Final answer:
Yes, for a given quantity of energy, the increase in entropy is greater when the surroundings are cool because the entropy change is inversely proportional to temperature. This is consistent with the second law of thermodynamics, where heat transfer from hot to cold results in an increase in entropy of the system.
Step-by-step explanation:
For a given quantity of energy, the increase in entropy is indeed greater when the surroundings are cool than when they are hot. This is explained by the relationship between entropy (ΔS), heat transferred (Q), and temperature (T), which is expressed as ΔS = Q/T. Because entropy is the measure of randomness or disorder within a system, when energy is transferred as heat, the change in entropy is inversely proportional to the temperature of the system receiving the heat. Therefore, when the surroundings are cooler, they have a lower T value, which results in a larger increase in entropy for the same amount of energy transferred compared to when the surroundings are hotter.
An important principle to consider is the second law of thermodynamics, which states that the total entropy of a system either increases or remains constant in any spontaneous process. Heat naturally transfers from a hotter object to a cooler one, leading to an increase in entropy. The lower temperature of the cooler surroundings causes a more significant change in entropy than the decrease in entropy of the hotter object, thus producing an overall increase in entropy for the system. This result is true for any system undergoing an irreversible process.