A heat engine operates with the cold reservoir at temperature $$324$$ K. The minimum temperature of the hot reservoir, if the heat engine takes $$300$$ J heat from the hot reservoir and delivers $$180$$ J heat to the cold reservoir per cycle, is ______ K.

We need to find the minimum temperature of the hot reservoir for a heat engine.

Since the cold reservoir temperature is $$T_C = 324$$ K, the heat absorbed from the hot reservoir is $$Q_H = 300$$ J, and the heat delivered to the cold reservoir is $$Q_C = 180$$ J, the engine must operate at maximum (Carnot) efficiency to achieve the minimum hot reservoir temperature.

For a Carnot engine, the relationship between heat exchanged and reservoir temperatures is given by

$$\frac{Q_C}{Q_H} = \frac{T_C}{T_H}$$

Substituting the given values into this expression, we have

$$\frac{180}{300} = \frac{324}{T_H}$$

From this,

$$\frac{3}{5} = \frac{324}{T_H}$$

This gives

$$T_H = \frac{324 \times 5}{3}$$

Next, simplifying the right-hand side yields

$$T_H = \frac{1620}{3} = 540$$ K

The minimum temperature of the hot reservoir is 540 K.