Let's analyze the decay chain step-by-step. We start with a nucleus
X having atomic number
Z and mass number
A. The decay chain is as follows:
XX1In alpha decay, an alpha particle (helium nucleus) is emitted, which has a mass number of 4 and an atomic number of 2 .
Therefore:
X1: atomic number
=Z−2,‌‌ mass number
=A−4X1X2In beta decay, a neutron is converted into a proton, increasing the atomic number by 1 while the mass number remains unchanged.
Therefore:
‌X2: atomic number
=(Z−2)+1=Z−1, mass number
=A−4‌X2X3Again, alpha decay reduces the atomic number by 2 and the mass number by 4 .
Therefore:
X3: atomic number
=(Z−1)−2=Z−3,‌‌ mass number
=(A−4)−4=A−8X3X4Gamma decay does not change either the atomic number or the mass number.
Therefore:
X4: atomic number
=Z−3, mass number
=A−8We are given that
X4 has a mass number of 172 and an atomic number of 69 . This gives us two equations:
‌Z−3=69‌A−8=172 Now, solve these equations:
From the first equation:
Z=69+3=72.From the second equation:
A=172+8=180. Thus, the original nucleus
X has an atomic number of 72 and a mass number of 180 .