Yesterday I randomly started thinking about Newton's Law of Cooling. The problem I realized is that it assumes the ambient temperature stays constant over time, which is obviously not true. So what I tried doing was to modify the differential equation into a system of differential equations, and taking the heat capacity of each into account.
$$
\frac{dT_1(t)}{dt} = C_2(T_2(t)-T_1(t))\\
\frac{dT_2(t)}{dt} = C_1(T_1(t)-T_2(t))\\
T_1(0) = T_{10}\\
T_2(0) = T_{20}
$$
which had the following solution:
$$
T_1(t)={\frac {C_{{2}}T_{{20}}+C_{{1}}T_{{10}}}{C_{{1}}+C_{{2}}}}+{\frac {C_{
{2}} \left( -T_{{20}}+T_{{10}} \right) {{\rm e}^{- \left( C_{{1}}+C_{{
2}} \right) t}}}{C_{{1}}+C_{{2}}}}\\
T_2(t)={\frac {C_{{2}}T_{{20}}
+C_{{1}}T_{{10}}}{C_{{1}}+C_{{2}}}}-{\frac {C_{{1}} \left( -T_{{20}}+T_{{10}} \right) {{\rm e}^{ -\left(C_{{1}}
+C_{{2}} \right) t}}}{C_{{1}}+C_{{2}}}}
$$
When I plotted the two equations out they seem to be right. They also follow the fact that at any point in time $C_1\Delta T_1=C_2\Delta T_2$. The end behaviour also seem to be correct. However, in cases where $C_1\rightarrow\infty$ you would expect it to behave similarly to Newton's Law of Cooling, but in reality $T_2$ drops its temperature in a very short period of time, which doesn't seem to be right. I tried looking this up but couldn't find much on this topic. If anyone can point out what I did wrong that would be great. I've had very little experience with thermodynamics but I do realize how limited these models are at describing real world scenarios. This was more for the fun of playing with differential equations.
Best Answer
The flaw in your reasoning seems to be that $C$ is not in fact heat capacity. In Newton's Law of Cooling, the proportionality constant would be related inversely to the heat capacity of the two heated liquids/gasses/materials, and directly to the heat conductance of the object separating the two materials. A material with a higher heat capacity would have a smaller temperature change for a given temperature difference, and a thin piece of metal separating the materials would result in a much larger $\frac{dT}{dt}$ than a thick piece of styrofoam would.
When bringing $C_1$ to $\infty$, you are actually decreasing the heat capacity and increasing the conductance, both of which would cause $T_2$ to drop quickly as you observed.