1. Let the masses of oxygen and nitrogen in the mixture be in the ratio $m_{O_2} : m_{N_2} = 1:4$

2. Number of molecules is proportional to number of moles: $n = \frac{m}{M}$ where $m$ is mass and $M$ is molar mass.

3. Molar masses:

   • For oxygen gas, $O_2$: $M_{O_2} = 32$
   • For nitrogen gas, $N_2$: $M_{N_2} = 28$

4. Therefore, the ratio of number of molecules is $\frac{N_{O_2}}{N_{N_2}} = \frac{n_{O_2}}{n_{N_2}} = \frac{m_{O_2}/32}{m_{N_2}/28}$

5. Using $m_{O_2}:m_{N_2} = 1:4$, $\frac{N_{O_2}}{N_{N_2}} = \frac{1/32}{4/28} = \frac{1}{32}\cdot\frac{28}{4} = \frac{28}{128} = \frac{7}{32}$

6. Hence, $N_{O_2} : N_{N_2} = 7:32$

7. Checking options:

   • A: $1:8$ ❌
   • B: $3:16$ ❌
   • C: $1:4$ ❌
   • D: $7:32$ ✅

So the correct option is D.