The isotopes of \({ U }^{ 238 }\ and\ { U }^{ 235 }\) occur in nature in the ratio 140 : 1. Assuming that at the time of earth's formation, they were present in equal ratio, make an estimate of the age of the earth. The half lives of \({ U }^{ 238 }\ and\ { U }^{ 235 }\) are \(4.5\times { 10 }^{ 9 }\) years and \(7.13\times { 10 }^{ 8 }\) years respectively.
Given : \(\log _{ 10 }{ 140 } =2.1461,\log _{ 10 }{ 2 } =0.3010\)

Two radioactive nuclei A and B, in a given sample disintegrates into a stable nucleus C. At time t = 0, number of A species are 4N₀ and that of B are N₀. Half-life of A (for conversion to C) is 1 min whereas, that of B is 2 min. Initially, there are no nuclei of C present in the sample. When number of nuclei of A and B are equal, then what would be the number of nuclei of C present in the sample?

(a) Deduce the expression ,N = \({ N }_{ 0 }{ e }^{ -\lambda t }\), for the law of radioactive decay
(b) (i) Write symbolically the process expressing the \({ \beta }^{ + }\) decay of \(_{ 11 }^{ 22 }{ Na }\) . Also write the basic nuclear process underlying this decay.
(ii) Is the nucleus formed in the decay of the nucleus \(_{ 11 }^{ 22 }{ Na }\), an isotope or an isobar?

Write symbolically the process expressing the \({ \beta }^{ + }\) decay of \(_{ 11 }^{ 22 }{ Na }\). Also write the basic nuclear process underlying this decay.

Is the nucleus formed in the decay of the nucleus \(_{ 11 }^{ 22 }{ Na }\), an isotope or an isobar?