Is it more cost effective to pour milk into hot coffee and then heat it up or to heat the milk separately?
Let’s see what Gemini has to say about it. Not being that good at maths, I kept the volume and temperature at nice round numbers, because I kind of guessed Gemini would go all Einstein in the response.
I asked Gemini. If 300ml of coffee is at 100 degree centigrade, and 200ml of milk is at 10 degree centigrade, and it takes the milk one minute to heat to 100 degrees in the microwave, and I pour the cold milk into the hot coffee, how long would it take the mixture to heat to 100 degree when heated in the microwave. Would it still be one minute?
Naturally Gemini gave me an answer worthy of that wonderful video podcast Veritasium, and took a deep dive into the time, energy and temperature as variables.
Here’s Googles response:-
No, it would not still be one minute. It would take longer than one minute to heat the mixture to 100 degrees Celsius in the microwave.
Here’s why:
- Initial Temperature of the Mixture: When you pour the cold milk into the hot coffee, they will mix and reach an equilibrium temperature. This temperature will be significantly lower than the initial 100 degrees Celsius of the coffee. We calculated it to be approximately 64 degrees Celsius.
- Increased Volume: You now have a larger volume of liquid to heat: 300 ml (coffee) + 200 ml (milk) = 500 ml of mixture. The microwave heated only 200 ml of milk in one minute.
- Temperature Differential: The microwave’s efficiency in heating depends on the amount of energy required to raise the temperature.
- In the original scenario, the microwave raised 200 ml of milk from 10 °C to 100 °C (ΔT=90 °C).
- In the new scenario, the microwave needs to raise 500 ml of the mixture from approximately 64 °C to 100 °C (ΔT=36 °C).
While the temperature difference is smaller in the second case (ΔT=36 °C vs 90 °C), the much larger volume of liquid (500 ml vs 200 ml) will dominate the heating time.
Let’s quantify it roughly:
A microwave heats by transferring energy. The energy (Q) required to heat a substance is given by:
Q=m⋅c⋅ΔT
Where:
- m = mass
- c = specific heat capacity
- ΔT = change in temperature
Assuming the specific heat capacity (c) is similar for both milk and the coffee-milk mixture, the energy required is proportional to m⋅ΔT. The heating time in a microwave is proportional to the energy required.
Original Scenario (Milk only):
- Mass (proportional to volume): 200 ml
- ΔT: 100 °C−10 °C=90 °C
- Energy units (proportional): 200×90=18000 units
- Time: 1 minute
New Scenario (Mixture):
- Mass (proportional to volume): 500 ml
- Initial mixture temperature: ≈64 °C
- ΔT: 100 °C−64 °C=36 °C
- Energy units (proportional): 500×36=18000 units
Interestingly, based on these rough calculations, the energy required to heat the mixture from 64°C to 100°C is approximately the same as the energy required to heat the milk from 10°C to 100°C.
However, this calculation assumes the microwave is operating at a constant power output and that the heating efficiency is the same regardless of the initial temperature or composition, which isn’t entirely accurate. Microwaves interact with water molecules. While the total energy needed might be similar in this simplified model, the rate at which the microwave can deliver that energy and how it’s absorbed by the larger, hotter mass can differ.
Key point for real-world scenarios:
- Larger Volume: Heating a larger volume of liquid always takes more time with the same power input.
- Initial Temperature: While the initial temperature is higher for the mixture, the total energy needed still involves bringing a substantial volume up.
Therefore, while the simplified calculation above suggests similar energy requirements (and thus similar time), real-world microwave performance and the increased volume would generally mean it takes longer than one minute. The difference might not be as drastic as if the starting temperature was still 10°C, but it won’t be the same.
A more precise answer would require knowing the microwave’s power output and more exact specific heat capacities of coffee and milk, but the general principle holds: more mass to heat means more time.
So my next question was:
Would it cost me less to heat the milk in the microwave before pouring it into the coffee?