[Peter Stanford]

Peter Stanford  asked the Naked Scientists:
   
Hello Chris.

If I make a hot, extra cup of tea at breakfast and put it on the table in my kitchen, will it ever exactly drop down to the same temperature as the kitchen? If I measure the temperature of the tea and the room every five minutes and record my findings on a graph, even if the lines appear to touch, if I zoom in on them with a magnifying glass, then a microscope, then an electron microscope, won't the lines always never quite touch, even if I measure the temperature until the end of the universe (or until I have to put the cat out, before I go to bed. Which ever one happens first)?
 
Many thanks,
Peter.

What do you think?

[Madidus_Scientia]

Yes, after a while it will reach the same temperature as the room. Just that the room will become slightly hotter for it.

[Geezer]

Perhaps it will be slightly colder than the air temperature because of evaporation?

[syhprum]

I think that what you are asking are the increments of temperature infinite in number or are they very small but finite in number.
I think the general view is that temperature is quantised i.e the latter. 

[geo driver]

never leave tea to go to room temperature.  you can beat a good cup of hot tea.

the smallest thing can be infinitely big if you zoom in on it enough

[Geezer]

never leave tea to go to room temperature.  you can beat a good cup of hot tea.

the smallest thing can be infinitely big if you zoom in on it enough

Wow! Someone in France actually knows that tea should be hot. This proves without a shadow of doubt that Geo is not French []

(I should point out that a great many Americans fail to grasp the subtleties in the art of tea making. On the subject of tea making, the tale about my mum in a Spanish cafe will have to wait.)

[lightarrow]

Peter Stanford  asked the Naked Scientists:
   
Hello Chris.

If I make a hot, extra cup of tea at breakfast and put it on the table in my kitchen, will it ever exactly drop down to the same temperature as the kitchen? If I measure the temperature of the tea and the room every five minutes and record my findings on a graph, even if the lines appear to touch, if I zoom in on them with a magnifying glass, then a microscope, then an electron microscope, won't the lines always never quite touch, even if I measure the temperature until the end of the universe (or until I have to put the cat out, before I go to bed. Which ever one happens first)?

If what you say were true, then you couldn't even have a temperature for the tea or for the room: different portions of the tea (or of the room) are initially at different temperatures, so they would never reach thermical equilibrium, and this would also be true for every macroscopic object in your room and in the universe and the concept of temperature wouldn't exist at all...
(P.S. temperature cannot be defined for microscopic objects, that is for a small number of particles, in the sense that it would be very inaccurate).