The fundamental concept you need here is called specific heat
capacity. Speaking loosely, this is a measure of how hard it is to
heat something up or cool it down. More precisely, it measures how
much energy it takes to raise the temperature of the indicated
substance, per unit of mass and temperature increase. In the SI world,
we measure it in J/kg/K (Joules (energy) divided by the mass in
kilograms and the temperature change in Kelvin).
When a uniform substance changes temperature uniformly, the amount of
energy required or released is given by the formula:
Q = m.c.dT
where Q is the energy (traditionally written as Q in thermodynamics),
m is the mass, c the specific heat capacity and dT (actually the d
should be a Delta) is the temperature change.
The second basic concept is that in these situations, heat is not
"lost" (neglecting losses to the environment); instead, it is
transferred from the hotter object to the cooler object. So the heat
given up by the iron is absorbed by the water, and both wind up at
some intermediate temperature.
Now, suppose the water starts off at some temperature T_w, and the
iron at T_i. They will both wind up at the same intermediate
temperature, T. The mass of water will be L kg (since the density of
water is 1 kg/L), and the mass of iron X kg as specified. The specific
heats of iron and water are respectively 444 and 4186 J/kg/K. (At
standard temperature and pressure - the values change with
temperature, which is obviously a complicating factor, but we shall
blithely ignore it. ;-))
So, heat given up by the iron = m.c.dT = X.444.(T_i - T)
and heat absorbed by the water = L.4186.(T - T_w)
and these are equal, so we have
4186.L.T - 4186.L.T_w = 444.X.T_i - 444.X.T
Collecting T terms gives us
T(4186L + 444X) = 444.X.T_i + 4186.L.T_w
and so
T = (444.X.T_i + 4186.L.T_w) / (4186L + 444X)
Obviously you need to measure T, T_w and T_i in the same units, and
for these values to work you need them to be either in degrees
Centigrade (Celsius) or in Kelvin. Fahrenheit definitely will not
work!
Well, actually Fahrenheit *will* work in this case, because you're
comparing two substances and all that's important here is the ratio of
the masses and the ratio of the specific heats - note that the
equation above can be rewritten as a weighted average of T_i and T_w,
with the weights being the product of the mass and the specific heats.
But if you try to do other problems you will need to make sure that
you use the right units - if your specific heats are in J/kg/K, you
need to use those units; if they are given in cal/g/deg C, use those;
and so on. |
