Question about Partial Differentials from my Thermo homework

[yecko]

Homework Statement

partial differential

Relevant Equations

chain rule / quotient rule

From the solution of my thermodynamics homework,

$$
({\frac{\partial h_{fg}/T}{\partial T}})_P \\ = ({\frac{\partial h_{g}/T}{\partial T}})_P - ({\frac{\partial h_{f}/T}{\partial T}})_P = \frac{1}{T} ({\frac{\partial h_{g}}{\partial T}})_P - \frac {h_g}{T^2} - \frac{1}{T} ({\frac{\partial h_{f}}{\partial T}})_P + \frac {h_f}{T^2}
$$

Isn't $({\frac{\partial h_{g}/T}{\partial T}})_P = \frac {h_g}{T^2}$?
And $({\frac{\partial h_{f}/T}{\partial T}})_P = \frac {h_f}{T^2}$

Why is there the other two parts: $\frac{1}{T} ({\frac{\partial h_{g}}{\partial T}})_P$ and $- \frac{1}{T} ({\frac{\partial h_{f}}{\partial T}})_P$ ?
Thank you

 

[George Jones]

I am a little confused by your notation. When you write, for example, $\left( \frac{\partial h_g /T}{\partial T} \right)_P$, do you mean $\left[ \frac{\partial}{\partial T} \left( h_g /T \right) \right]_P$?

Also, do $h_g$ and $h_f$ have $T$ dependence? If so, then

Why is there the other two parts: $\frac{1}{T} ({\frac{\partial h_{g}}{\partial T}})_P$ and $- \frac{1}{T} ({\frac{\partial h_{f}}{\partial T}})_P$ ?

come from the product rule for differentiation,

$$\left[ \frac{\partial}{\partial T} \left( h_g /T \right) \right]_P = h_g \left[ \frac{\partial}{\partial T} \left(\frac{1}{T} \right) \right]_P + \frac{1}{T} \left[ \frac{\partial h_g}{\partial T} \right]_P$$

 

[haruspex]

There seems to be some significance to the forms h_fg, h_f, h_g that needs to be explained to the reader.