Step 1: Write the given reaction and identify the metal.
$2MNO_3 \rightarrow M_2O + 2NO_2 + \frac{1}{2}O_2$. The nitrate of metal M decomposes to give the metal OXIDE (M2O), nitrogen dioxide, and oxygen. We need to find which alkali metal M behaves this way.
Step 2: Recall thermal decomposition of alkali metal nitrates.
When heated: Na, K, Rb, Cs nitrates decompose to give the metal NITRITE and oxygen: \[ 2MNO_3 \xrightarrow{\Delta} 2MNO_2 + O_2 \]
Step 3: Identify the exception - Lithium.
Lithium nitrate is unique among alkali metal nitrates. Due to the anomalous behavior of lithium (high charge density, resemblance to Mg - diagonal relationship), it decomposes more extensively: \[ 4LiNO_3 \xrightarrow{\Delta} 2Li_2O + 4NO_2 + O_2 \]
Step 4: Match with the given equation.
The given equation $2MNO_3 \rightarrow M_2O + 2NO_2 + \frac{1}{2}O_2$ matches exactly with lithium nitrate decomposition (scaled down from 4 mol to 2 mol). M = Li (Lithium).
Step 5: Explain why Li behaves differently.
Li+ has the smallest size among alkali metal ions, giving it very high charge density. This high polarizing power causes the nitrate ion to decompose completely to the oxide, rather than just losing one oxygen to give nitrite.
Step 6: Conclusion.
The metal M is Lithium. LiNO3 is the only alkali metal nitrate that decomposes to give the metal oxide (Li2O), NO2, and O2.
\[ \boxed{M = \text{Li (Lithium) - option 4}} \]