A certain quantity of real gas occupies a volume of $$0.15$$ dm$$^3$$ at $$100$$ atm and $$500$$ K when its compressibility factor is $$1.07$$. Its volume at $$300$$ atm and $$300$$ K (When its compressibility factor is $$1.4$$) is _____ $$\times 10^{-4}$$ dm$$^3$$ (Nearest integer)

A real gas occupies 0.15 dm$$^3$$ at 100 atm and 500 K with compressibility factor $$Z_1 = 1.07$$. We need to find its volume at 300 atm and 300 K with $$Z_2 = 1.4$$.

For the two states of the same gas (same $$n$$):

$$\frac{P_1 V_1}{Z_1 T_1} = nR = \frac{P_2 V_2}{Z_2 T_2}$$

$$V_2 = \frac{P_1 V_1 Z_2 T_2}{Z_1 T_1 P_2}$$

$$V_2 = \frac{100 \times 0.15 \times 1.4 \times 300}{1.07 \times 500 \times 300}$$

$$V_2 = \frac{100 \times 0.15 \times 1.4}{1.07 \times 500}$$

$$V_2 = \frac{21}{535} = 0.039252 \text{ dm}^3$$

$$V_2 = 0.039252 \text{ dm}^3 = 392.52 \times 10^{-4} \text{ dm}^3$$

Rounding to the nearest integer: $$\approx 393 \times 10^{-4}$$ dm$$^3$$.

The answer is $$\mathbf{393}$$.