عمر النصف

هذا المقال هو عن المفهوم العلمي والرياضي. إذا كنت تريد الاستخدامات الأخرى، انظر عمر النصف (توضيح).

عدد أعمار النصف المنقضية	الكسر المتبقي	النسبة المتبقية
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	.5625
7	1⁄128	0	.78125
n	1⁄2n	100⁄2n

فترة عمر النصف { Half-life ؛ ويُرمَز لها: t_½) لمادة نشيطة إشعاعيا هو الزمن اللازم لنصف العينة المأخوذة من المادة ليحدث له تحلل إشعاعي .

وللتعميم ، فإنه في الدراسة الكمية للتحلل الأسي ، فإن فترة عمر النصف هو الزمن اللازم لكمية المادة لتصبح نصف قيمتها الأصلية . ( لن يتم مناقشة هذه النقطة بالتفصيل هنا ويمكن مراجعة موضوعات متعلقة بالأسفل )

 معادلات لعمر النصف في التحلل الأسي

المقالة الرئيسية: تضاؤل أسي

التحلل الأسي يمكن وصفه بأي من المعادلات (الصيغ) الأربع التالية:^:109–112

$${\displaystyle{\begin{aligned} \displaystyle N(t)&\displaystyle=N_{0}\left({\frac{1}{2}}\right)^{\frac{t}{t_{1/2}}}\\ \displaystyle N(t)&\displaystyle=\left(2^{-{\frac{t}{t_{1/2}}}}\right)N_{0}\\ \displaystyle N(t)&\displaystyle=N_{0}e^{-{\frac{t}{\tau}}}\\ \displaystyle N(t)&\displaystyle=N_{0}e^{-\lambda t}\end{aligned}}}$$

${\displaystyle{\displaystyle N(t)=N_{0}e^{-\lambda t}\,}}$

حيث

• ${\displaystyle{\displaystyle N_{0}}}$ هى القيمة الأصلية للN (عند t=0)
• λ ثابت موجب(ثابت التحلل).

عندما تكون t=0, يكون الوغاريتم قيمته تساوى 1, ويكون Nt مساوية لـN0. حيث t تقترب من اللانهاية, يقترب اللوغاريتم من الصفر.

وبالتحديد ، فإنه يوجد وقت ${\displaystyle{\displaystyle t_{1/2}\,}}$ تصبح :

${\displaystyle{\displaystyle N(t_{1/2})=N_{0}\cdot{\frac{1}{2}}}}$

ووبالتعويض في المعادلة السابقة نحصل على :

${\displaystyle{\displaystyle N_{0}\cdot{\frac{1}{2}}=N_{0}e^{-\lambda t_{1/2}}\,}}$

${\displaystyle{\displaystyle e^{-\lambda t_{1/2}}={\frac{1}{2}}\,}}$

${\displaystyle{\displaystyle-\lambda t_{1/2}=\ln{\frac{1}{2}}=-\ln{2}\,}}$

${\displaystyle{\displaystyle t_{1/2}={\frac{\ln 2}{\lambda}}\,}}$

وعلى هذا فإن فترة عمر النصف تكون 69.3% من متوسط عمر النصف.

 عمر النصف ودرجات التفاعل

In chemical kinetics, the value of the half-life depends on the reaction order:

 الحركة من الدرجة الصفرية

The rate of this kind of reaction does not depend on the substrate concentration, [A]. Thus the concentration decreases linearly.

$$d[{\ce {A}}]/dt=-k$$

The integrated rate law of zero order kinetics is:

$$[{\ce {A}}]=[{\ce {A}}]_{0}-kt$$

In order to find the half-life, we have to replace the concentration value for the initial concentration divided by 2:

$$[{\ce {A}}]_{0}/2=[{\ce {A}}]_{0}-kt_{1/2}$$

and isolate the time:

$$t_{1/2}={\frac {[{\ce {A}}]_{0}}{2k}}$$

This t_½ formula indicates that the half-life for a zero order reaction depends on the initial concentration and the rate constant.

 الحركة من الدرجة الأولى

In first order reactions, the rate of reaction will be proportional to the concentration of the reactant. Thus the concentration will decrease exponentially.

$$[{\ce {A}}]=[{\ce {A}}]_{0}\exp(-kt)$$

as time progresses until it reaches zero, and the half-life will be constant, independent of concentration.

The time t_½ for [A] to decrease from [A]₀ to 1/2[A]₀ in a first-order reaction is given by the following equation:

$$[{\ce {A}}]_{0}/2=[{\ce {A}}]_{0}\exp(-kt_{1/2})$$

It can be solved for

$$kt_{1/2}=-\ln \left({\frac {[{\ce {A}}]_{0}/2}{[{\ce {A}}]_{0}}}\right)=-\ln {\frac {1}{2}}=\ln 2$$

For a first-order reaction, the half-life of a reactant is independent of its initial concentration. Therefore, if the concentration of A at some arbitrary stage of the reaction is [A], then it will have fallen to 1/2[A] after a further interval of

${\displaystyle{\displaystyle{\tfrac{\ln 2}{k}}.}}$

Hence, the half-life of a first order reaction is given as the following:

$$t_{1/2}={\frac{\ln 2}{k}}$$

The half-life of a first order reaction is independent of its initial concentration and depends solely on the reaction rate constant, k.

