Answer
(a) $ X = \,^{226}\text{Ra} $
(b) $ X = \,^{35}\text{Cl} $
(c) $ X = \,^{40}\text{Ca} $
(d) $ X = \,^{24}\text{Mg} $
Work Step by Step
let's analyze each decay process step-by-step.
$$\color{blue}{\bf [a]}$$
$$ \,^{230}\text{Th} \rightarrow X + \alpha $$
Thorium-230 undergoes alpha decay, which means it emits an alpha particle (helium nucleus, $ \,^{4}\text{He} $). The alpha particle has a mass number of 4 and an atomic number of 2.
Subtract these from thorium-230, so the mass number is then $ 230 - 4 = 226 $, and the atomic number is $ 90 - 2 = 88 $.
Thus, the element with atomic number 88 is radium, so
$$\boxed{ X = \,^{226}\text{Ra}} $$
$$\color{blue}{\bf [b]}$$
$$ \,^{35}\text{S} \rightarrow X + e^- + \bar{\nu} $$
Sulfur-35 undergoes beta-minus decay, where it emits an electron ($ e^- $) and an antineutrino ($ \bar{\nu} $). In beta-minus decay, the atomic number increases by 1, while the mass number remains the same. So the mass number is 35 (unchanged) and the atomic number is $ 16 + 1 = 17 $.
Thus, the element with atomic number 17 is chlorine, so
$$\boxed{ X = \,^{35}\text{Cl} }$$
$$\color{blue}{\bf [c]}$$
$$ X \rightarrow \,^{40}\text{K} + e^+ + \nu $$
This decay process involves the emission of a positron ($ e^+ $) and a neutrino ($ \nu $), indicating beta-plus decay. In beta-plus decay, the atomic number decreases by 1, while the mass number remains the same. So the mass number is 40 (unchanged) and the atomic number of potassium ($ K $) is 19. Thus, the original atomic number is then $ 19 + 1 = 20 $
Therefore, the element with atomic number 20 is calcium, so
$$\boxed{ X = \,^{40}\text{Ca} }$$
$$\color{blue}{\bf [d]}$$
$$ \,^{24}\text{Na} \rightarrow \,^{24}\text{Mg} + e^- + \bar{\nu} + \gamma $$
Sodium-24 undergoes beta-minus decay, emitting an electron ($ e^- $), an antineutrino ($ \bar{\nu} $), and a gamma photon ($ \gamma $). In beta-minus decay, the atomic number increases by 1, while the mass number remains the same. So, the mass number is 24 (unchanged) and the atomic number of magnesium ($ \text{Mg} $) is 12, which matches the expected product. Thus,
$$\boxed{ X = \,^{24}\text{Mg} }$$
