Step 1: Analyze Statement I
Dimethyl ether (\(CH_3OCH_3\)) demonstrates moderate, though incomplete, solubility in water. While it engages in hydrogen bonding, its molecular size limits its solubility. Diethyl ether (\(C_2H_5OC_2H_5\)) exhibits only slight solubility due to its larger hydrophobic ethyl groups. This statement is accurate regarding diethyl ether's very limited solubility.
Therefore, Statement I is considered true as it accurately characterizes the solubility of both ethers, with dimethyl ether being more soluble than diethyl ether.
Step 2: Analyze Statement II
Sodium metal is employed to dry diethyl ether, reacting with any residual water to produce sodium hydroxide and hydrogen gas. Conversely, ethyl alcohol (ethanol) reacts with sodium metal, yielding sodium ethoxide and hydrogen, rendering it unsuitable for drying with sodium.
Consequently, Statement II is deemed true because sodium is effective for drying diethyl ether but not for drying ethyl alcohol.