# Thermal energy of a system

If I had a hot cup of tea and I added to it cold milk, what would happen is that tea will lose some of its thermal energy to the milk. But why do we notice that our liquid is no longer as hot as before adding milk? After all, the total thermal energy of the system (the cup) is conserved even if there has been a transfer of heat between the two. Suppose the cup is an isolated system.

• The mixed milk and tea is colder than the original tea, and hotter than the original milk. What's the confusion? Jun 26, 2016 at 17:08
• For the same reason we do not notice the system as cold as was the milk. Jun 26, 2016 at 17:09
• What do you mean "our liquid is no longer as hot"? Our liquid is hotter than it was before we added the tea to it! Jun 28, 2016 at 1:34

Before mixing the average kinetic energy of the molecules which make up the tea is greater than the average kinetic energy of the molecules which make up the milk.
This is restatement of the fact that the tea is hotter than the milk as the temperature of a substance depends on the average kinetic energy of the molecules which make up the substance.

When you mix the tea and the milk together the molecules of the tea and the milk collide and eventually reach an average kinetic energy (temperature) which is less than the average kinetic energy (temperature) of the tea before mixing and is greater than the average kinetic energy (temperature) of the milk before mixing.
So the collisions between the molecules redistribute the kinetic energy amongst the molecules and what is called thermal equilibrium is reached.

If there is no loss of heat or chemical reaction then the kinetic energy of the hot tea plus the kinetic energy of the cold milk is equal to the kinetic energy of the mixture but on average the tea molecules have less kinetic energy after mixing and the milk molecules have more kinetic energy after mixing.

You are talking about two different cases. At first you say:

We notice that our liquid is no longer as hot as before adding milk.

Then you say:

The total thermal energy of the system (the cup) is conserved.

The total thermal energy of tea and milk is same before and after mixing. But, we cannot talk about temperature of the tea by considering to change of thermal energy of both of tea and milk. If we want to determine if temperature of tea increase or decrease, we should consider to its own thermal energy before and after mixing.

In addition, as it has been mentioned in Sammy's answer, you cannot compare temperature of whole of the tea and milk before and after mixing. Because, we cannot define a determined temperature for the system before mixing.

When you say that the system (ie the contents of the cup) is not as 'hot' as before, you are talking about temperature. However, the system was initially at two different temperatures (hot tea, cold milk). There is no obvious way of comparing the two initial temperatures with one final temperature. To decide if there has been a change in temperature, you have to wait until the system has reached thermal equilibrium. Then it is at a single, uniform temperature.

You can of course compare the total thermal energies before and after, to see if there is any loss (or gain!) caused by mixing. This can happen if there is a chemical reaction between the two ingredients. For example, dissolving a salt or sugar in water involves a (reversible) chemical reaction; for some salts there is a change in temperature.

• Thank you , but aren't we supposed to have in this case an increase in thermal energy because another liquid is added and though it's cold but it contains a thermal energy , even if tea has lost some of it's thermal energy to the milk , but the system as a whole has finally thermal energies of both tea and milk , and as thermal energy is proportional to temperature, so the temperature should increases as well ,,
– user65035
Jun 26, 2016 at 19:05
• @ZaKh : Yes, what you say is correct - until the last statement. Thermal energy is proportional to temperature for a particular substance. In this case you are talking about 3 different substances - tea, milk, and a mixture of tea and milk. The proportionality does not apply when you change substances, or change the proportions of a mixture. Jun 27, 2016 at 2:08
• You mean you never had this happen to you: drinking a cup of nice hot tea, and someone bumps into you, spilling half, leaving you with ice cubes? I hate it when that happens. Jun 27, 2016 at 9:47