Specific heat capacity for solids of changing temperature

and I understand that the total energy needed is simply integrating - but how could I calculate the red line in his diagram for a solid? I found the equations for constant volume or pressure but I'm not sure which (if either) would be a good approximation for a solid.

My specific scenario is I would like to calculate the temperature of a wire (let's say copper) as a function of time while a current is passed through it (let's assume this is somehow kept constant, because in actuality the resistance would also be changing as a function of temperature). I know the power dissipated by a wire (or any resistor) is $I^2R$, and I realize using known specific heat capacity of copper at $25{}^{\circ}\mathrm{C}$ would probably give a decent estimate - but if possibly I would like to account for the changing heat capacity.

• Well, constant volume seems unlikely (thermal expansion and all that). Constant pressure is a pretty good bet, particularly since for almost all metals you need a great deal of pressure to actually do much to the Gibbs free energy (and hence the heat capacity). Apr 6, 2016 at 21:10
• That's what I was thinking, volume is must more likely to be changing
– Eric
Apr 6, 2016 at 21:11
• To clarify - do you mean how to calculate what the heat capacity of a solid is? Starting with what knowledge? Apr 6, 2016 at 21:26
• What the heat capacity is as a function of time - I know the material and hence anything known about that material. I have found charts with known capacities at incremental temperatures for copper for example, but nowhere have I found an actual equation. Is it just something that must be measured?
– Eric
Apr 6, 2016 at 21:29
• Yes, you find mostly tabulated values. Some times they will fit some polynomial form to it. For example, the low temperature folks like some kind of functional form for <10K temperatures. Apr 6, 2016 at 21:59

For a large temperature change, specific heat is a function of temperature. Once that function has been characterized, one can obtain the total heat input as an enthalpy change,$$\Delta H = \int_{T_1}^{T_2} c\,\mathrm{d}T.$$Note that it is common to curve fit specific heat as a function of temperature, such that specific heat is a polynomial function of temperature (which makes it very easy to integrate).

• Upon noticing @Nat editing post I realize I never selected an answer, this seems the clearest in my opinion - notably that a fitted line based off measured data seems the best approach at which point integration is trivial.
– Eric
Apr 9, 2018 at 4:13

Solids are nearly incompressible, so it matters very little if you use the heat capacity at constant pressure or constant volume. How much work do you think an expanding solid sample can do if held at a constant pressure, even if the pressure is high? At constant pressure , $\Delta PV$ is going to be very small, so the change in internal energy will be nearly equal to the change in enthalpy. Have you actually compared the data for constant volume vs constant pressure (particularly at a pressure close to 1 atm, where your wire will be sitting)?

• Sorry for the late response. Looking at the following NIST table: nist.gov/data/PDFfiles/jpcrd263.pdf the second page shows the values for C(p) and C(v) and they appear to average around a 3% difference at room temp (~300K).
– Eric
Apr 13, 2016 at 16:34
• Most tables seem to use constant pressure for copper, so I decided to just take those data points from ~200-500K from a few different tables and calculate a line of best fit. This seemed to give a pretty fair approximation in the ranges I'm concerned with.
– Eric
Apr 13, 2016 at 16:37