To determine which displacement reaction is most likely to occur among the given options, let's analyze the reactivity of the elements involved in the reactions:
1. Reactivity Series:
The reactivity series is a list of elements ordered by their ability to displace other elements from compounds. The reactivity series relevant to this question includes:
[tex]\[\text{Na > Ca > Mg > Sr > Ba}\][/tex]
2. Displacement Reactions:
A more reactive element can displace a less reactive element from its compound. Therefore, we need to compare the reactivity of the elements involved in each reaction with barium (Ba).
Option A:
[tex]\[BaCl_2 + Ca \rightarrow CaCl_2 + Ba\][/tex]
- Calcium (Ca) is more reactive than barium (Ba), so this reaction can occur.
Option B:
[tex]\[BaCl_2 + Sr \rightarrow SrCl_2 + Ba\][/tex]
- Strontium (Sr) is less reactive than barium (Ba), so this reaction cannot occur.
Option C:
[tex]\[BaCl_2 + 2 Na \rightarrow 2 NaCl + Ba\][/tex]
- Sodium (Na) is more reactive than barium (Ba), so this reaction can occur.
Option D:
[tex]\[BaCl_2 + Mg \rightarrow MgCl_2 + Ba\][/tex]
- Magnesium (Mg) is less reactive than barium (Ba), so this reaction cannot occur.
From the above analysis, we can see that both options A and C are possible reactions because the elements Ca and Na are more reactive than Ba. However, we are asked to find the most likely reaction.
To decide between options A and C, we compare the reactivity of [tex]\( \text{Na} \)[/tex] and [tex]\( \text{Ca} \)[/tex]. Sodium (Na) is more reactive than calcium (Ca) because in the reactivity series, Na is higher than Ca.
Therefore, the displacement reaction that is most likely to occur is:
[tex]\[
\boxed{C}
\][/tex]
