A mixture of carbon dioxide and oxygen has volume 8310 cm$$^3$$, temperature 300 K, pressure 100 kPa and mass 13.2 g. The number of moles of carbon dioxide and oxygen gases in the mixture respectively are _____.
(Assume both gases behave like ideal gases) [$$R = 8.31$$ J/mol.K]

The ideal-gas equation is $$PV = nRT$$, where $$n$$ is the total number of moles present.

Convert the given data to SI units:
Volume : $$V = 8310\ \text{cm}^3 = 8310 \times 10^{-6}\ \text{m}^3 = 8.31 \times 10^{-3}\ \text{m}^3$$
Pressure : $$P = 100\ \text{kPa} = 100 \times 10^{3}\ \text{Pa} = 1.00 \times 10^{5}\ \text{Pa}$$
Temperature : $$T = 300\ \text{K}$$
Gas constant : $$R = 8.31\ \text{J mol}^{-1}\text{K}^{-1}$$

Calculate the total moles in the mixture:

$$n_{\text{total}} = \frac{PV}{RT} = \frac{(1.00 \times 10^{5}) (8.31 \times 10^{-3})}{8.31 \times 300}$$

Numerator : $$1.00 \times 10^{5} \times 8.31 \times 10^{-3} = 8.31 \times 10^{2} = 831$$
Denominator : $$8.31 \times 300 = 2493$$

$$n_{\text{total}} = \frac{831}{2493} \approx 0.333\ \text{mol}$$

Let $$n_1$$ be the moles of $$CO_2$$ (molar mass $$M_{CO_2}=44\ \text{g mol}^{-1}$$) and $$n_2$$ be the moles of $$O_2$$ (molar mass $$M_{O_2}=32\ \text{g mol}^{-1}$$).

Number-of-moles relation:
$$n_1 + n_2 = 0.333 \quad -(1)$$

Mass relation (total mass given is $$13.2\ \text{g}$$):
$$44 n_1 + 32 n_2 = 13.2 \quad -(2)$$

From $$(1)$$, $$n_2 = 0.333 - n_1$$. Substitute into $$(2)$$:

$$44 n_1 + 32 (0.333 - n_1) = 13.2$$

$$44 n_1 + 10.666 - 32 n_1 = 13.2$$

$$12 n_1 + 10.666 = 13.2$$

$$12 n_1 = 13.2 - 10.666 = 2.534$$

$$n_1 = \frac{2.534}{12} \approx 0.211\ \text{mol}$$

Using $$(1)$$ again:
$$n_2 = 0.333 - 0.211 \approx 0.122\ \text{mol}$$

Hence, moles of $$CO_2 \approx 0.21\ \text{mol}$$ and moles of $$O_2 \approx 0.12\ \text{mol}$$.

Option C which is: $$0.21$$ and $$0.12$$