Quantities and Conditions:
I measured a typical teacup from my cupboard and found
that it massed just over 100 gm, and held 200 ml of water.
Neglecting the density change of water, consider that to
be 200 gm of hot or cold water. I also neglected the difference
between water and tea.
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Answer #1:
First, we solve by engineering calculation:
Water has a specific heat of 1 Cal/gm*K (by definition)
A typical ceramic material (silicon dioxide)
has a specific heat of (45 J/mol-K) or 0.168 Cal/gm*C.
Many consider an ideal tea brewing temperature to be 85-90C.
Let's choose 90 C. And suppose the teacup starts at room temperature
of 25 C.
If 200 gm of water at 90 C is added to 100 gm of ceramic at 25 C,
the combination is in equilibrium when the two temperatures become equal.
Together they have a heat content (relative to 0 C) of ...
100gm * 25C * 0.168 cal/gm*C + 200gm * 90C * 1.0 cal/gm*C = 18420 cal
at the equilibrium temperature, the above equation becomes
100gm * T * 0.168 cal/gm*C + 200gm * T * 1.0 cal/gm*C = 18420 cal
Solving for T gives 84 C.
Take 84 C to the temperature of the pre-heated teacup.
Grinding through the equations once more, gives
Before mixing:
100gm * 84C * 0.168 cal/gm*C + 200gm * 90C * 1.0 cal/gm*C = 19410 cal
After equilibrating:
100gm * T * 0.168 cal/gm*C + 200gm * T * 1.0 cal/gm*C = 18420 cal
And finally:
T = 89.5 C
So the answer is:
Preheating the teacup increases the equilibrium tea temperature
from 84 C to 89.5 C
[ Both are too hot for my palate, so then I would cool
it down to a reasonable drinking temperature of say 60 C ]
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Answer #2:
Using the same quantities and temperatures as in the above example,
I measured the actual temperature after five minutes, and found 79 C.
The missing 6 C is probably due to convective cooling and evaporation
from the surface. I didn't think to measure the mass lost due to evaporation,
but that would have verified the evaporative heat loss.
Then, dumping the 79 C water, and pouring in another 200 ml of 90 C water,
and again waiting five minutes, I measured a temperature of 83 C.
In the experimental case, the preheated cup gave 83 C
compared to the cool cup at 79 C.
The slightly smaller temperature difference seen experimentally is probably
due to the higher evaporation rate at higher temperatures. The hotter the
tea, the faster it will cool by evaporation, so the preheated cup cooled
more than the cool cup.
[And once again, all the cups are too hot for my taste, so I would wait
for them to cool further before drinking them]
I believe that answers your question, and also gives you the tools
to calculate the equilibrium temperatures for other size teacups
and brewing temperatures should you feel the need. |
