3E1 :  Hypothetical Process Paths and the Latent Heat of Vaporization  7 pts 

Use the hypothetical process path (HPP) shown below to help you determine the DH in J/mole for propane (C_{3}H_{8}) as it changes from a subcooled liquid at P_{1} = 300 kPa and T_{1} = 250 K to a superheated vapor at P_{5} = 100 kPa and  
T_{5} = 300 K. Calculate the molar DH for each step in the HPP. Assume the propane vapor behaves as an ideal gas and a constant heat capacity of 69.0 J/moleK.  


Do not use tables of thermodynamic properties, except to check your answers. Use the Antoine and ClausiusClapeyron Equations to estimate the heat of vaporization of propane at T_{1}.  
Note: The molar volume of saturated liquid propane at 250 K is 7.8914 x 10^{5} m^{3}/mole.  
Read :  Step
12 is straightforward because we will assume that
the liquid propane is incompressible. We can use the Antoine Equation with the ClausiusClapeyron Equation to estimate DH_{vap} for step 23. Step 34 is easy because we were instructed to assume the propane is an ideal gas and the enthalpy of an ideal gas is not a function of pressure. Step 45 is straightforward because the problem instructs us to use a constant C_{p} value. 

Diagram:  The diagram in the problem statement is adequate.  
Given:  P_{1}  300  kPa  Find:  DH_{12}  ???  J/mol  
T_{1}  250  K  DH_{23}  ???  J/mol  
T_{5}  300  K  DH_{34}  ???  J/mol  
P_{5}  100  kPa  DH_{45}  ???  J/mol  
V_{liq}  7.8914E05  m^{3}/mole  DH_{15}  ???  J/mol  
C^{o}_{P}  69.0  J/moleK  
Assumptions:  1  ClausiusClapeyron applies:  
 The saturated vapor is an ideal gas  
 The molar volume of the saturated vapor is much, much greater than the molar volume of the saturated liquid.  
 The latent heat of vaporization is constant over the temperature range of interest.  
2  The superheated vapor also behaves as an ideal gas.  
3  Liquid propane is incompressible.  
Equations / Data / Solve:  
Step 12 involves a change in pressure on an incompressible liquid at constant temperature.  
Since neither the internal energy nor the molar volume of an incompressible liquid are functions of pressure :  

Eqn 1  
We can use the Antoine Equation to determine the vapor or saturation pressure of propane at T_{1}.  
Log_{10}(P*) = A  (B / (T + C))  Eqn 2  
P is in bar  T is in Kelvin  
The Antoine constants from the NIST WebBook are:  A =  4.53678  
B =  1149.36  
C =  24.906  
P_{2} = P*(T_{1})  Eqn 3  P_{2}  226.9  kPa  
Now, we can plug numbers into Eqn1, but be careful with the units.  
DH_{12}  5.768  J/mole  
Next, we can observe that DH_{23} = Latent Heat of Vaporization at 250 K.  
We can estimate the heat of vaporization using the Clausius Clapeyron Equation.  

Eqn 4  
If we plot Ln P* vs. 1/T(K), the slope is  DH_{vap}/R.  
We can calculate the vapor pressures at two different temperatures using the Antoine Equation. Use temperatures near the temperature of interest, 250 K. Use the two points to estimate the slope over this small range of temperatures.  

Eqn 5  
From the Antoine Equation:  
T_{a}  249.9  K  P_{a}  226.12  kPa  
T_{b}  250.1  K  P_{b}  227.71  kPa  
Slope =  2188.7  K  
Next we use this slope with Eqn 4 to determine the latent heat of vaporization at 250 K :  
R =  8.314  J/mol K  DH_{vap}  18197  J/mole  
DH_{23}  18,197  J/mole  
Next, we need to determine the enthalpy change from state 3 to 4, in which the pressure of the saturated vapor is reduced. This causes the vapor to become a superheated vapor.  
Recall the assumption that the vapor behaves as an ideal gas. Because enthalpy is only a function of T for ideal gases, and since T_{3} = T_{4} :  
DH_{34}  0  J/mole  
Next, let's consider the enthalpy change from state 4 to 5.  
Because we assumed the vapor phase is an ideal gas with constant C_{P}, we can evaluate DH using:  

Eqn 6  
Plugging numbers into Eqn 6 yields :  DH_{45} =  3,450  J/mole  
Finally, put them all together:  DH_{15} = DH_{12} + DH_{23} + DH_{34} + DH_{45} =  21,641  J/mole  
Notice that DH_{12} is very small compared to ΔH_{23} and ΔH_{45}. In fact ΔH_{12} is negligible.  
This shows why it is often acceptable to approximate the enthalpy of a subcooled liquid using the enthalpy of the saturated liquid at the same TEMPERATURE. It is NOT accurate to approximate the enthalpy of a subcooled liquid using the enthalpy of the saturated liquid at the same PRESSURE.  
Verify:  1  We can test the validity of the ideal gas assumption for state 3 as follows.  

V_{3}  9.160  L/mol  
Because V_{3} < 20 L/mole, the ClausiusClapeyron Equation is not very accurate. This issue makes the results from this analysis somewhat unreliable.  
It is not as easy to test the 2nd assumption that underpins the ClausiusClapeyron Equation.  
We can use the NIST Webbook to determine the molar volume of saturated liquid and saturated vapor at 250 K.  
V_{sat vap}  8.9258  L/mol  
V_{sat liq}  0.078977  L/mol  V_{sat vap} / V_{sat liq =}  113.02  
Since V_{sat vap} is more than 100 times greater than V_{sat liq} this assumption underpinning the use of the ClausiusClapeyron Equation is valid.  
Because we considered a very narrow temperature range, just 0.2°C, the last assumption underpinning the use of the ClausiusClapeyron Equation is almost certainly valid.  
2  Is the superheated vapor be accurately treated as an ideal gas?  

V_{3}  20.785  L/mol  
Because V_{4} > 20 L/mole, the ClausiusClapeyron Equation can be applied.  
3  Since ΔH_{12} is negligible, this assumption is not very important.  
Nonetheless, we can use the NIST Webbook to determine the molar volume of liquid at P_{1} = 100 kPa and at P_{2} = 226.9 kPa at 250 K. and see if the molar volume changes significantly.  
V_{1}  0.0625  L/mol  10.887059  V_{2}  0.0693  L/mol  
We find that V_{2} differs from V_{1} by about 11%. So it is not very accurate to treat liquid propane as an incompressible liquid under these conditions.  
This may be ok in this problem since ~ΔH_{12} is so small that an 11% error in its value will still not matter.  
Answers :  DH_{12}  5.77  J  
DH_{23}  18,197  J  
DH_{34}  0  J  
DH_{45}  3,450  J  
DH_{15}  21,600  J  (Rounded to 3 significant digits)  
The assumption that the saturated vapor can be accurately treated as an ideal gas is not valid and, as a result, ΔH_{23} and ΔH_{15} are not reliable. 