Two radioactive nuclei P and Q, in a given sample decay

Home » Modern Physics » Nuclei » Question

Two radioactive nuclei P and Q, in a given sample decay into a stable nucleolus R. At time t = 0, number of P species are 4 N₀ and that of Q are N₀ . Half-life of P (for conversion to R) is 1 minute where as that of Q is 2 minutes. Initially there are no nuclei of R present in the sample. When number of nuclei of P and Q are equal, the number of nuclei of R present in the sample would be

Initially P → 4N₀
Q → N₀
Half life T_P = 1 min.
T_Q = 2 min.
Let after time t number of nuclei of P and Q are equal, that is

	4N₀	=	N₀
	2^t/1		2^t/2

⇒	4N	=	1
	2^t/1		2^t/2

⇒ 2^t/1 = 4.2^t/2
2².2^t/2 = 2^{(2 + t/2)}

⇒	t	= 2 +	t
	1		2

⇒	t	= 2
	2

⇒ t = 4 min

N_P =	4N₀	=	N₀
	2^4/1		4

at t = 4 min.

N₀ =	N₀	=	N₀
	4		4

or population of R

4N₀ -	N₀	+ N₀ -	N₀
	4		4

=	9N₀
	2