Calculation of $C_v$ and $C_p$

Question

In book "Element of Gasdynamics" (authors Liepmann and Roshko) at paragraph "1.7 The First Law Apllied to Reversible Process. specific Heats." There is written (specifying that I'm going to report only some formulas and not the entire test between each formula and other) :

$$\mathrm{d}e=\mathrm{d}q-p\cdot \mathrm{d}v \quad(1.15) $$

$$c=dq/dt \quad(1.16)$$

$$c_v=(dq/dT)_v \quad(1.17a)$$

$$c_p=(dq/dT)_p \quad(1.17b)$$

$$\mathrm{d}e=\frac{\partial(e)}{\partial(v)}\cdot dv+\frac{\partial(e)}{\partial(T)}\cdot dT=dq-p\cdot dv$$

$$c_v=\frac{\partial(e)}{\partial(T)} \quad(1.18)$$

$$c_p=\frac{\partial(e)}{\partial(T)}+(\frac{\partial(e)}{\partial(v)}+p)\cdot(\frac{\partial(v)}{\partial(T)})_p \quad(1.19)$$

Why in equation (1.18) is not indicated that $\frac{\partial(e)}{\partial(T)}$ is keeping "$v$" constant? Why in equation (1.19) is not indicated that $\frac{\partial(e)}{\partial(T)}$ is keeping "$p$" constant?

Answer 1

This notation is really bad but seems mathematically correct.

$e$ is defined as only dependent of $v$ and $T$. As such, eq (1.15) gives the total derivative of $e$ regarding these variables.

Eq (1.19) takes for granted that $v$ does not vary and as such equates the PARTIAL derivative $\frac{\partial e}{\partial T}$ with $c_v$. A better notation would be $\left.\frac{\mathrm{d}e}{\mathrm{d}T}\right|_v=c_v$ precising that your equating the total variation of energy relative to any variation of temperature without volume change with $c_v$.

The partial derivative notation just says that for any transformation, you're equating a not measurable part of the change in energy (the part related to temperature change) to $c_v$. Both are correct but one is obviously less understandable at first sight.

In the same spirit you don't have to precise that $\frac{\partial e}{\partial T}$ is taken with $p$ constant because $\mathrm{d}e$ does not depend on $p$.

Answer 2

About first question: "Why in equation (1.18) is not indicated that ∂(e)/∂(T) is keeping "v" constant?" I can answer that the author writes an important note before the formula de=∂(e)/∂(v)⋅dv+∂(e)/∂(T)⋅dT=dq−p⋅dv. in the note it's written: "It is customary in thermodynamics to indicate the variables which are kept constant in forming a partial derivative by a subscript, e.g., $$(∂e/∂v)_T $$ means that T is kept constant. We shall use this notation only where there could be any doubt about the variable that is considered constant. " So it's considered implicit in equation (1.18) that $$(∂e/∂v)$$ is keeping v constant. About second question: "Why in equation (1.19) is not indicated that ∂(e)/∂(T) is keeping "p" constant?" the objection (my objection !) is wrong because the addend should be $$(\frac{\partial(e)}{\partial(T)})_v\cdot(\frac{\partial(T)}{\partial(T)})_p $$, so the book considers, without any doubt about the variable, that $$(\frac{\partial(e)}{\partial(T)})_v$$ can be written $$\frac{\partial(e)}{\partial(T)}$$ and $$(\frac{\partial(T)}{\partial(T)})_p$$ is equal to 1 in any case.

Answer 3

What literary translation is given to the notation : $c_{v}$ and $c_{p}$?

c: specific heat capacity.

v, p index : at constant volume, pressure.

What is the use of adding the same symbol in the right member of the equation!.

PS: when you 'pour' physical or mathematical ideas into a 'symbolic mold', it must express (contain) all the ideas, a good nation facilitates the comprehension and avoids the useless clutter what makes understand Alexander Grothendieck to its students from the beginning .