# Encyclopedia

Number of
half-lives
elapsed
Fraction
remaining
Percentage
remaining
0 1/1 100
1 1/2 50
2 1/4 25
3 1/8 12 .5
4 1/16 6 .25
5 1/32 3 .125
6 1/64 1 .563
7 1/128 0 .781
... ... ...
n 1/(2n) 100/(2n)
.Half-life is the period of time it takes for a substance undergoing decay to decrease by half.^ YouTube: Half-Life 2 Part 10: Welcome to Black Mesa (Time: 7:26) Head downriver.
• Half-Life 2 Walkthrough 14 January 2010 7:20 UTC www.mahalo.com [Source type: General]

^ YouTube: Half Life 2 Highway 17 Part 3 (Time: 9:48) Continue up the road.
• Half-Life 2 Walkthrough 14 January 2010 7:20 UTC www.mahalo.com [Source type: General]

^ YouTube: Half Life 2 Highway 17 Part 4 (Time: 9:58) Sandtraps .
• Half-Life 2 Walkthrough 14 January 2010 7:20 UTC www.mahalo.com [Source type: General]

The name originally was used to describe a characteristic of unstable atoms (radioactive decay), but may apply to any quantity which follows a set-rate decay.
The original term, dating to 1907, was "half-life period", which was later shortened to "half-life" sometime in the early 1950s.[1]
.Half-lives are very often used to describe quantities undergoing exponential decay—for example radioactive decay—where the half-life is constant over the whole life of the decay, and is a characteristic unit (a natural unit of scale) for the exponential decay equation.^ I am naturally a man who hates deception, and wishes above all things to lead a life with its windows open and shades up, but I have been forced into the very reverse.
• "Doc." Gordon 2 February 2010 14:49 UTC home.comcast.net [Source type: Original source]

.However, a half-life can also be defined for non-exponential decay processes, although in these cases the half-life varies throughout the decay process.^ These take place immediately after the end of Half-Life 2.
• Half-Life 2 Walkthrough 14 January 2010 7:20 UTC www.mahalo.com [Source type: General]

For a general introduction and description of exponential decay, see the article exponential decay. For a general introduction and description of non-exponential decay, see the article rate law.
.The converse of half-life is doubling time.^ Originally posted by misterdurp Originally posted by Gameloading Originally posted by misterdurp Originally posted by Gameloading Half Life 2 is one of the overrated games of all time.
• HL2 - best game of the decade. - General Gaming - General Discussion - MMORPG.com Forums 14 January 2010 7:20 UTC www.mmorpg.com [Source type: General]

^ Originally posted by Gameloading Originally posted by misterdurp Originally posted by Gameloading Half Life 2 is one of the overrated games of all time.
• HL2 - best game of the decade. - General Gaming - General Discussion - MMORPG.com Forums 14 January 2010 7:20 UTC www.mmorpg.com [Source type: General]

^ YouTube: Half Life 2 Highway 17 Part 4 (Time: 9:58) Sandtraps .
• Half-Life 2 Walkthrough 14 January 2010 7:20 UTC www.mahalo.com [Source type: General]

The table at right shows the reduction of a quantity in terms of the number of half-lives elapsed.

## Probabilistic nature of half-life

.A half-life often describes the decay of discrete entities, such as radioactive atoms.^ Using a grenade in a game like Half Life 2 is such a chore.
• HL2 - best game of the decade. - General Gaming - General Discussion - MMORPG.com Forums 14 January 2010 7:20 UTC www.mmorpg.com [Source type: General]

.In that case, it does not work to use the definition "half-life is the time required for exactly half of the entities to decay". For example, if there is just one radioactive atom with a half-life of 1 second, there will not be "half of an atom" left after 1 second.^ There is a ramp just to the left of it.
• Half-Life 2 Walkthrough 14 January 2010 7:20 UTC www.mahalo.com [Source type: General]

^ When we look at the amount of impact a game like Halo had when compared to Half Life 2 both culturally and in the genre, its clear that Halo has done more for the genre than Half Life 2 did, not that I'm suggesting that Halo should be game of the decade, I just used it as an example because it also happens to be an FPS. If Half Life 2 is an old school and plays it safe, that's fine.
• HL2 - best game of the decade. - General Gaming - General Discussion - MMORPG.com Forums 14 January 2010 7:20 UTC www.mmorpg.com [Source type: General]

^ No complaints there, but from a gameplay perspective, Half Life 2 did nothing that pushed the genre forward.
• HL2 - best game of the decade. - General Gaming - General Discussion - MMORPG.com Forums 14 January 2010 7:20 UTC www.mmorpg.com [Source type: General]