 الحركة من الدرجة الثانية

In second order reactions, the rate of reaction is proportional to the square of the concentration. By integrating this rate, it can be shown that the concentration [A] of the reactant decreases following this formula:

$${\frac {1}{[{\ce {A}}]}}=kt+{\frac {1}{[{\ce {A}}]_{0}}}$$

We replace [A] for 1/2[A]₀ in order to calculate the half-life of the reactant A

$${\frac {1}{[{\ce {A}}]_{0}/2}}=kt_{1/2}+{\frac {1}{[{\ce {A}}]_{0}}}$$

and isolate the time of the half-life (t_½):

$$t_{1/2}={\frac {1}{[{\ce {A}}]_{0}k}}$$

This shows that the half-life of second order reactions depends on the initial concentration and rate constant.

 التحلل بعمليتين أو أكثر

Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life T_½ can be related to the half-lives t₁ and t₂ that the quantity would have if each of the decay processes acted in isolation:

$${\displaystyle{\frac{1}{T_{1/2}}}={\frac{1}{t_{1}}}+{\frac{1}{t_{2}}}}$$

For three or more processes, the analogous formula is:

$${\displaystyle{\frac{1}{T_{1/2}}}={\frac{1}{t_{1}}}+{\frac{1}{t_{2}}}+{\frac{1}{t_{3}}}+\cdots}$$

For a proof of these formulas, see Exponential decay § Decay by two or more processes.

العنصر النشيط إشعاعيا يمكن أن يتحلل بطريقتين أو أكثر . وهذه الطرق لها إمكنيات مختلفة لحدوثها ، ولذا فإن لكل منها فترة عمر نصف خاصة بها .

فمثلا لنظامين من أنظمة التحلل ، فإن كمية المادة المتبقية بعد زمن قدره t يتم حسابها من المعادلة :

${\displaystyle{\displaystyle N(t)=N_{0}e^{-\lambda_{1}t}e^{-\lambda_{2}t}=N_{0}e^{-(\lambda_{1}+\lambda_{2})t}}}$

وبنفس النظام المتبع في القسم السابق ، يمكن حساب عمر النصف النهائي الجديد ${\displaystyle{\displaystyle T_{1/2}\,}}$ كالتالى :

${\displaystyle{\displaystyle T_{1/2}={\frac{\ln 2}{\lambda_{1}+\lambda_{2}}}\,}}$

أو بالتعبير عنه بواسطة فترتي عمر النصف :

${\displaystyle{\displaystyle T_{1/2}={\frac{t_{1}t_{2}}{t_{1}+t_{2}}}\,}}$

حيث ${\displaystyle{\displaystyle t_{1}\,}}$ فترة عمر النصف بالطريقة الأولى ${\displaystyle{\displaystyle t_{2}\,}}$ فترة عمر النصف بالطريقة الثانية .

 أمثلة

للمزيد من المعلومات: تضاؤل أسي § تطبيقات وأمثلة

There is a half-life describing any exponential-decay process. For example:

• As noted above, 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. See List of nuclides.
• The current flowing through an RC circuit or RL circuit decays with a half-life of ln(2)RC or ln(2)L/R, respectively. For this example the term half time tends to be used rather than "half-life", but they mean the same thing.
• In a chemical reaction, the half-life of a species is the time it takes for the concentration of that substance to fall to half of its initial value. In a first-order reaction the half-life of the reactant is ln(2)/λ, where λ (also denoted as k) is the reaction rate constant.

 في الأحياء والصيدلة

انظر أيضاً: عمر النصف الحيوي

A biological half-life or elimination half-life is the time it takes for a substance (drug, radioactive nuclide, or other) to lose one-half of its pharmacologic, physiologic, or radiological activity. In a medical context, the half-life may also describe the time that it takes for the concentration of a substance in blood plasma to reach one-half of its steady-state value (the "plasma half-life").

The relationship between the biological and plasma half-lives of a substance can be complex, due to factors including accumulation in tissues, active metabolites, and receptor interactions.

While a radioactive isotope decays almost perfectly according to first order kinetics, where the rate constant is a fixed number, the elimination of a substance from a living organism usually follows more complex chemical kinetics.

For example, the biological half-life of water in a human being is about 9 to 10 days, though this can be altered by behavior and other conditions. The biological half-life of caesium in human beings is between one and four months.

The concept of a half-life has also been utilized for pesticides in plants, and certain authors maintain that pesticide risk and impact assessment models rely on and are sensitive to information describing dissipation from plants.

In epidemiology, the concept of half-life can refer to the length of time for the number of incident cases in a disease outbreak to drop by half, particularly if the dynamics of the outbreak can be modeled exponentially.

 موضوعات متعلقة

• التحلل الأسي
• متوسط عمر النصف
• تحلل إشعاعي
• جداول النظائر :
  • جدول نظائر(مقسوم)
  • جدول نظائر(كامل)

 انظر أيضا

• قائمة النيوكليدات المشعة حسب عمر النصف
• Mean lifetime

 وصلات خارجية

ابحث عن half-life في
قاموس المعرفة.

• Nucleonica.net, Nuclear Science Portal
• Nucleonica.net, wiki: Decay Engine
• Bucknell.edu, System Dynamics - Time Constants
• Subotex.com, Half-Life elimination of drugs in blood plasma - Simple Charting Tool