## Problems and Solutions

Chapter 2

PvT Behavior of Pure Components

### Textbook Examples:

XPS-file display is available in Internet Explorer. In case of Firefox, select Internet Explorer when asked for the software to open the file.

02.05
**
Amount of Ethylene in a Bottle if Given Volume at T, P Using a High Precision Equation of State.
**

02.06
**
Isobaric Heating of Ethylene Using a High Precision Equation of State.
**

(the link to the reference
"02.EOS-ethylene.mcd" has to be deleted and added new via Insert -
Reference !)

02.07
**
Zeno line, Joule-Thomson Inversion Curve and Boyle Curve Using a High Precision Equation of State.
**

Missing due to convergence problems that need to be solved.

02.08
**
Pressure of Steam in a Vessel from Different Equations of State.
**

Part a-c

Part d

02.09
**
Density of Propylene from the Peng-Robinson Equation of State.
**

### Additional Problems:

P02.01
**Compressibility Factor and Molar Volume of Methanol Steam
**

Calculate the compressibility factor and the molar volume of
methanol steam at 200°C and 10 bar

a) using the ideal gas law

b) with the virial equation truncated after the 3^{rd}
virial coefficient

(virial coefficients: B=-219 cm^{3}/mol; C=-17,300 cm^{6}/mol^{2})

P02.02 ** Pressure
and Compressibility Factor of Ethylene Using the Virial and the
Peng-Robinson Equation of State**

A container with a volume of V=0.1 m^{3} is filled with m
= 10 kg ethylene at a temperature of T = 300 K. What will be the
pressure and compressibility factor of the gas? Use the virial equation
with only two coefficients and the Peng-Robinson equation of state to
describe the PVT-behavior. (virial coefficient: B=-138 cm^{3}/mol,
all other properties are given in Appendix A)

Mathcad (2001) - Solution (zip)

Mathcad (2001) - Solution as XPS

P02.03 ** Calculation
of the 2 ^{nd} Virial Coefficient from the VdW EOS**

Derive an expression for the 2

^{nd}virial coefficient based on the van der Waals equation of state.

Mathcad (2001) - Solution (zip)

Mathcad (2001) - Solution as XPS

P02.04 **Volume
Dependence of the Internal Energy from the VdW EOS**

Derive an expression for the volume dependence of the internal
energy U at constant temperature based on the van der Waals equation of
state.

Mathcad (2001) - Solution as XPS

P02.05 **Pressure
of CO _{2} from Ideal Gas Law,
Virial Equation and Redlich-Kwong EOS**

Calculate the pressure of 1 mol CO

_{2}in a container of 2.5 dm

^{3}at 40°C via the

a) ideal gas law

b) virial equation

c) Redlich-Kwong-equation

(virial coefficient: B=-110 cm

^{3}/mol; for all other required properties see Appendix A)

Mathcad (2001) - Solution as XPS

P02.06
** Derivation of Residual Functions Using the Redlich-Kwong
EOS **

Derive the expressions for the residual functions (h-h^{id}),
(s-s^{id}), (g-g^{id}) using the Redlich-Kwong equation
of state.

Mathcad (2001) - Solution as XPS

P02.07 ** Change
in Enthalpy During Isothermal Compression**

Calculate the change in enthalpy of 1 mol of liquid ethanol
during isothermal compression from 1 bar to 100 bar at a temperature of
25°C. The compressibility coefficient (1.14·10^{-4}
bar^{-1}), the thermal expansion coefficient (1.1·10^{-3}
K^{-1}) and the molar volume (58.04 cm^{3}/mol) are
regarded as constant.

Mathcad (2001) - Solution as XPS

P02.08 ** Virial
Coefficient Data Via the DDB**

Search for experimental 2^{nd} virial coefficient data
for nitrogen in the DDB. Compare the values to the estimation results
from the Tsonopoulos method via DDBSP-Predict. Estimate the Boyle
temperature from the experimental findings.

DDB Explorer Version video
(large),
(medium),
(small)

P02.09 **VBA
Program to Calculate Vapor Pressure Curve, Comparison to DDB Data**

Retrieve the vapor pressure and the liquid density data for
methane from the DDBSP Explorer Version and export the values to Excel.
Implement a liquid vapor pressure curve calculation for the van der
Waals equation of state in Excel-VBA and compare the results along the
vapor-liquid coexistence curve to the experimental data.

Step 1: DDB Explorer Version - Data retrieval
and Export (large), (medium), (small)

Step 2: Excel document containing data and VBA code
planned for June 2012

P02.10 E**stimation
of the Azentric Factor from Critical Data and Normal Boiling Temperature**

Estimate the acentric factor of methane, propane, pentane and
heptane using the critical data and normal boiling temperatures given in
Appendix A and discuss the results.

Mathcad (2001) - Solution as XPS

P02.11 **Vapor
Pressure Calculation and Slope of the Vapor Pressure Curve via SRK**

Calculate the vapor pressure of benzene between 280 and 540 K
using the Soave-Redlich-Kwong equation of state with critical data and
acentric factor given in Appendix A. Compare the slope of the vapor
pressure curve in the log(P) vs. 1/T diagram with the slope calculated
via the vapor pressure correlation also given in Appendix A.

