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I mean how can we say that in 5730 years, 1/2 the no. of C14 nucleus will decay because in reality we don't know when a particular nucleus will decay

Qmechanic
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Souhardya Mondal
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    Related: http://physics.stackexchange.com/q/102222/2451 , http://physics.stackexchange.com/q/7584/2451 and links therein. – Qmechanic Dec 25 '14 at 14:12
  • This was already asked anyway: http://physics.stackexchange.com/q/102222/ – DanielSank Dec 25 '14 at 14:24
  • Another decay question by OP: http://physics.stackexchange.com/q/154964/2451 – Qmechanic Dec 25 '14 at 14:38
  • How can you tell the temperature of a macroscopic object, when heat is just particles buzzing and wiggling about at "random" velocities? – Luaan Apr 30 '15 at 11:10

3 Answers3

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It is quite simple actually. We don't know when an individual carbon atom will decay for sure. However, in $1Kg$ of carbon, I have over $10^{25}$ atoms over there. Using the Law of large numbers we actually say that $10^{25}$ is a "large number" and then correctly infer the probability of decay of a single carbon atom, by observing the decay frequency of the whole group of $1Kg$ of carbon.

Physicist137
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It is similar to inferring of the life expectancy of a country. Some may live for 80 years, some might be 20 years, some one for few months. If the average life of several thousands of people considered, that approximately gives the life expectancy of a country.

In same way in a substance after an instant ie t = 0, the life time of an atom is considered. It is the time interval before decay. The total life time of N number of atoms divided by the number of atoms gives the average life. Average or mean life is actually reciprocal of decay constant. Once the decay constant is known, the half life of the substance can be evaluated, so also activity of the sample.

Radha Krishna
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The nuclear decay follows an inverse exponential law, so by plotting the measured rate of decay against the natural logarithm of time, you get a straight line whose slope relates to the half-life.

See http://en.wikipedia.org/wiki/Exponential_decay

EDIT: Although the decay of an individual unstable nucleus is entirely random and it is impossible to predict when a particular atom will decay, what we can say is that at any point in time, it is equally likely to decay. This implies that for a mass containing a large number of nuclides, the rate of decay is proportional to the total number of atoms in the mass. That is, the nuclear decay process follows an exponential decay law. This is a statistical statement about the the mass not about individual atoms.

Statistically, we can not say exactly when a given C14 atom will decay. What we can say is that each C14 atom has a (fixed) probability of decaying within a given time interval. The 'half-life' is a measure of this time frame for 50% chance of decay (ie: half the number of nuclides will have decayed within this time). Saying 'half the number of C14 atoms will decay in 5730 years' does not mean that 'each C14 atom has a 50% chance of decay' but it means 'each C14 atom has a 50% chance of decay after 5730 years'.

The number (5730 years for the half-life of C14) is generally determined by experimental measurement.

If you are looking for a deeper physical explanation involving sub-nuclear interactions and quantum chromodynamics, then you'll can try to simulate the C14 nucleus using high-precision supercomputer to solve a billion-by-billion element matrix, like this research team at Iowa State University.

theo
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