Derivatives are a common math subject in science and engineering but it is perhaps in thermodynamics where these powerful tools find some application. However, in thermodynamics one needs to go a little deeper than single variable derivatives.
State functions, for example, involve dependence on more than one variable. This in turn, would mean that partial and total derivatives would be needed. Here, a brief explanation of these two types of derivatives shall be considered.
Two features defining partial and total derivatives
Margenau and Murphy, in their book Mathematics of physics and chemistry, state that there are in fact three main features that characterize two further types of derivatives (apart from the single variable derivatives). For a function $f(x,y,z)$ these features can be stated, in a simplified way, as follows,
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| Fig. 1 Common types of derivatives found in thermodynamics |
Feature 1.- Only one independent variable vary
- $x$ vary while $y$ and $z$ remain constant,
- $y$ vary while $x$ and $z$ remain constant,
- $z$ vary while $x$ and $y$ remain constant.
Feature 2.- All independent variables vary
This feature would mean that all: $x$, $y$ and $z$, vary and none remain constant. This feature is to be expected in a total derivative. In practice, a total derivative is in fact the result of contributions made of partial derivatives.
How does a partial derivative looks like
The operator of these type of derivatives is usually represented with: $\partial$.
Considering the case of the same function as before: $f(x,y,z)$. The following partial derivatives can be written,
- $\dfrac{\partial f}{\partial x}$. This reads: partial derivative of $f$ with respect to $x$. Remember that all other independent variables remain constant. Another way of saying the same would be: $\left( \dfrac{\partial f}{\partial x}\right)_{yz}$.
- $\dfrac{\partial f}{\partial yx}$. This reads: partial derivative of $f$ with respect to $y$. Remember that all other independent variables remain constant. Another way of saying the same would be: $\left( \dfrac{\partial f}{\partial y}\right)_{xz}$.
- $\dfrac{\partial f}{\partial z}$. This reads: partial derivative of $f$ with respect to $z$. Remember that all other independent variables remain constant. Another way of saying the same would be: $\left( \dfrac{\partial f}{\partial z}\right)_{xy}$.
In the above expressions the subscripts outside the parenthesis indicate the independent variables that are kept constant during the derivation indicated inside the parenthesis.
How do you write a total derivative
Since in a total derivative all independent variables may change at once, writing one single operation, to express this, can be challenging. However, this can be done as the summation of all the contributions of variations of all independent variables, one by one.
Let us now consider a thermodynamical variable such as the volume $V$ which is a function of the pressure and temperature $T$, according to some state equations, as that for ideal gases for example. In other words,
$V(P,T)$
or
$V=\mathbf{fn}(P,T)$
where $\mathbf{fn}$ means function.
What would be the total derivative of $V$?
Let us first write down, all, the partial derivatives of $V$,
The using the expressions in Eq. (01) we may write the total derivative of $V$ as follows
where as you may see that variations of all independent variables has been included. Writing a total derivative may seem mechanical but two further special cases may arise.
How would you write a total derivative of $V$ if the independent variables $P$ and $T$ depend on another independent variable $u$?
In other words, how would you rewrite Eq. (02) for $P=\mathbf{fn}(u)$ and $T=\mathbf{fn}(u)$?
Let us not worry about the meaning of $u$ and just accept that it is another physical property or parameter of the system capable of making $P$ and $T$ change.
Since $P$ depends solely on $u$ any derivative related to it would be the traditional single variable derivative. The same applies for $T$.
Also, since $P$ and $T$ depend on $u$ so does $V$. This implies that variations with respect to $u$ must be considered not only for the derivatives of $P$ and $T$ but for $V$ too.
With this in mind, Eq. (02) can be rewritten as,
How would you write the total derivative of $V$ if only one of the independent variables, say $P$, depend on another independent variable $u$?
The simplicity of Eq. (04) could be striking but you should be aware that this equation states the total derivative of $V$ with respecto to $u$. Of course, one way of keeping the changes on $T$ is already given in Eq. (02).
How would you write the total derivative of $V$ if one of the independent variables depend on the other, say $P=\mathbf{fn}(T)$?
where as you can easily see a simplification can be made because,
Any question? Write in the comments and I shall try to help.
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