There will be either zero atoms left or one atom left, depending on whether or not the atom happens to decay.
Instead, the half-life is defined in terms of probability. It is the time when the expected value of the number of entities that have decayed is equal to half the original number. For example, one can start with a single radioactive atom, wait its half-life, and measure whether or not it decays in that period of time. Perhaps it will and perhaps it will not. .But if this experiment is repeated again and again, it will be seen that it decays within the half life 50% of the time.^ YouTube: Half Life 2 Highway 17 Part 4 (Time: 9:58) Sandtraps .
• Half-Life 2 Walkthrough 14 January 2010 7:20 UTC www.mahalo.com [Source type: General]

^ YouTube: Half Life 2 A Red Letter Day (Time: 9:27) Follow Alyx into the lab.
• Half-Life 2 Walkthrough 14 January 2010 7:20 UTC www.mahalo.com [Source type: General]

^ YouTube: Half Life 2 Point Insertion (Time: 8:37) Move forward.
• Half-Life 2 Walkthrough 14 January 2010 7:20 UTC www.mahalo.com [Source type: General]

.In some experiments (such as the synthesis of a superheavy element), there is in fact only one radioactive atom produced at a time, with its lifetime individually measured.^ Besides the firelight there was only one great bronze lamp to illuminate the room.
• "Doc." Gordon 2 February 2010 14:49 UTC home.comcast.net [Source type: Original source]

^ There is only one consolation which I have with regard to him; unless my diagnosis was entirely at fault, he would have had that attack of erysipelas anyway.
• "Doc." Gordon 2 February 2010 14:49 UTC home.comcast.net [Source type: Original source]

^ I dare say there was no danger this time, only he came up behind like a cat, and —” .
• "Doc." Gordon 2 February 2010 14:49 UTC home.comcast.net [Source type: Original source]

.In this case, statistical analysis is required to infer the half-life.^ HLstatsX - Realtime Player Statistics for Half-Life 1 and Half-Life 2!
• HLstatsX - Realtime Player Statistics for Half-Life 1 and Half-Life 2! - Global Ranking 14 January 2010 7:20 UTC www.hlstatsx.com [Source type: General]

In other cases, a very large number of identical radioactive atoms decay in the time-range measured. In this case, the central limit theorem ensures that the number of atoms that actually decay is essentially equal to the number of atoms that are expected to decay. In other words, with a large enough number of decaying atoms, the probabilistic aspects of the process can be ignored.
There are various simple exercises that demonstrate probabilistic decay, for example involving flipping coins or running a computer program. See the following websites: [1], [2], [3].

## Formulae for half-life in exponential decay

An exponential decay process can be described by any of the following three equivalent formulae:
$N_t = N_0 \left(\frac {1}{2}\right)^{t/t_{1/2}}$
$N_t = N_0 e^{-t/ au} \,$
$N_t = N_0 e^{-\lambda t} \,$
where
• N0 is the initial quantity of the thing that will decay (this quantity may be measured in grams, moles, number of atoms, etc.),
• Nt is the quantity that still remains and has not yet decayed after a time t,
• t1 / 2 is the half-life of the decaying quantity,
• τ is a positive number called the mean lifetime of the decaying quantity,
• λ is a positive number called the decay constant of the decaying quantity.
The three parameters t1 / 2, τ, and λ are all directly related in the following way:
$t_{1/2} = \frac{\ln (2)}{\lambda} = au \ln(2)$
where ln(2) is the natural logarithm of 2 (approximately 0.693).
By plugging in and manipulating these relationships, we get all of the following equivalent descriptions of exponential decay, in terms of the half-life:
$N_t = N_0 \left(\frac {1}{2}\right)^{t/t_{1/2}} = N_0 2^{-t/t_{1/2}} = N_0 e^{-t\ln(2)/t_{1/2}}$
$t_{1/2} = t/\log_2(N_0/N_t) = t/(\log_2(N_0)-\log_2(N_t)) = (\log_{2^t}(N_0/N_t))^{-1} = t\ln(2)/\ln(N_0/N_t)$
Regardless of how it's written, we can plug into the formula to get
• Nt = N0 at t=0 (as expected—this is the definition of "initial quantity")
• $N_t=\left(\frac {1}{2}\right)N_0$ at t = t1 / 2 (as expected—this is the definition of half-life)
• Nt approaches zero when t approaches infinity (as expected—the longer we wait, the less remains).

