Trying to estimate the temperature rise ΔT of an object, Q = mcΔT, BUT using a temperature dependent specific heat capacity c(T).

Question

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.

Answer 1

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