Algeba/ Temperature

Question

A thermometer reading 10°C is brought into a room with a constant temperature of 32°C. If the thermometer reads 13°C after 3 ​minutes, what will it read after being in the room for 5 ​minutes? For 10 ​minutes?

Answer 1

Hi KristaLea K.,

You need two things here. 1=assumptions required about heat flow and 2= math necessary to set up and solve the actual problem. Let's take those in order.1) Heat flow into or out of items is assumed (unless you have other data) to be instantaneously proportional to the difference between the instantaneous temperature of the object and the temperature of the surroundings. Here, the surroundings are assumed to be of infinite heat capacity -- they don't cool any as the thermometer heats up. So -- "instantaneous change" should immediately trigger your mental response: "exponential decay process", that is, the thermometer temperature will approach 32C as an exponential-decay-type asymptote. Good so far?

2) Then, set up the equation. You could do this as one master equation for thermometer actual temperature. I advise against that, it's too easy to mess up on the setup. So set it up to calculate the difference to 32C instead, that's a simpler expression. Call that difference to 32C == "deltat" for obvious reasons.

So, deltat(t) = deltat(0)*exp(-kt)

Substitute in the given data:

deltat(3) = (32-13) = 9 = deltat(0)*exp(-3k) = (32-10)*exp(-3k) .

Solve for k, then set that constant back into the equation above and solve for deltat(5), then for deltat(10).

Lastly, check your results (which you should calculate exactly, then round iff your instructor wishes!). You might expect t(5) to be near, but not exactly, 15C, and t(10) to be slightly less than exactly 20C.

That would be assuming none of your classmates blunder into, and break, your thermometer, as you are waiting around.

-- Cheers, -- Mr. d.