A container with 1 kg of water in it is kept in sunlight, which causes the water to get warmer than the surroundings. The average energy per unit time per unit area received due to the sunlight is 700 Wm$$^{−2}$$ and it is absorbed by the water over an effective area of 0.05 m$$^2$$. Assuming that the heat loss from the water to the surroundings is governed by Newton’s law of cooling, the difference (in $$^\circ C$$) in the temperature of water and the surroundings after a long time will be _____________. (Ignore effect of the container, and take constant for Newton’s law of cooling = 0.001 s$$^{−1}$$, Heat capacity of water = 4200 J kg$$^{−1}$$ K$$^{−1}$$)

The water receives energy from sunlight at a constant rate and loses energy to the surroundings according to Newton’s law of cooling. After a long time the system reaches steady‐state, so the rate of heat gain equals the rate of heat loss.

Heat gained from sunlight
Average solar flux = $$700\;\text{W m}^{-2}$$
Effective absorbing area = $$0.05\;\text{m}^2$$
Power absorbed, $$P_{\text{in}} = 700 \times 0.05 = 35\;\text{W}$$ $$-(1)$$

Heat lost by Newton’s law of cooling
For a body of total heat capacity $$C_{\text{tot}}$$, Newton’s law can be written as
$$\frac{dT}{dt} = -k\,(T - T_{\text{surr}})$$
where $$k = 0.001\;\text{s}^{-1}$$ is the given cooling constant.

The instantaneous rate of heat loss (power) is then
$$P_{\text{out}} = C_{\text{tot}}\;k\,(T - T_{\text{surr}})$$ $$-(2)$$
because $$P_{\text{out}} = C_{\text{tot}}\left|\,\frac{dT}{dt}\right|$$ and $$|dT/dt| = k\,(T - T_{\text{surr}})$$.

Total heat capacity of 1 kg water:
$$C_{\text{tot}} = m c = 1\;\text{kg} \times 4200\;\text{J kg}^{-1}\text{K}^{-1} = 4200\;\text{J K}^{-1}$$ $$-(3)$$

Steady‐state condition
At long times $$P_{\text{in}} = P_{\text{out}}$$.
Using $$(1)$$, $$(2)$$ and $$(3)$$:
$$35 = 4200 \times 0.001 \times (T - T_{\text{surr}})$$

Therefore
$$T - T_{\text{surr}} = \frac{35}{4.2} = 8.33^\circ\text{C}$$

Hence, after a long time, the water settles at about $$\mathbf{8.3^\circ\text{C}}$$ warmer than the surroundings.