Graphing M(water)*C(water)*ΔT = M(metal)*C(metal)*ΔT

Question

i'm currently doing an experiment, the goal of which is to determine the specific heat capacity of a certain metal. We are using the calorimeter method to do this. I want to graph the equation of Heat (Q=mcΔT) for both the water inside the calorimeter and the metal, meaning I would graph:

m(water)c(water)ΔT(water) = m(metal)c(metal)ΔT(metal)

where: m=mass, c=specific heat capacity, ΔT=change in temperature.

My question is if I graph the first half of the equation: m(water)c(water)ΔT(water) on the y-axis, is there a way I can get c(metal) as my gradient?

I know every value except for c(metal).
Multiple trials were done with different starting temperatures of the metal, therefore I would like to create a graph. From my knowledge it should be possible to graph:

m(water)c(water)ΔT(water) = m(metal)c(metal)ΔT(metal)

where the first half of the equation is the Y-axis
m(metal)*c(metal) is the gradient and
ΔT(metal) is the x-axis.

Could anyone verify if this is possible or if my thinking is correct? It's been a while since I've had to do anything like this.
Thanks!

Chet Miller · Answer

You should make a graph of the final temperature of the water and metal as y and the starting water temperature as x.  The slope of the line will be related to the c of the metal, and you can determine the c from the slope (knowing the other parameters).

ADDENDUM

$$T=frac{m_{water}c_{water}}{(m_{metal}c_{metal}+m_{water}c_{water})}T_{0,water}+frac{m_{metal}c_{metal}T_{0,metal}}{(m_{metal}c_{metal}+m_{water}c_{water})}$$

I assume you are using the same initial metal temperature in all cases and varying the initial water temperature.

Another form of the equation is:
$$(T-T_{0,water})=frac{m_{metal}c_{metal}}{(m_{metal}c_{metal}+m_{water}c_{water})}(T_{0,metal}-T_{0,water})$$

So plot $(T-T_{0,water})$ vs $(T_{0,metal}-T_{0,water})$.  You should get a straight line through the origin with a slope of $frac{m_{metal}c_{metal}}{(m_{metal}c_{metal}+m_{water}c_{water})}$

Graphing M(water)C(water)ΔT = M(metal)C(metal)ΔT

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