Error using constant specific heat

In the last post we looked at picking a value to use for c_p when we make the constant specific heat approximation.  In this post, we’ll look more closely at the error involved in that approximation in order to have a feel for when it is appropriate, or not, to use it.
This figure shows the variation in c_p with temperature for air. 

Recall that in the constant specific heat assumption, we ignore this variation, and assume that a single value of c_p is “good enough” over the full range of temperature.  First, let’s look at the so-called “cold gas” approximation which just consists of using c_p at a single temperature, regardless of the temperature range involved.

The figure below shows the percent error in Δh assuming that the specific heat is constant at its value at 300 K, and T₁ is constant at 220K and  the following definitions are used for Δh:

As would be expected, the error increases as the temperature range increases.  Notice that even over a temperature range of over 2000K, the error is only about 13%, which may be good enough for some applications.  This applies only to air.
Now, let’s look at the error associated with using either the trapezoidal rule, or c_p at the exact center of the temperature range, as shown here:

This figure shows the percent error for these two approaches:

Even over a span of 2000K, the maximum error in either method is only about 2%.  Again, this applies only to air.

Finally,  we will look at the error associated with smaller spans of temperature.  In this figure, each colored line represents the error associated with a  different temperature span: 100K, 200K, or 300K, plotted as a function of the center point of the span.

As might be expected, the error is very small for the smallest (100K) span, and somewhat higher for the others.
All of these results were only for air. 
This figure shows c_p for CO₂. Notice that the range of c_p is larger than for air.

These figures show the exact same error results for CO₂ that were presented above for air.
Cold gas:

 Trapezoidal, and exact center:

Smaller ranges:

The trends for CO₂ are similar to those for air, but the numbers are considerably different.