A diatomic gas $$(\gamma = 1.4)$$ does $$400$$ J of work when it is expanded isobarically. The heat given to the gas in the process is ______ J.

Since for an isobaric (constant pressure) process, the work done by the gas is $$W = P\Delta V = nR\Delta T$$.

Meanwhile, the heat supplied is $$Q = nC_p\Delta T$$.

Therefore, the ratio of heat to work is $$\frac{Q}{W} = \frac{nC_p\Delta T}{nR\Delta T} = \frac{C_p}{R}$$. For an ideal gas, $$C_p = \frac{\gamma R}{\gamma - 1}$$, which gives $$\frac{Q}{W} = \frac{\gamma}{\gamma - 1}$$.

Substituting the given values, $$\gamma = 1.4$$ and $$W = 400$$ J, we get $$Q = W \times \frac{\gamma}{\gamma - 1} = 400 \times \frac{1.4}{1.4 - 1}$$, hence $$Q = 400 \times \frac{1.4}{0.4} = 400 \times 3.5 = 1400 \text{ J}$$.

The answer is $$\boxed{1400}$$ J.