Why can we differentiate this entropy total derivative with repect to Temperature?

[zenterix]

Homework Statement

I am going through a thermodynamics course and there is a section about natural variables.

Relevant Equations

In that section, there is the following derivation

Ignoring chemical potential for now, the natural variables of $U$ are $S$ and $V$. Thus

$$dU=\left (\frac{\partial U}{\partial S}\right )_VdS+\left (\frac{\partial U}{\partial V}\right )_SdV=TdS-pdV\tag{1}$$

which we can rewrite for $dS$ as

$$dS=\frac{dU}{dT}+\frac{pdV}{T}\tag{2}$$

We wish to determine how the entropy depends on temperature.

We can differentiate the expression for $dS$ with respect to temperature at constant volume to obtain $\left (\frac{\partial S}{\partial T}\right )_V$.

Apply the derivative to obtain

$$\left (\frac{\partial S}{\partial T}\right )_V=\frac{1}{T}\left (\frac{\partial U}{\partial T}\right )_V+\frac{p}{T}\left (\frac{\partial V}{\partial T}\right )_V$$

$$=\frac{1}{T}\left (\frac{\partial U}{\partial T}\right )_V=\frac{C_V}{T}$$

Mathematically, what is happening exactly?

In other words, why/how does $dS$ become $\left (\frac{\partial S}{\partial T}\right )_V$?

Here is my current understanding.

$dS$ is the total derivative function for $S(T,V)$.

$dS$ is a linear function of $dT$ and $dV$ and this linear function is a function of $T$ and $V$.

That is, at each $T$ and $V$ we have a different function $dS$.

If we differentiate $dS$ relative to $T$ we are determining the rate of change of $dS$ relative to $T$ at fixed $V$.

Notation-wise, it seems we would have $\frac{\partial (dS)}{\partial T}$, which is a notation I have never seen before.

If this is correct so far, then it seems that $\frac{\partial (dS)}{\partial T}$ is written as just $\left (\frac{\partial S}{\partial T}\right )_V$.

But this latter step isn't very clear to me.

So my question is essentially how to understand what is going on from a mathematical perspective in this derivation.

 

[DrDu]

No, don't differentiate dS, rather use the differential of S with variables T and V, $dS(T,V)=\left(\partial S/\partial T\right)_V dT + \left (\partial S /\partial V\right)_T dV$ and a similar expression for dU(T,V).