Mathcad (2001) - Solution as XPS

P02.12 **Cooling
Duty for a Gaseous Propylene Stream Using Peng-Robinson**

In a heat exchanger, gaseous propylene is cooled down from
J_{1}
= 90°C, P_{1} = 20 bar to
J_{2}=60°C.
The pressure drop across the heat exchanger is
DP
= 2 bar. How much cooling water is necessary? The supply and return
temperatures of the cooling water are
J_{CWS}
= 30°C and
J_{CWR}
= 40°C, respectively. Use the Peng-Robinson equation of state for
propylene and the function given for c_{P}^{L} in
Appendix A.

Mathcad (2001) - Solution as XPS

P02.13 **Change
of Pressure when Heating Liquid Water in a Constant Volume Using a High
Precision EOS**

A closed vessel filled completely with liquid water at
J_{1
}= 25°C, P_{1} = 1 bar. Due to solar radiation, it is
heated up to
J_{2}=60°C.
What pressure P_{2} is built up? Use a high-precision equation
of state. Calculate the transferred heat.

Mathcad (2001) - Solution as XPS

P02.14 **Ideal
Gas Enthalpy Calculation for Temperature and Pressure Change**

An ideal gas is heated up from T_{1} to T_{2} in
a heat exchanger. The pressure drop is
DP
= P_{2}-P_{1} > 0. Why can the duty be calculated with
the isobaric heat capacity by q_{12} = c_{P}^{id}
×(T_{2}-T_{1)},
although a pressure drop occurs?

P02.15 **Relationship
to Calculate the Heat Capacity at Constant Pressure from the Second
Derivative of the Gibbs Energy with Temperature **

Show that the relationship

P02.16 ** Heat
Duty for Isobaric and Isochoric Heating of a Gas Using a High Precision
Equation of State**

A vessel is filled with nitrogen at
J_{1
}= 20°C and P_{1} = 1 bar. With the help of a high
precision equation of state, calculate the duty

a) if the drum is
heated up isobarically to
J_{2
}= 100°C.

b) if the drum is
heated up isochorically to
J_{2
}= 100°C.

Interpret the results.

Mathcad (2001) - Solution as XPS

P02.17 ** Relationship
to Calculate the Entropy from the Derivative of the Gibbs Energy with
Temperature at Constant Pressure**

Show that the relationship

P02.18 ** Vapor
Pressure of Acetone from 7 Different EOS**

Calculate the vapor pressure of acetone at T_{1} = 260 K,
T_{2} = 360 K and T_{3} = 450 K using

a) the vapor
pressure equation listed in Appendix A

b) a high-precision
equation of state

c) the Peng-Robinson
equation of state

d) the
Redlich-Kwong-Soave equation of state

e) the Redlich-Kwong
equation of state

f) the PSRK equation
of state

g) the VTPR equation
of state

What are the conclusions of the results ?

(All required parameters can be found in the appendix.)

Mathcad (2001) - Solution (zip)

Mathcad (2001) - Solution as XPS

P02.19 **Density
of Liquid Methanol from 7 Different EOS**

Calculate the saturated density of liquid methanol at T_{1}
= 300 K and T_{2} = 430 K using

a) the density
equation listed in Appendix A

b) a high-precision
equation of state

c) the Peng-Robinson
equation of state

d) the
Redlich-Kwong-Soave equation of state

e) the Redlich-Kwong
equation of state

f) the PSRK equation
of state

g) the VTPR equation
of state

What are the conclusions of the results ?

(All required parameters can be found in the appendix.)

Mathcad (2001) - Solution (zip) (planned for June 2012

Mathcad (2001) - Solution as XPS

P02.20 ** Compressibility
Factor of Gaseous Propylene Via Tsonopoulos, Peng-Robinson and a High
Precision Equation of State **

Calculate the compressibility factor z of gaseous propylene at P_{1}
= 2 bar and P_{2} = 10 bar at T = 293.15 K, using the
Tsonopoulos and the Peng-Robinson equations of state. Check the results
with a high-precision equation of state. All required parameters can be
found in the appendix.

Mathcad (2001) - Solution (zip)

Mathcad (2001) - Solution as XPS

P02.21 ** Required
Input Parameters for the Calculation of Saturated Vapor Enthalpy**

Make a list of all the input parameters necessary for the
calculation of the enthalpy of the saturated vapor of a pure substance
if

a) the Peng-Robinson
equation of state

b) the vapor
pressure equation in combination with the Peng-Robinson equation of
state

c) the VTPR equation
of state

is used. The vapor pressure itself shall not be an input parameter.

P02.22 **Saturated
Vapor Fugacity, Vapor and Liquid Volumes and Heat of Vaporization Using
Soave-Redlich-Kwong**

Calculate

a) the fugacity f^{s}
at the saturation state

b) the molar volumes
v^{L} and v^{V} in the saturation state

c) the enthalpy of
vaporization

for n-butane, benzene and water at
J_{1}=30°C,
J_{2}=80°C
and
J_{3}=130°C
using the Soave-Redlich-Kwong equation of state. All required parameters
can be found in the Appendix.

Mathcad (2001) - Solution as XPS

P02.23 **Standard
Gibbs Energy of Formation for the Liquid Phase from the Value for the
Ideal Gas State at 1 atm**

In Appendix A, the Standard Gibbs energy of formation at
J=25°C
and P = 1 atm is reported to be

Mathcad (2001) - Solution as XPS