To create a balanced overall reaction from the given unbalanced half-reactions, we can follow the following steps:
### Step 1: Write the given unbalanced half-reactions
The unbalanced half-reactions provided are:
1. Oxidation half-reaction for Calcium (Ca):
[tex]\[ \text{Ca} \longrightarrow \text{Ca}^{2+} \][/tex]
2. Reduction half-reaction for Sodium (Na):
[tex]\[ \text{Na}^{+} \longrightarrow \text{Na} \][/tex]
### Step 2: Add electrons to balance each half-reaction
For the oxidation half-reaction (loss of electrons):
[tex]\[ \text{Ca} \longrightarrow \text{Ca}^{2+} + 2e^- \][/tex]
For the reduction half-reaction (gain of electrons):
[tex]\[ \text{Na}^{+} + e^- \longrightarrow \text{Na} \][/tex]
### Step 3: Balance the electrons in each half-reaction
To make sure the electrons gained in reduction are equal to the electrons lost in oxidation, we need to balance the number of electrons. We notice that the oxidation half-reaction has 2 electrons, while the reduction half-reaction has 1 electron. To balance them, we need to multiply the reduction half-reaction by 2.
So, the balanced reduction half-reaction becomes:
[tex]\[ 2\text{Na}^{+} + 2e^- \longrightarrow 2\text{Na} \][/tex]
### Step 4: Combine the balanced half-reactions
Now, we can combine the balanced oxidation and reduction half-reactions. When we add them together, the electrons cancel out:
[tex]\[ \text{Ca} \longrightarrow \text{Ca}^{2+} + 2e^- \][/tex]
[tex]\[ 2\text{Na}^{+} + 2e^- \longrightarrow 2\text{Na} \][/tex]
Combining them gives:
[tex]\[ \text{Ca} + 2\text{Na}^{+} \longrightarrow \text{Ca}^{2+} + 2\text{Na} \][/tex]
Therefore, the balanced overall reaction is:
[tex]\[ \boxed{\text{Ca} + 2\text{Na}^{+} \longrightarrow \text{Ca}^{2+} + 2\text{Na}} \][/tex]
