For a dimerization reaction,
$$2A(g) \to A_2(g)$$
at 298K, $$\Delta U^- = -20\,\text{kJ mol}^{-1}$$, $$\Delta S^- = -30\,\text{JK}^{-1}\text{mol}^{-1}$$, then the $$\Delta G^-$$ will be .....J.

We are given the dimerization reaction: $$2A(g) \rightarrow A_2(g)$$ at $$298$$ K, with $$\Delta U^\circ = -20 \text{ kJ mol}^{-1}$$ and $$\Delta S^\circ = -30 \text{ J K}^{-1} \text{mol}^{-1}$$.

We need to find $$\Delta G^\circ$$ in joules. We use the relation $$\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$$.

First, we find $$\Delta H^\circ$$ using the relation $$\Delta H^\circ = \Delta U^\circ + \Delta n_g RT$$, where $$\Delta n_g$$ is the change in the number of moles of gaseous species.

For the reaction $$2A(g) \rightarrow A_2(g)$$: moles of gaseous products $$= 1$$, moles of gaseous reactants $$= 2$$. So $$\Delta n_g = 1 - 2 = -1$$.

Converting $$\Delta U^\circ$$ to joules: $$\Delta U^\circ = -20 \times 1000 = -20000$$ J mol$$^{-1}$$.

Now, $$\Delta H^\circ = -20000 + (-1)(8.314)(298) = -20000 - 2477.57 = -22477.57$$ J mol$$^{-1}$$.

Rounding: $$\Delta H^\circ \approx -22478$$ J mol$$^{-1}$$.

Now we calculate $$\Delta G^\circ$$:

$$\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ = -22478 - 298 \times (-30)$$

$$\Delta G^\circ = -22478 + 8940 = -13538$$ J.

Therefore, $$\Delta G^\circ = -13538$$ J.