Matching Nuclear Decay and Reaction Types with Parent Nuclei Reactions

To determine the correct matching between the nuclear processes in List I and the nuclear reactions in List II, we analyze each reaction by applying the conservation laws of mass number ($A$) and atomic number ($Z$).

1. Reaction 2: ${}^{238}_{92}\text{U} \rightarrow {}^{234}_{90}\text{Th} + {}^{4}_{2}\text{He}$

   • The mass number decreases by $4$ ($238 \rightarrow 234$).
   • The atomic number decreases by $2$ ($92 \rightarrow 90$).
   • This change corresponds to the emission of an $\alpha$-particle (${}^{4}_{2}\text{He}$).
   • Therefore, P (Alpha decay) matches with 2.

2. Reaction 1: ${}^{15}_{8}\text{O} \rightarrow {}^{15}_{7}\text{N} + {}^{0}_{1}e^{+} + \nu_e$

   • The mass number remains unchanged ($15 \rightarrow 15$).
   • The atomic number decreases by $1$ ($8 \rightarrow 7$).
   • This change corresponds to $\beta^+$ decay (positron emission).
   • Therefore, Q ($\beta^+$ decay) matches with 1.

3. Reaction 4: ${}^{239}_{94}\text{Pu} \rightarrow {}^{140}_{57}\text{La} + \dots$

   • A heavy parent nucleus (${}^{239}_{94}\text{Pu}$) splits into lighter daughter nuclei of intermediate mass numbers along with neutrons.
   • This process represents nuclear fission.
   • Therefore, R (Fission) matches with 4.

4. Reaction 3: ${}^{185}_{83}\text{Bi} \rightarrow {}^{184}_{82}\text{Pb} + {}^{1}_{1}\text{p}$

   • The mass number decreases by $1$ ($185 \rightarrow 184$).
   • The atomic number decreases by $1$ ($83 \rightarrow 82$).
   • This change corresponds to the emission of a proton (${}^{1}_{1}\text{p}$).
   • Therefore, S (Proton emission) matches with 3.

Combining the results: $\text{P} \rightarrow 2, \quad \text{Q} \rightarrow 1, \quad \text{R} \rightarrow 4, \quad \text{S} \rightarrow 3$

Thus, the correct option is C.