### Decay by two or more processes

Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life T1/2 can be related to the half-lives t1 and t2 that the quantity would have if each of the decay processes acted in isolation:
$\frac{1}{T_{1/2}} = \frac{1}{t_1} + \frac{1}{t_2}$
For three or more processes, the analogous formula is:
$\frac{1}{T_{1/2}} = \frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + \cdots$
For a proof of these formulae, see Decay by two or more processes.

### Examples

There is a half-life describing any exponential-decay process. For example:
• The current flowing through an RC circuit or RL circuit decays with a half-life of RCln(2) or ln(2)L / R, respectively.
• In a first-order chemical reaction, the half-life of the reactant is ln(2) / λ, where λ is the reaction rate constant.
• In radioactive decay, the half-life is the length of time after which there is a 50% chance that an atom will have undergone nuclear decay. It varies depending on the atom type and isotope, and is usually determined experimentally.

## Half-life in non-exponential decay

Many quantities decay in a way not described by exponential decay—for example, the evaporation of water from a puddle, or (often) the chemical reaction of a molecule. In this case, the half-life is defined the same way as before: The time elapsed before half of the original quantity has decayed. However, unlike in an exponential decay, the half-life depends on the initial quantity, and changes over time as the quantity decays.
As an example, the radioactive decay of carbon-14 is exponential with a half-life of 5730 years. A quantity of carbon-14 will decay to half of its original amount (on average) after 5730 years, regardless of how big or small the original quantity was. After another 5730 years, one-quarter of the original will remain. On the other hand, the time it will take a puddle to half-evaporate depends on how deep the puddle is. Perhaps a puddle of a certain size will evaporate down to half its original volume in one day. .But by waiting a second day, there is no reason to expect that precisely one-quarter of the puddle will remain; in fact, it will probably be much less than that.^ You yourself know how many men I have seen here, and you know I never see men at Annie's.’ There is no one else.
• "Doc." Gordon 2 February 2010 14:49 UTC home.comcast.net [Source type: Original source]

^ He said to himself that he did not care if she were old enough to be his mother, his grandmother even, there was no one in the whole world like her.
• "Doc." Gordon 2 February 2010 14:49 UTC home.comcast.net [Source type: Original source]

^ He, in fact, began to consider that the day was waning, and what a wild-goose chase it would probably be for him to attempt to follow the man.
• "Doc." Gordon 2 February 2010 14:49 UTC home.comcast.net [Source type: Original source]

This is an example where the half-life reduces as time goes on. (In other non-exponential decays, it can increase instead.)
A biological half-life is also a type of half-life associated with a non-exponential decay, namely the decay of the activity of a drug or other substance after it is introduced into the body.
The decay of a mixture of two or more materials that each have different half-lives is not a simple exponential, as each material decays at a rate independent of the other. Mathematically, the sum of two exponential functions is not a single exponential function, A common example of such a situation is the waste of nuclear power stations, which is a mix of substances with vastly different half-lives. Consider a sample containing a rapidly decaying element A, with a half-life of 1 second, and of slowly decaying element B, with a half-life of one year. After a few seconds, almost all atoms of element A have decayed after repeated halving of the initial total number of atoms but very few atoms of element B will have decayed yet as not even one half-life has elapsed. Thus, the mixture taken as a whole does not decay by halves.

## References

1. ^ John Ayto, "20th Century Words" (1989), Cambridge University Press.

# Study guide

Up to date as of January 14, 2010

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## Half life

.The half life of any object is the amount of time it takes for half of the material to decay or disintegrate.^ Originally posted by misterdurp Originally posted by Gameloading Originally posted by misterdurp Originally posted by Gameloading Half Life 2 is one of the overrated games of all time.
• HL2 - best game of the decade. - General Gaming - General Discussion - MMORPG.com Forums 14 January 2010 7:20 UTC www.mmorpg.com [Source type: General]

^ Originally posted by Gameloading Originally posted by misterdurp Originally posted by Gameloading Half Life 2 is one of the overrated games of all time.
• HL2 - best game of the decade. - General Gaming - General Discussion - MMORPG.com Forums 14 January 2010 7:20 UTC www.mmorpg.com [Source type: General]

^ YouTube: Half Life 2 Highway 17 Part 4 (Time: 9:58) Sandtraps .
• Half-Life 2 Walkthrough 14 January 2010 7:20 UTC www.mahalo.com [Source type: General]

• Carbon-14

# Simple English

Half life can mean different things:

# Citable sentences

Up to date as of December 30, 2010

Here are sentences from other pages on Half-Life 2, which are similar to those in the above article.