To determine which element does not directly react with oxygen, we must consider the typical reactivity of each option with oxygen.
Phosphorus (P): Phosphorus is known to react readily with oxygen to form phosphorus oxides, such as P_4O_{10}. This is a vigorous reaction, especially in its white form.
Sulfur (S): Sulfur also reacts with oxygen to form sulfur dioxide (SO_2) and with excess oxygen, sulfur trioxide (SO_3), further combining to form sulfuric acid in water.
Sodium (Na): Sodium is a highly reactive metal and reacts with oxygen at room temperature to form sodium oxide (Na_2O). This reaction is quite rapid and exothermic.
Chlorine (Cl): Unlike the other elements listed, chlorine does not react directly with oxygen. Chlorine is a relatively inert element towards oxygen under standard conditions.
Therefore, the element that does not directly react with oxygen is Chlorine (Cl).