# Hot coffee

hot coffee!!

i went with my friend and his wife to the coffee shop. the coffee was brought when my friend's wife was in the wash room. it was steaming hot while the milk was cold.we mixed the coffee and milk and started drinking. (considering the fact that i prefer my coffee without milk, take this case as an exception.)

now, i wanted to keep the coffee and milk mixed for his wife but my friend said it would be better to keep it separate till she came back as it would be hotter if it was mixed after she came than it were mixed right then.

who is right?

## Answers and Replies

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Well, you are right: the steam would increase the heat loss.

Gokul43201
Staff Emeritus
Gold Member
:: There are 4 important cases to be considered :

The milk pot is larger than your coffee cup. (eg: it really is a pot.)

The milk pot is smaller than your coffee cup. (eg: it's one of those tiny single-serving things.)

The milk is warm compared to room temperature.

The milk is cold compared to room temperature.

The solutions are :

Add warm milk in large pots LATER.

Add warm milk in small pots NOW.

Add cold milk in large pots NOW.

Add cold milk in small pots LATER.
::

SGT
The speed of cooling of the coffee or warming of the milk depends not only on the mass of the liquids, as Gokul has pointed, but on the difference in temperature between the liquid and the ambient air and on the area of the surface of the liquid in contact with the air.
Without knowing all this variables, it is impossible to give an answer, but intuitively I think the mixed coffee will be only slightly above ambient air temperature and will cool slowly.

NateTG
Homework Helper
Actually, there are even more factors:
The type of container that the cofee and milk are in is also important. Clearly, a thermos is going to hold the heat of cofee better than a paper cup.
Is there a lid on the cofee, and do you need to remove the lid to pour the milk in? Keeping the lid on can make a big difference over time.

Now, the traditional version of this question is that there is a cup of cofee which is too hot to drink, and a container of room-temperature milk. And the question is whether it is faster to wait for the cofee to cool before putting the milk in, or to put the milk in right away. In that case, it's quite clear that the better choice is to put the milk in so that the temperature after adding it is just drinkable.

In your example, it's a bit tricky since the milk is cold, and the cofee is hot. That means that while they are seperate, the milk is gaining heat from the room, and the cofee is loosing heat to the room. As long as the milk is (proportionally) gaining heat more rapidly than the coffee is loosing it, its clearly better to keep them seperate.

For example, if the milk is at liquid nitrogen temperatures (frozen solid, I know), and the cofee is tepid, then you're better off waiting for the milk to thaw. Alternatively, if the milk is mildly chilled, and the cofee is almost boiling, then you're probably better off mixing them right away to reduce net cooling.

If
1) the specifics heats (coffee & milk) are the same, and
2) the coeficients (coffee & milk) of energy exchange with the ambient are the same,
then, it doesn't matter if you mix coffee and milk and wait 10 minutes, or if you wait 10 minutes and then mix them : the final temperature will be the same.

DaveC426913
Gold Member
Well, maybe this is being overanalyzed, and maybe not.

I believe in practice, all these factors cancel out, except this:

Heat exchange occurs as a 4th exponent of the diff in temp. i.e. The hotter it is, the faster it will cool. (Additionally, the cold milk will warm faster, but the ratio of milk to coffee is normally quite small.)

By combining the two liquids, you will have a liquid that is (at least relatively) closer to ambient room temperature. Thus, it will actually cool slower.

So, add the milk, it will stay hotter longer.

DaveC426913 said:
The hotter it is, the faster it will cool.
.
.
By combining the two liquids, you will have a liquid that is (at least relatively) closer to ambient room temperature. Thus, it will actually cool slower.
So, the hotter it is, the sooner it reaches the room temperature!? :yuck:

SGT
Kittel Knight said:
So, the hotter it is, the sooner it reaches the room temperature!? :yuck:
No, the speed of change will be greater. The time to reach room temperature is the difference in temperatures divided by the speed of change.
Of course, a hot coffee will start to cool very quickly, but to reach room temperature it must pass by lower temperatures, when it will cool slowly. So, of course, hot coffee will take longer to reach room temperature.
The problem posted involves mixing cool milk to the coffee. If you mix both liquids, the coffee-milk will be closer to room temperature and will cool slowly.
If you let the hot coffee cool alone, it will reach very quickly a temperature slightly superior to the rooom's temperature. If then you mix cold milk the final temperature will be lower.
We must remember that coffee is almost pure water, so has a greater specific heat then milk, wich is a mixture of water, fat and protein (mainly). The coffee will cool more fast than the milk will warm.

SGT said:
No, the speed of change will be greater. The time to reach room temperature is the difference in temperatures divided by the speed of change.
Of course, a hot coffee will start to cool very quickly, but to reach room temperature it must pass by lower temperatures, when it will cool slowly. So, of course, hot coffee will take longer to reach room temperature.
The problem posted involves mixing cool milk to the coffee.
Agreed.

If you mix both liquids, the coffee-milk will be closer to room temperature and will cool slowly.
It's ok. The problem is you can not conclude anything about the final temperature from this. See below.

If you let the hot coffee cool alone, it will reach very quickly a temperature slightly superior to the rooom's temperature. If then you mix cold milk the final temperature will be lower.
We must remember that coffee is almost pure water, so has a greater specific heat then milk,...
So, the milk would be much more closer to the rooms temperature. And, due the greater coffes's specific heat, the mix would be closer to the coffee's temperature.

We must remember that coffee is almost pure water, so has a greater specific heat then milk, wich is a mixture of water, fat and protein (mainly). The coffee will cool more fast than the milk will warm.
If the coffee's specific heat is greater, it will cool slower than the milk will warm.

DaveC426913 said:
I believe in practice, all these factors cancel out, except this:
Heat exchange occurs as a 4th exponent of the diff in temp...
It is useless in this problem. What is the relationship between temperature, time and specific heat?

(Clue: the temperature follows an exponential decay along the time)

SGT
Kittel Knight said:
If the coffee's specific heat is greater, it will cool slower than the milk will warm.
You are right!

NateTG
Homework Helper
This will soon devolve into a discussion of the Mpemba effect....

Gokul43201
Staff Emeritus
Gold Member
I thought mpemba was involved only in the vicinity of a phase transition, and under non-equilibrium conditions.

DaveC426913
Gold Member
"It is useless in this problem. What is the relationship between temperature, time and specific heat?"

I am not convinced that these play a significant factor in the problem without making it more complicated than it needs to be. All that is needed is to know that cooling is very dependent of the relative difference in temps. A big temp diff means a big cooling factor.

Diagram

Time/temp/specific heat factors will transform both curves on the graph proportionally, but will not change their relationship.

All other things aside, when measured after any arbitrary amount of time (presuming both have had their milk added), Cup A will be hotter than Cup B.

Moonbear
Staff Emeritus
Gold Member
The solution is simple...tell your friend's wife not to take so darn long in the washroom that the coffee risks getting cold while waiting for her! :rofl:

Gokul43201
Staff Emeritus