[Thermo] Derivation of compressibility factor vs reduced pressure

In summary, the conversation discusses the derivation of a compressibility factor graph using the Van der Waals equation of state and reduced pressure. The speaker mentions attempting to find a solution for 12 hours and wondering if it can be done without using values for critical pressure or temperature. The other person suggests using experimental data from gases such as propane and plotting the data to generate the graph. The conversation also briefly mentions the calculation of the critical temperature, pressure, and volume and solving the Van der Waals equation for the compressibility factor. Finally, the speaker asks if they are supposed to solve a specific equation for the cubic root.
  • #1
-Nita
2
0

Homework Statement


derivation of compressibility factor vs. reduced pressure
I am supposed to derive the graph by solving equations
Diagramma_generalizzato_fattore_di_compressibilit%C3%A0.jpg



Homework Equations


Van der Waals equation of state
compressibility factor, Z = (Pv)/(RT)
reduced pressure = P/critical pressure
Z = f(Tr, Pr)


The Attempt at a Solution


I sat for 12 hours attempting to find a solution but just spent time trying to understand what I was doing instead.
Is there a way to get the graph mathematically without using any values for critical pressure or temperature?

Thank you!
 
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  • #2
Compressibility charts are derived from experimental data from 10 gases such as propane, Nitrogen, Carbon Dioxide etc. Select a gas you have properties for such as propane. Tc propane is 370 K. Pc propane is 42.7 bar. Assume a constant temperature of let's say 555 K to generate the Tr=1.5 line. Vary pressure from 42.7 to 300 bars (Pr 1 to 6). Use propane tables to find v (specific volume) solve for Z. Plot Tr(Z,Pr)
 
  • #3
After solving the Van der Waals equation for the compressibility factor and
deriving the critical temperature, pressure, and volume

knowing that a horizontal inflection point occurs on the isotherm at critical point

am I supposed to solve PV3-(Pb+RT)V2+aV-ab=0
for the cubic root?
 

1. What is the compressibility factor in thermodynamics?

The compressibility factor, also known as the Z-factor, is a dimensionless quantity used in thermodynamics to measure the deviation of a real gas from an ideal gas. It is defined as the ratio of the gas's actual volume to its ideal volume at the same temperature and pressure.

2. How is the compressibility factor related to reduced pressure?

The compressibility factor is related to reduced pressure through the reduced pressure equation: Z = P/RT, where Z is the compressibility factor, P is the pressure, R is the gas constant, and T is the absolute temperature. This equation is used to calculate the compressibility factor for a given gas at a specific temperature and pressure.

3. What is the significance of the compressibility factor in thermodynamic calculations?

The compressibility factor is important in thermodynamic calculations because it provides a measure of how real gases deviate from ideal gases. It allows for more accurate predictions of gas behavior at different temperatures and pressures, which is essential for many industrial and scientific applications.

4. How is the compressibility factor vs reduced pressure curve derived?

The compressibility factor vs reduced pressure curve is derived by plotting the compressibility factor Z against the reduced pressure Pr (Pr = P/Pc, where Pc is the critical pressure of the gas). This curve can be obtained experimentally or calculated using various thermodynamic equations, such as the van der Waals equation of state.

5. What does the shape of the compressibility factor vs reduced pressure curve indicate?

The shape of the compressibility factor vs reduced pressure curve indicates the behavior of a gas under different conditions. For ideal gases, the curve is a horizontal line at Z=1, indicating that the compressibility factor does not change with pressure. However, for real gases, the curve is not a straight line and varies depending on the gas's properties and the temperature and pressure it is subjected to.

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