Which cp to use?

In an earlier post we talked about the variation of specific heat with temperature.  Today we’ll explore that variation in more depth and consider the choice of temperature at which to evaluate specific heat.
In that earlier post, we considered the variation of the specific heat ratio, c_p/c_v with temperature.  This time, we’ll look exclusively at c_p, although we could do the exact same thing for c_v.  For an ideal gas, the constant pressure specific heat, c_p can be expressed as:

where “h” is the enthalpy.  Or we can write:

Graphically, this looks like this:

Now, as an approximation, it is common practice to assume that the variation of specific heat with temperature is slight enough that over small temperature ranges it can be treated as a constant which makes the integral really easy to evaluate:

This immediately raises the question of which temperature to use to evaluate c_p.  If we use c_p(T₁), then we miss the area in red for our estimation of Δh:

On the other hand, if we use c_p(T₂), then we overestimate Δh:

So, it is pretty obvious that the best choice is to pick some temperature in-between the two extremes, call it T_cent.  Now, the two triangular-shaped areas tend to cancel each other out, mostly.

Now, as a pragmatic matter, there isn’t any point in spending a lot of time getting to the exact center for T_cent:

because the function c_p(T) is non-linear, anyway.  So, for most practical applications which use the “constant specific heat” approximation, it is sufficient to pick a value for c_p somewhere near the center of the T₁ to T₂ range, and proceed happily.