Question

derive relation between half life and average life of a radioactive sample

Answer 1

To get the mean (or average) lifetime of a radioactive element, you take a sample of the radioactive atoms and wait for all of them to decay away, and keep track of how long each atom lasts. The sum of all the lifetimes of the atoms, divided by the original number of atoms, is the mean lifetime. In other words, the mean lifetime is simply the arithmetic average of the lifetimes of the individual atoms. The lifetime units are time units such as seconds or years. 

The half-life of a radioactive element is the amount of time it takes for half of a sample of the element to decay away. It is smaller than the mean lifetime by a factor of ln(2), the natural logarithm of 2. 

half-life = (mean lifetime)*ln(2)mean lifetime = (half-life) / ln(2)      ln(2) = 0.6931      
The half-life units are time units such as seconds or years. 

The decay rate or decay constant is the fraction of the total mass that decays in one unit of time. It is equal to the inverse of the mean lifetime: 

decay rate = 1 / (mean lifetime)mean lifetime =  1 / (decay rate)
For example, radium-226 has a half-life of 1,602 years, an mean lifetime of (1,602)/ln2 = 2,311 years, and a decay rate of 1/(2,311) = 0.000433 per year. In other words, 0.0433 percent of the radium decays away each year, or 433 parts per million per year. The decay rate units are the inverse of time units, for example, years-1 ("per year") or sec-1 ("per second").  

To calculate the decay rate in becquerels(atoms per second) for a given mass of a radioactive element sample, do the following: 

Take the half-life and divide by ln2 (0.6931) to get the mean lifetime; convert the time units to seconds; and take the inverse to get the decay rate per second. For radium-226, the decay rate is 0.000433 per year / [ (365.25 days/year) * (24 hours/day) * (60 minutes/hour) * (60 seconds/minute) ] = 1.37 x 10-11 per second 

Take the mass of the element sample in grams, divide by the atomic mass to get moles, then multiply by Avagadro's number (6.02 x 1023) to get the number of atoms. For example, for 1.00 gram of radium-226, the number of atoms is [1.00 gram / (226 grams/mole)] x (6.02 x 1023 atoms/mole) =  2.66 x 1021 atoms

Multiply the number of atoms by the decay rate per second to get the number of atoms decaying per second. This is the decay rate in becquerels. For 1.00 gram of radium-226, the decay rate is 2.66 x 1021 atoms x 1.37 x 10-11 per second = 3.65 x 1010 atoms per second 

To convert the decay rate in becquerels to curies, divide by 3.7 x 1010. For example, for 1.00 gram of radium-226, the activity is 3.65 x 1010 / 3.7 x 1010 =  0.99 curie. The fact that this is close to 1.00 is not a coincidence. The original definition of the curie unit was based on the activity of radium emanation (radon) in equilibrium with 1 gram of radium. The difference is a rounding error.