Question

Heat in the amount of \(100 \mathrm{kJ}\) is transferred directly from a hot reservoir at \(1200 \mathrm{K}\) to a cold reservoir at \(600 \mathrm{K}\) Calculate the entropy change of the two reservoirs and determine if the increase of entropy principle is satisfied.

Short answer

Answer: Yes, the increase of entropy principle is satisfied, as the total entropy change is positive (83.34 J/K).

Step-by-step solution

Identify the given data

We are given: - Heat transferred, Q = \(100 \mathrm{kJ}\) - Temperature of the hot reservoir, \(T_H = 1200 \mathrm{K}\) - Temperature of the cold reservoir, \(T_C = 600 \mathrm{K}\)

Calculate the entropy change for the hot reservoir

Using the formula for entropy change, we will calculate the change in entropy for the hot reservoir: \(\Delta S_H = -\frac{Q}{T_H}\) Using the given values: \(\Delta S_H = -\frac{100 \times 10^3\,\mathrm{J}}{1200\,\mathrm{K}} = -\frac{100}{1.2}\,\mathrm{J}\,\mathrm{K}^{-1} = -83.33\,\mathrm{J}\,\mathrm{K}^{-1}\) The negative sign indicates a decrease in entropy for the hot reservoir.

Calculate the entropy change for the cold reservoir

Using the formula for entropy change, we will calculate the change in entropy for the cold reservoir: \(\Delta S_C = \frac{Q}{T_C}\) Using the given values: \(\Delta S_C = \frac{100 \times 10^3\,\mathrm{J}}{600\,\mathrm{K}} = \frac{100}{0.6}\,\mathrm{J}\,\mathrm{K}^{-1} = 166.67\,\mathrm{J}\,\mathrm{K}^{-1}\)

Calculate the total entropy change

Now, we will find the total entropy change by adding the individual entropy changes for both reservoirs: \(\Delta S_{total} = \Delta S_H + \Delta S_C\) \(\Delta S_{total} = -83.33\,\mathrm{J}\,\mathrm{K}^{-1} + 166.67\,\mathrm{J}\,\mathrm{K}^{-1} = 83.34\,\mathrm{J}\,\mathrm{K}^{-1}\) The total entropy change is positive.

Determine if the increase of entropy principle is satisfied

According to the increase of entropy principle, the total entropy change must be greater than or equal to zero for any process or the sum of all entropy changes must be positive. In this case, the total entropy change is positive (\(\Delta S_{total} = 83.34\,\mathrm{J}\,\mathrm{K}^{-1}\)), which means that the increase of entropy principle is satisfied.