# EoS: Departure Functions

### Departure Functions

In this section, we learned how to handle $PvT$ data for non-ideal-gas materials; however, the only way that we have handled non-measurable properties for such materials was to use the steam tables (which, obviously, only works for water).

One conceptually satisfying way to alleviate that shortcoming is to exploit the fact that we need to choose a reference state when calculating fundamental or derived properties. In this approach, we choose as our reference state, the material of interest in its "ideal gas state".

In this way, if we could calculate the difference between the material in its "ideal gas state" versus in its "real state", we could make use of the ideal-gas relations that we have thus far used.

##### Definition:

Departure functions (here denoted as $M^{dep}$ for a generic property, $M$) are defined as the difference between the property of interest in its real state (designated here as $M$) and its ideal gas state (designated here $M^{ig}$):
$M^{dep} = M - M^{ig}$

In this way, we can write the actual value of a property as:

$M = M^{ig}+M^{dep}$

Using this approach, we can then express our differences in any state function (for example entropy) between states 1 and 2 as:

$s_2 - s_1 = (s_2^{ig} + s_2^{dep}) - (s_1^{ig} + s_1^{dep})$

$s_2 - s_1 = (s_2^{dep} - s_1^{dep}) + (s_2^{ig} -s_1^{ig}) = (s_2^{dep} - s_1^{dep}) + \Delta s_{1 \to 2}^{ig}$

We have already discussed $\Delta s^{ig}$, so we can replace that to get:

##### IMPORTANT:

$s_2 - s_1 = (s_2^{dep} - s_1^{dep}) + \int_{T_1}^{T_2} \frac{c_P dT}{T} - R \ln \frac{P_2}{P_1}$

##### Note:

The way that we "convert" a real fluid to an ideal gas is to drop the pressure fro $P$ down to $P=0$. So that (as can be seen in our schematic), we can get the departure function at $T_i$ and $P_i$ by going from ($T_i,P_i$) to ($T_i, P=0$ -- ideal gas) to ($T_i, P=P_i$ -- ideal gas). Doing this for both $T_1$ and $T_2$ allows us to calculate $s_2^{dep} - s_1^{dep}$.

We note that our analysis of the fundamental property relations told us the exact differential for $s$ at constant $T$:

$(ds)_T = - \left (\frac{\partial v}{\partial T} \right )_P dP$

So that our steps for calculating $s^{dep}$ yield:

$s_i^{dep} = \int_0^{P_i} - \left (\frac{\partial v}{\partial T} \right )_P dP - \left [ \int_0^{P_i} - \left (\frac{\partial v}{\partial T} \right )_P dP \right ]^{ig}$

$s_i^{dep} = \int_0^{P_i} - \left (\frac{\partial v}{\partial T} \right )_P dP + \int_0^{P_i} \frac{R dP}{P}$

A similar analysis for $h$ gives:

##### IMPORTANT:

$h_2 - h_1 = (h_2^{dep} - h_1^{dep}) + \int_{T_1}^{T_2} c_P dT$

where $h_i^{dep}$ can be calculated from (noting that $h^{ig}$ is independent of $P$):

$h_i^{dep} = \int_0^{P_i} \left [- T\left (\frac{\partial v}{\partial T} \right )_P + v \right ]dP$

##### Note:

The easiest way to calculate $s_i^{dep}$ or $h_i^{dep}$ is to look them up on Lee-Kesler tables in the same way that we did for $z$ values!

##### Outcome:

Use equations of state to calculate departure functions

##### Outcome:

Calculate departure functions from Lee-Kesler charts