1. **Mug A (Cools First, Ice Added Later)**: When you let the coffee cool for 10 minutes, it loses heat to the surrounding environment. The rate of heat loss is initially high because the larger temperature difference between the coffee and the surroundings, but it decreases as the coffee cools down. When you add an ice cube after 10 minutes, the remaining heat in the coffee is used to melt the ice and then further cool the coffee. Since the coffee is not as hot when the ice is added, the ice cube will melt more slowly and will be more effective in cooling the coffee.
2. **Mug B (Ice Added First, Then Cools)**: Adding an ice cube to boiling hot coffee will cause rapid melting of the ice due to the high temperature difference. This rapid melting absorbs a significant amount of heat from the coffee quickly, but because the ice melts quickly, it may not be as effective in cooling the coffee over time. After the ice melts, the now warmer coffee continues to cool down for 10 minutes.
In theory, the amount of heat lost by the coffee in both scenarios should be similar, assuming the environment is the same for both mugs. However, in practice, the cooling curve might be slightly different due to factors like how quickly the ice melts and how heat transfer occurs in each scenario. Your intuition that Mug A might end up slightly cooler could be correct, especially if the ice cube in Mug B melts too quickly to efficiently cool the coffee over the duration of the 10 minutes.
While the basic principle of energy conservation suggests that both mugs should eventually reach a similar temperature, the dynamics of the cooling process (rate of heat loss to the environment, rate of ice melting) can lead to different cooling curves. Mug A might end up slightly cooler after the entire process due to the more efficient use of the ice cube's cooling potential, as the ice melts more slowly and steadily in the already somewhat cooled coffee. However, the difference in final temperatures might be small and also depends on various factors like the size of the ice cube, the initial temperature of the coffee, and the ambient temperature.
source: ChatGPT