For a given chemical reaction $$\gamma_1 A + \gamma_2 B \to \gamma_3 C + \gamma_4 D$$. Concentration of C changes from $$10$$ mmol dm$$^{-3}$$ to $$20$$ mmol dm$$^{-3}$$ in $$10$$ s. Rate of appearance of D is $$1.5$$ times the rate of disappearance of B which is twice the rate of disappearance of A. The rate of appearance of D has been experimentally determined to be $$9$$ mmol dm$$^{-3}$$ s$$^{-1}$$. Therefore the rate of reaction is ______ mmol dm$$^{-3}$$ s$$^{-1}$$. (Nearest Integer)

For the reaction $$\gamma_1 A + \gamma_2 B \to \gamma_3 C + \gamma_4 D$$, we need to find the rate of reaction.

The concentration of C changes from 10 mmol dm$$^{-3}$$ to 20 mmol dm$$^{-3}$$ in 10 s, so $$\frac{d[C]}{dt} = \frac{20 - 10}{10} = 1 \text{ mmol dm}^{-3} \text{s}^{-1}$$.

The rate of appearance of D is given as 9 mmol dm$$^{-3}$$ s$$^{-1}$$:
$$\frac{d[D]}{dt} = 9 \text{ mmol dm}^{-3} \text{s}^{-1}$$ Since the rate of appearance of D equals 1.5 times the rate of disappearance of B, it follows that
$$\frac{d[D]}{dt} = 1.5 \times \left(-\frac{d[B]}{dt}\right) \implies -\frac{d[B]}{dt} = \frac{9}{1.5} = 6 \text{ mmol dm}^{-3} \text{s}^{-1}$$ Furthermore, as the rate of disappearance of B is twice the rate of disappearance of A, we have
$$-\frac{d[B]}{dt} = 2 \times \left(-\frac{d[A]}{dt}\right) \implies -\frac{d[A]}{dt} = \frac{6}{2} = 3 \text{ mmol dm}^{-3} \text{s}^{-1}$$.

For the general reaction $$\gamma_1 A + \gamma_2 B \to \gamma_3 C + \gamma_4 D$$, the rate of reaction $$r$$ is defined as:
$$r = \frac{1}{\gamma_1}\left(-\frac{d[A]}{dt}\right) = \frac{1}{\gamma_2}\left(-\frac{d[B]}{dt}\right) = \frac{1}{\gamma_3}\frac{d[C]}{dt} = \frac{1}{\gamma_4}\frac{d[D]}{dt}$$ Since all these expressions must be equal, the coefficients are proportional to the individual rates:
$$\gamma_1 : \gamma_2 : \gamma_3 : \gamma_4 = 3 : 6 : 1 : 9$$

Using any of the equivalent expressions yields:
$$r = \frac{1}{\gamma_3}\frac{d[C]}{dt} = \frac{1}{1} \times 1 = 1 \text{ mmol dm}^{-3} \text{s}^{-1}$$ Verification with other species confirms consistency:
$$r = \frac{1}{\gamma_1}\left(-\frac{d[A]}{dt}\right) = \frac{1}{3} \times 3 = 1$$ ✓
$$r = \frac{1}{\gamma_2}\left(-\frac{d[B]}{dt}\right) = \frac{1}{6} \times 6 = 1$$ ✓
$$r = \frac{1}{\gamma_4}\frac{d[D]}{dt} = \frac{1}{9} \times 9 = 1$$ ✓

The correct answer is $$\mathbf{1}$$ mmol dm$$^{-3}$$ s$$^{-1}$$.