If the Earth is treated as a spherical black body of radius \(6371 \mathrm{~km}\), absorbing heat from the Sun at a rate given by the solar constant (1370. W/m \(^{2}\) ) and immersed in space with an approximate temperature of \(T_{s p}=50.0 \mathrm{~K},\) it radiates heat back into space at an equilibrium temperature of \(278.9 \mathrm{~K}\). (This is a slight refinement of the model in Chapter 18.) Estimate the rate at which the Earth gains entropy in this model.

To determine the heat absorbed by Earth, we need its cross-sectional area, and to find the heat emitted back to space, we need its surface area. Let's find both of these with the given Earth's radius \(R = 6371\,\text{km}\). Cross-sectional area: \(A_c = \pi R^2\) Surface area: \(A_s = 4\pi R^2\)

Given the solar constant \(S = 1370\,\text{W/m}^2\), the rate of heat absorbed by Earth can be computed as follows: Heat absorption rate: \(Q_{abs} = A_c \times S\) The equilibrium temperature of Earth is \(T_{Earth} = 278.9\,\text{K}\). Given the temperature of space \(T_{space} = 50.0\,\text{K}\), we can use the Stefan-Boltzmann law to find the rate at which Earth radiates heat back to space. The Stefan-Boltzmann constant \(\sigma = 5.67\times10^{-8} \,\mathrm{W} \cdot \mathrm{m}^{-2} \cdot \mathrm{K}^{-4}\). Heat emission rate: \(Q_{em} = A_s \times \sigma \times (T_{Earth}^4 - T_{space}^4)\)

Now, we will compute the rate at which Earth gains entropy. To do this, we need to find the entropy gain due to heat absorption and the entropy loss due to heat emission. Entropy change can be computed using the formula: \(\Delta S = \frac{Q}{T}\) So, the entropy gain due to heat absorption: \(\Delta S_{abs} = \frac{Q_{abs}}{T_{space}}\) And the entropy loss due to heat emission: \(\Delta S_{em} = \frac{Q_{em}}{T_{Earth}}\) The net rate at which Earth gains entropy is the difference between the entropy gain due to heat absorption and the entropy loss due to heat emission. Entropy gain rate: \(\Delta S_{net} = \Delta S_{abs} - \Delta S_{em}\) With all the steps outlined above, you can now substitute the given values and compute the rate at which Earth gains entropy in this model.