Radiant energy from the Sun reaches the Earth at a rate of 1.70 × 1017 W. When 30% of the radiant energy is reflected and 70% absorbed, the average temperature of the Earth's atmosphere is 253 K. If the atmosphere absorbed 100% of the solar radiant energy, then what would be the average temperature of the Earth's atmosphere?

To solve this problem, we need to understand that the power absorbed by the Earth is proportional to the fourth power of its temperature (T^4). This is known as the Stefan-Boltzmann law.

Step 1: Calculate the power currently absorbed by the Earth.

Given that the Earth reflects 30% of the radiant energy, it absorbs 70%. So, the power absorbed by the Earth is 70% of 1.70 × 10^17 W, which equals 1.19 × 10^17 W.

Step 2: Calculate the ratio of the power absorbed when 100% of the energy is absorbed to the power currently absorbed.

If the Earth absorbed 100% of the solar radiant energy, the power absorbed would be 1.70 × 10^17 W. The ratio of this to the current power absorbed is (1.70 × 10^17 W) / (1.19 × 10^17 W) = 1.43.

Step 3: Calculate the new temperature.

Since the power absorbed is proportional to T^4, the new temperature would be the fourth root of the ratio of the powers multiplied by the current temperature.

So, the new temperature T' = T * (1.43)^(1/4) = 253 K * (1.43)^(1/4) = 288 K.

Therefore, if the Earth's atmosphere absorbed 100% of the solar radiant energy, the average temperature of the Earth's atmosphere would be 288 K.