Answer to Question #87537 in Atomic and Nuclear Physics for Shivam Nishad

Question

The disintegration constant of ²³⁸U is 4.87×10⁻¹⁸s⁻¹. Calculate its half life(in years). Also calculate the number of disintegration per second from 1 gram of Uranium. It is given that Avogadro's number= 6.02×10²³

Answer 1

To calculate half-life for the given isotope let's use formula:

$$T_{1/2}=\frac {ln(2)} {\lambda} = \frac {0.693} {4.87*10^{-18} s^{-1}} = 1.42 * 10^{17} s;$$

where T_1/2 - half-life period, ln - natural logarithm, lambda - disintegration constant.

To convert this value into years, we need to divide it by 60*60*24*365=31,536,000:

$$T_{1/2}=\frac {1.42 * 10^{17}} {31536000} = 4.5028 *10^9 years$$

The total formula for activity of a given sample of radionuclide is:

$$A=\lambda \frac {m} {M} * N_A*2^{-t/T_{1/2}};$$

where A - number of disintegration per second (activity of the sample), m - mass of the sample, M - molar mass of the Uranium-238, N_A - Avogadro number, t - period of time.

Or numerically:

$$A = 4.87*10^{-18} s^{-1} * \frac {1 g} {238 g/mol} * 6.02*10^{23} mol^{-1} * 2^{-1 s/1.42∗10^{17} s};$$ $$A \approx 12318 s^{-1};$$

Answer: the half-life period for the ²³⁸U is 4.5028*10⁹ years. Activity of the sample is 12318 disintegarations per second.