# Thermodynamics Questions

1. Oct 7, 2012

### cmmcnamara

Hey all, I have a question that arose out of some thermodynamics review I'm going over for an upcoming course. It relates to different properties surrounding ideal gases. There are some other various ideas too.

Firstly, I want to talk about specific heat and their relations between other gas properties as this is driving me nuts on some of the variable specific heat problems.

I know that based off experimentation that u=u(t) and h=h(t) so that when we take the partial derivatives of these functions we get:

$$\left(\frac{\partial u}{\partial T}\right)_V=\left(\frac{\partial Q}{\partial T}\right)_V=c_v$$

$$\left(\frac{\partial u}{\partial T}\right)_P=\left(\frac{\partial Q}{\partial T}\right)_P=c_p$$

This also shows that the different specific heats are variable with temperature. My book also lists that

$$R=c_p-c_v$$

which seems to me that R=R(t).

When I work on problems with variable specific heat I realize that usually the problem requires that tabulated data is used for the cyclic integral of $\frac{\partial Q}{\partial T}$ because it puts the entropy on an absolute scale and allows relative pressure/volume ratios to be used to find other data about the problem. However the problem I'm having with the book is that when solving they don't seem to use a "variable" gas constant? The difference between solutions is usually negligible (1%-5% error) but it is making me wonder:

Is R actually R=R(T)? Or is R actually a constant? That the difference between $c_p$ and $c_v$ is actually always constant? I cannot seem to tell if this is true or not simply based off available tabulated data. I made an excel sheet from tabulated variations in each type of specific heat but the difference for each substance always seemed to be one of two values both within .001 difference between one another. Therefore I can't really tell if R=R(t) where $\frac{dR}{dT}<<1$ or if $R=c_p-c_v=constant$ , $\forall T$.

2. Oct 7, 2012