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{{for|other uses|Half time (disambiguation)}}
[[File:Half times.png|right]]
The '''half time''' is the time taken by a quantity to reach one half of its final value, where the rate of change is proportional to the difference between the present value and the final value (i.e. in [[exponential decay]] processes). It is synonymous with [[half-life]], but used in slightly different contexts.


The diagram shows the increase in the quantity (red) in response to a step-change in the motive force that changes it (blue). The time-axis is in multiples of the half time. It can be seen that the quantity increases to one-half of its final value after one half time, to three-quarters after two half times, to seven-eighths after three half times, and so on.


The relationship between the quantity (''Q'') and time (''t'') is described by the mathematical formula:
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: <math> Q(t) = Q_\mathrm{f}(1 - e^{-\lambda t}) \, </math>
 
where ''Q''<sub>f</sub> is the final value and {{lambda}} is a constant, approximately equal to 0.69 divided by the half time – more precisely: log<sub>e</sub>(2) / (half time).
 
Where the quantity decreases in response to a step-decrease in the motive force that changes it, the curve is mirrored in the time-axis and may be referred to as [[exponential decay]].
 
The concept of half time is used in [[Scuba diving|diving]] physiology where [[Tissue (biology)|body tissues]] take up and release [[inert gases]] (usually [[nitrogen]]) following changes in depth. Different tissue types have different half times for a given inert gas, and modelling the uptake and release of gases by the tissues is important to avoid [[decompression sickness]].
 
==Examples==
*[[File:RC charging.svg|right|thumb|The voltage (v) on the capacitor (C) changes with time as the capacitor is charged or discharged via the resistor (R)]]In electronics, when a [[capacitor]] is charged or discharged via a [[resistor]], the voltage on the capacitor follows the above formula, with the half time approximately equal to 0.69 times the [[time constant]], which is equal to the product of the resistance and the capacitance.
 
*The first models of nitrogen uptake and release in the body of a diver used five parallel compartments with half times from 5 minutes to 75 minutes.<ref name="BE2003-439">{{cite book |title=Bennett and Elliott's physiology and medicine of diving, 5th Revised edition |author=Tikuisis, Peter; Gerth, Wayne A. |editor=Brubakk, Alf O; Neuman, Tom S |year=2003 |publisher=Saunders Ltd |location=United States |chapter=10.1: Decompression Theory |page=439 |isbn=0-7020-2571-2 |oclc=51607923}}</ref><ref name=Haldane1908>{{cite journal |author=Boycott A. E., Damant G. C. C., [[John Scott Haldane|Haldane John Scott]] |title=Prevention of compressed air illness |journal=Journal of Hygiene |volume=8 |issue=3 |pages=342–443 |date=1908 |url=http://archive.rubicon-foundation.org/7489 |pmid=20474365 |accessdate=2009-06-15 |pmc=2167126 |doi=10.1017/S0022172400003399 }}</ref> Later models refined this by considering more compartments and a wider range of half times. The U.S. Navy tables used six compartments with half times of 5, 10, 20, 40, 80 and 120 minutes.<ref name="DiD2005-215">{{cite book |author=Lippmann, John; [[Simon Mitchell|Mitchell, Simon J]] |title=Deeper into Diving |publisher=J.L. Publications |location=Victoria, Australia |month=October |year=2005 |edition=2 |page=215 |chapter=15 |isbn=0-9752290-1-X |oclc=66524750 }}</ref><ref name=NEDU57>{{cite journal |author=des Granges, M |title=Standard Air Decompression Table |journal=[[United States Navy Experimental Diving Unit]] Technical Report |volume=NEDU-RR-5-57 |date=1957 |url=http://archive.rubicon-foundation.org/3331 |accessdate=2009-06-15 }}</ref> The [[Bühlmann tables]] use twelve of the sixteen compartments in the ZH-L<sub>16</sub> algorithm, which uses half times from 4 to 635 minutes.<ref name="DiD2005-226">{{cite book |author=Lippmann, John; Mitchell, Simon |title=Deeper into Diving |publisher=J.L. Publications |location=Victoria, Australia |month=October |year=2005 |edition=2 |page=226 |chapter=17 |isbn=0-9752290-1-X |oclc=66524750 }}</ref><ref name=decoDCS>{{cite book |title=Decompression-Decompression Sickness |author=[[Albert A. Bühlmann|Bühlmann, Albert A.]] |year=1984 |publisher=Springer-Verlag |location=Berlin New York |isbn=0-387-13308-9 }}</ref>
 
==References==
{{reflist}}
 
==See also==
*[[Half-life]]
*[[Dive tables]]
*[[Bühlmann tables]]
*[[Dive computer]]
 
[[Category:Concepts in physics]]
[[Category:Exponentials]]

Revision as of 11:53, 8 February 2014


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