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

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.

• Are you trying to find $\Delta T$ for $Q= 1$ Joule ?
– WW1
Commented Apr 27, 2021 at 3:37
• Yup! Got a result! (Myself too :D) Commented Apr 27, 2021 at 3:49
• @WW1 What's odd is I get 293.15100179617, while I expected 293.15025 or so. Any ideas? Commented Apr 27, 2021 at 3:52

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

• Yup! I figured it out. But oddly, I got Tf ≈ 293.1489982040342603 which is odd, because I expected a temperature rise from 293.15. Any idea why? Edit: Oh I messed up when I subtracted 1 from both sides. Thanks! Commented Apr 27, 2021 at 3:44
• What's odd is I get 293.15100179617, while I expected 293.15025 or so. Any ideas? Commented Apr 27, 2021 at 3:51
• Why were you expecting 0.25 mK? In the first case - with a $c$ of $4000$ you got $0.24 mK$ , whereas in the second case you have a $c$ of ~$1000$, which means your $\Delta T$ should be roughly 4x, hence around $1 mK$. This makes sense to me Commented Apr 27, 2021 at 4:08