To determine which displacement reaction is most likely to occur between different metals and [tex]\( \text{BaCl}_2 \)[/tex], we need to consider the reactivity series of metals. In a displacement reaction, a more reactive metal can displace a less reactive metal from a compound.
Here's a summary of the reactivity of the metals involved in the given options (from most reactive to least reactive):
1. Sodium ([tex]\( \text{Na} \)[/tex])
2. Calcium ([tex]\( \text{Ca} \)[/tex])
3. Strontium ([tex]\( \text{Sr} \)[/tex])
4. Barium ([tex]\( \text{Ba} \)[/tex])
5. Manganese ([tex]\( \text{Mn} \)[/tex])
Given the options, we examine the possibility of displacement:
Option A:
[tex]\[ \text{BaCl}_2 + \text{Ca} \rightarrow \text{CaCl}_2 + \text{Ba} \][/tex]
- Calcium is more reactive than Barium and can displace Barium.
Option B:
[tex]\[ \text{BaCl}_2 + \text{Sr} \rightarrow \text{SrCl}_2 + \text{Ba} \][/tex]
- Strontium is more reactive than Barium and can displace Barium.
Option C:
[tex]\[ \text{BaCl}_2 + 2\text{Na} \rightarrow 2\text{NaCl} + \text{Ba} \][/tex]
- Sodium is much more reactive than Barium and can easily displace Barium.
Option D:
[tex]\[ \text{BaCl}_2 + \text{Mn} \rightarrow \text{MnCl}_2 + \text{Ba} \][/tex]
- Manganese is less reactive than Barium, so it cannot displace Barium.
Thus, the most likely displacement reaction is the one involving the most reactive metal, which is Sodium. Sodium ([tex]\(\text{Na}\)[/tex]) will react most vigorously with [tex]\(\text{BaCl}_2\)[/tex] to produce [tex]\(\text{NaCl}\)[/tex] and [tex]\(\text{Ba}\)[/tex].
The correct answer is:
[tex]\[ \boxed{C} \][/tex]
Therefore, the most likely displacement reaction to occur is:
[tex]\[ \text{BaCl}_2 + 2\text{Na} \rightarrow 2\text{NaCl} + \text{Ba} \][/tex]
