1. Identify each process from List II by checking the change in mass number $A$ and atomic number $Z$.

2. Process 1: ${}_8^{15}O \to {}_7^{15}N + \dots$

   • Mass number unchanged: $15 \to 15$
   • Atomic number decreases by 1: $8 \to 7$

   This is $\beta^+$ decay because in positron emission, $p \to n + e^+ + \nu$ so $Z$ decreases by 1 while $A$ remains same.

   Hence, $Q \to 1$

3. Process 2: ${}_{92}^{238}U \to {}_{90}^{234}Th + \dots$

   • Mass number decreases by 4: $238 \to 234$
   • Atomic number decreases by 2: $92 \to 90$

   This is alpha decay, since an alpha particle is ${}_2^4He$

   Hence, $P \to 2$

4. Process 3: ${}_{83}^{185}Bi \to {}_{82}^{184}Pb + \dots$

   • Mass number decreases by 1: $185 \to 184$
   • Atomic number decreases by 1: $83 \to 82$

   This corresponds to proton emission, since a proton is ${}_1^1H$

   Hence, $S \to 3$

5. Process 4: ${}_{94}^{239}Pu \to {}_{57}^{140}La + \dots$

   This is splitting of a heavy nucleus into lighter nuclei, so it is fission.

   Hence, $R \to 4$

6. Final matching

• $P$ (Alpha decay) $\to 2$
• $Q$ ($\beta^+$ decay) $\to 1$
• $R$ (Fission) $\to 4$
• $S$ (Proton emission) $\to 3$

So the correct option is: $\boxed{\text{C}}$

7. Comparison with stored answer

Stored correct answer: C

Our derived answer: C

So, the derived answer agrees with the stored answer.