Leaving this here for now... Will update with more information and references later.
Einstein and Debye showed that specific heat is a function of temperature, but is asymptotic at high* temperatures. Here is a simple explanation why:
Heat, with regard to everyday applications, is simply a measure of the motion of atoms and molecules. Let's start with gases. Classical theory tells us there are three types of motion for an atom: translational, rotational, and vibrational. Here's where quantum theory comes in. Quantum Mechanics tells us that there is a "temperature" threshold where each of these types of motion can begin to occur. We know from thermodynamics that there is a distribution of temperatures with regard to the atoms in a gas, so some atoms may have breached this temperature threshold and are in motion in more ways than others--let's call this having more degrees of freedom. This is where the temperature depends comes in. At low temperatures, there is a significant variance in degrees of freedom between atoms, and atoms continuously climb and fall below these thresholds. However, at high temperatures, most atoms have attained enough energy to gain all possible degrees of freedom. Therefore, there is little temperature dependence at high temperature for gases. In solids and liquids, magnetism must be considered, so they can exhibit different behavior at high temperatures.
The above discussion completely ignores pressure and volume, which are far more likely to have an impact on your calculations than temperature. Classical theory says polyatomic gases should have a constant specific heat at high temperatures, but I doubt it is that simple. I will do some reading on these issues and get back to you.
*Varies depending on the compound. Most gases have stable specific heats around room temperature, but behavior can vary wildly. See Einstein Temperature or Debye Temperature, specifically for Diamond.
You're right that the isentropic compressor work is the difference of enthalpy between the initial and final states (which are defined by the particular values of the state variables P, V, T).
And indeed, ΔH = Cp ΔT (assuming Cp does not depend much on T).
However, this formula is always valid for a perfect gas and does not assume that the pressure is constant. Ideed, Cp is called the heat capacity at constant pressure, but see it simply as a constant with a particular value. You do not necessarily need the transformation to be isobaric to use this constant at some point in your reasoning.
The "constant pressure" terminology just comes from the fact that for a perfect gas, for a transformation at constant pressure, the heat Q is equal to ΔH = Cp ΔT.
Of course, in the isentropic transformation, pressure is not constant, but the formula above is still correct.
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Not sure if this is correct. Reaction rate is temperature dependent. If you change constant volume to constant pressure, the predicted temperature will be much lower. So is the reaction rate.