases in which t(1/2) can be related to rate-law constants, see the bottom row of the table [10] 69.9.26.206 15:14, 26 June 2006 (UTC)

But note that these all relate to defining "half life" as just the time for the first 50% to disappear, but except in 1st order, it has nothing to do with how long it takes for the NEXT 50% to disappear, etc. So the table is misleading if you think of half-lives as per the exponential case, which is the way most people normally do (which is why all the problem here). As noted, the time for 50% to go to 25% is more than the "half-life" for second-order processes, and it's LESS than the half-life for 0 order processes (it's only half of the half-life). So it can be confusing. I'm sorry that somebody decided to generalize this idea.SBHarris 20:27, 16 July 2006 (UTC)

I don't see anything wrong with the definition[11][12]-- defining half-life based on the initial quantity and the time until half of it is left has the advantage that you don't need to understand the kinetics of the process to measure the half-life.

No, but using the term and doing the measurement has the disadvantage of making you think that maybe you think you know more than really do. For unless the "half-life" you're talking about is 1st order, it's completely dose-dependent (doesn't apply as a property of a drug per se), and also you know nothing about what happens to the drug after it gets down to 50%, except that it will keep going down more. So now you know the half-life for THAT dose of drug in THAT patient, or whatever. But so what? Without ability to generalize from the information, you have very little, unless you need to treat THAT patient with THAT dose of drug AGAIN...SBHarris

Many factors can influence the clearance of a drug and it is possible that a combination of factors are at play in the removal of a drug. It is possible that the first part of a drug removal process may be zeroth order... a latter part 1st order-- the definition of half-life could still be applied (though it would depend on the initial dose). Nephron  T|C 21:37, 16 July 2006 (UTC)

Agree. Not all drugs have a nice simple single half-life. But for those that don't, I think we should avoid the concept altogether. What good does it do us, except define a very narrow process, as above?SBHarris 21:45, 16 July 2006 (UTC)

The process is narrow, but that is its value. To be certain a process is 1st order you need many points on the time-concentration curve. Any case, drug removal isn't neat and tidy, even if the removal is 1st order-- there is still variation between patients. Also, I'll point-out that time to half the concentration may be all you need... if you're just giving one dose, after 1/2 the drug is cleared you're likely outside of the therapeutic window and likely don't care much about the concentration thereafter. Nephron  T|C 22:17, 16 July 2006 (UTC)

But if you're giving one dose, what do you care about all this? If you don't have first order kinetics, unless you happen to measure the drug level when it's EXACTLY 50%, you've just wasted your time measuring it. And even if you get it right, you're assuming it's a drug with a narrow window (factor of 2) where you need to dose on half-life, AND you're assuming you CAN dose on half life, even if you know it. But even so, you usually can't. You may find your lithium or dig patient has a half-life of 32 hours, but you would not dose them at that interval, even in the hospital. So where's the practical importance?

Your general statement I think is mostly wrong (ie, it's generally not useful to know drug "half life," unless it's first order. Some of the narrowest therapeutic window drugs like dig and lithium are actually approximated not too badly by first order kinetics. And as you many know, probably the most frequently measured drug levels in hospitals today, is a class which most certainly does NOT have a narrow therapeutic window, and is deliberately dosed to get very wide peak/traugh levels-- often 5:1 or more. That's the aminoglycosides. Which also are well-approximated by first-order kinetics. We once did these by hand, you know. Most of the individual patient variation for those drugs is in first order parameter differences-- Vd and k(elim), not in departures from first order (one compartment) modeling.

So your turn. I've given you examples. There are (to be sure) drugs with non-first order kinetics (theophyllin, phenobarb, etc) where we measure levels to adjust trough values. But for those drugs we don't really care about "half-life" and don't try to measure it, and don't even talk about it. And for good reason. What would we DO with the information? You see the point? Give me a clinically relevent example of use of a non-first order "half-life".SBHarris 23:15, 16 July 2006 (UTC)

Single place to discuss: RfC?

I know that Request for Comment tends to be a place where people go when they are fighting, but I agree with Kjkolb's comment on Half-life that there should be a single place to discuss this, and I can't think of a better place to do it. What do others think? Robert A.West (Talk) 11:55, 26 June 2006 (UTC)

Explain unitless nature of exponential argument (WHO is our audience)??

I've had an editor revert the statement that time-constants, either the exponential time τ = 1/λ or the half-life t(1/2), function to keep the argument of an exponential unitless. People with a lot of physics training know this already (it's obvious), but is this article really written for THEM? For the non-physicist and non-mathematician, it's helpful to point out that everything BUT the time in the argument needs to come out in units of 1/time, which is an easy way to see why λ must have units of 1/t. The edit says, why not just SEE this? My answer is, for that matter, why not just SEE that the solution of this extremely simple first order ordinary differential equation, is what it is? If you have much calculus, you don't NEED a derivation, so why not revert THAT?

What's OBVIOUS and what's TRIVIAL depends on how much experience you have. But WHO is our audience? It's only slightly more trivial mathematically that at any time t in a decay curve, τ = N(t)/[dN(t)/dt], so that you can quickly come up with τ, λ, t(1/2), etc. and thus you have these, any time you can evaluate your slope between two points, which is approximated by dN(t)/dt, the instantaneous decay rate evaluated at t.

In any case, please let us get a concensus on the question of who this article is written for (= target audience, both primary and secondary). I see it as a high school student or beginning pharmacy student who has had rudimentary calculus and maybe a little calculus based physics, or even none. But without deciding who our target audience is, we're destined to argue about this endlessly, because we're starting with different assumptions.SBHarris 18:06, 16 July 2006 (UTC)

alternate general formula for decay

i cant seem to see it mentioned anywhere here, so: Q(t)=a(1-r)^t where

• the coefficient a is the initial value of Q (at t =0)
• the base b is the growth factor where + b = 1 – r is decay (0 < b < 1) where r is the rate (as a decimal)
• the exponent t is the independent variable
• the dependent variable is the quantity Q

Mokaiba (talk) 07:15, 26 September 2006 (UTC)

Knowledge eroding?

QUOTING from the article: "In history of science, some believe that the body of knowledge of any particular science is gradually disproved according to an exponential decay pattern (see half-life of knowledge)."

This seems to me to suggest that (according to this belief) the sum of all scientific knowledge is decreasing over time. Is this what is meant? It seems counterintuitive.

Wanderer57 (talk) 14:44, 7 August 2008 (UTC)

I read it as that when a new field of science emerges, its beginning theories are quickly disproven, but over time, its theories remain stable. -- R'son-W _{(speak to me/breathe)} 07:56, 8 November 2008 (UTC)

Social sciences - glottochronology

The sentence on glottochronology was completely mistaken. The application of the exponential decay equation in glottochronology bears exclusively on the alleged steady decay of ONE language. To compute the time of split between TWO languages requires additional assumptions, which have nothing to do with exponential decay. HJJHolm (talk) 05:33, 18 May 2011 (UTC)

Median lifetime

Is it valid to say that the half-life is the median lifetime? Cesiumfrog (talk) 00:22, 6 February 2018 (UTC)