Using $Q = mcΔT$ I can calculate the temperature rise in an object.
I apply $1\ J$, to a $1\ kg$ mass of water which has $c = 4184\ \frac{J}{kg * K}$ and I get about $.24\ mK$.
Well now I want to estimate the heat rise $ΔT$ of an object using a temperature dependent specific heat capacity $c(T) = -0.2065*T + 1058.74$ , and I don't think I can use $Q = mcΔT$ and I'm not sure what to do.
I can interpret this equation $Q = mcΔT$ as the heat rise $ΔT$ that corresponds to $Q$ heat applied to a mass with a fixed $c$.
I can also interpret this equation as the heat required to change a mass $m$ with $c$ by $ΔT$.
Initial Temperature = $293.15\ K$ btw. I'm expecting about temperature rise of about $.24\ mK$, but I want to have a theoretical calculation.
Thanks.
You will need to use the differential equation form when heat capacity is a function of temperature
Observe that $Q = mC\Delta T$ is the integrated form the differential equation of heat transfer when $C$ is a constant. The actual equation is
$$dQ = mCdT$$
Hence, to solve for the temperature difference, we have
$$\int dQ = m\int_{T_1}^{T_2} CdT$$ $$\implies 1 = 1\cdot\int_{293.15}^{T_2}(-0.2065T + 1058.74)dT$$
This gives a quadratic in $T_2$, out of which only one root should make physical sense