Geometric and mathematical interpretation about a derivative of $H$ and related questions

Question

I would like to check whether my analogy is right and ask some related questions.

So, I am studying Thermodynamics right now and there are many calculus techniques are used for the thermodynamic relations between various state functions.

One example is

$ dH = \left(\frac{\partial{H}}{\partial{T}}\right)_{P}dT + \left(\frac{\partial{H}} {\partial{P}}\right)_{T}dP$

It is a total differential of enthalpy when considering $H(P,T)$.

By dividing ${dT}$, a derivative for $H$ can be earned like

$ \left(\frac{\partial{H}}{\partial{T}}\right)_{V} = \left(\frac{\partial{H}}{\partial{T}}\right)_{P} + \left(\frac{\partial{H}}{\partial{P}}\right)_{T}\left(\frac{\partial{P}}{\partial{T}}\right)_{V}$

For me, mathematically this process is clear and understandable.

For some 3-D space with $(H, P, T)$, a well-defined function $H(P,T)$ is a surface and it's total differential $dH$ is a natural change of $H$ earned by an infinitesimal change of $(dT, dP)$ and a gradient of $\left( \left(\frac{\partial{H}}{\partial{T}}\right)_{P}, \left(\frac{\partial{H}}{\partial{P}}\right)_{T}\right)$ at a specific point.

And $\left(\frac{\partial{H}}{\partial{T}}\right)_{V}$ is a directional derivative with a constraint $(V = \textrm{constant})$ therefore satisfying some relationship between $T$ and $P$ expressed by $\left(\frac{\partial{P}}{\partial{T}}\right)_{V}$ (and such direction is along a line also satisfying that constraint.).

So I would like to check that my mathematical analogy is correct.

Also, I really want to know

First, is there any way to explain physically and intuitively for each term of $\left(\frac{\partial{H}}{\partial{T}}\right)_{V}$?

Second, unlike typical geometry dimensions like $(x,y,z)$, definitely, thermodynamic variables like $(P,T)$ are inter-connected, I mean for me they are not independent like $(x, y, z)$. Then why such thermodynamics variables considered mathematically independent? Can I understand this because they are not truly independent so $H(P,T)$ only spans over a limited space (as like a 2-D surface)?

Answer 1

"Dividing by $dT$" isn't really a well-defined operation, especially when you "divide" $dH$ by $dT$ and end up with $(\partial H/\partial T)_V$. If you want to arrive at this expression, it's best to start with the definition of $H$, instead of $dH$. The enthalpy $H$ is derived from the internal energy $U$ via a Legendre transform:

$$H=U+PV$$

From here, you can differentiate directly:

$$\left(\frac{\partial H}{\partial T}\right)_V=\left(\frac{\partial U}{\partial T}\right)_V+V\left(\frac{\partial P}{\partial T}\right)_V$$

One of these quantities is well-known: $(\partial U/\partial T)_V=C_V$, the heat capacity at constant volume. Likewise, the other derivative corresponds to a physical quantity, namely: $\left(\partial P/\partial T\right)_V=\alpha/\kappa_T$, where $\alpha$ is the coefficient of thermal expansion and $\kappa_T$ is the bulk compressibility at constant temperature. So, we have:

$$\left(\frac{\partial H}{\partial T}\right)_V=C_V+V\frac{\alpha}{\kappa_T}$$

The values of these three coefficients are determined by the specific relation between volume, pressure, and temperature for this system; in other words, they are specified by the equation of state for your system.

Answer 2

And ($\frac{∂H}{∂T})_V $is a directional derivative with a constraint (V=constant) therefore satisfying some relationship between T and P expressed by $ (\frac{∂P}{∂T})_V$

Directional derivative is something different.. What you have here is a partial derivative. Consider the gradient vector of $H$ and dot that with a displacement vector of state variable to get the change in $H$ as you move from one point to another on the surface.

$$ dH(T,P) = \nabla H \cdot (dT, dP) = \frac{\partial H}{\partial T} dT + \frac{\partial H}{\partial P} dP$$

Now the above quantity is what is known as the differential or directional derivative(Ref)

(and such direction is along a line also satisfying that constraint.).

Not sure what you mean here.

Second, unlike typical geometry dimensions like (x,y,z), definitely, thermodynamic variables like (P,T) are inter-connected, I mean for me they are not independent like (x,y,z). Then why such thermodynamics variables considered mathematically independent? Can I understand this because they are not truly independent so H(P,T) only spans over a limited space (as like a 2-D surface)?

For a single substance homogenous system, then you can change $P$ and $T$ independently. Basically you can vary two out of three basic state variables at a time whilst the other is held constant. If you are varying pressure with temperature constant that means the volume is changing.

Given all of that, you could even write enthalpy as a function of temperature and volume if you wanted... though those variables are 'unnatural'

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More in-depth discussion of differentials:

https://www.youtube.com/watch?v=XGL-vpk-8dU&ab_channel=eigenchris

A post in which I wrote an answer about geometric nature of partials:

Explanation on natural variables concept