Equations of State
Chapter 2
Advanced Thermodynamics (M2794.007900)
Min Soo Kim
Seoul National University
2.2 Equation of State of an Ideal Gas
(2.1) m: mass of gas
M: molecular weight R: universal constant
(
(2.2)
• Since is the number of kilomoles of the gas, the equation of state of an ideal gas is
2.2 Equation of State of an Ideal Gas
• In Equation (2.2) we note that the extensive variable V divided by n, the number of kilomoles of the gas, is the specific volume .
• Thus the equation of state can be written
• The projections of the surface on the plane, plane, and the plane are shown in Figure 2.1
Figure 2.1 Diagrams for an ideal gas. (a) the isotherms are equilateral hyperbolae; (b) the isochores are straight lines; (c) the isobars are also straight lines.
2.3 Van Der Waals’ Equation for a Real Gas
𝟐 (2.3)
a and b: characteristic constants
• The term arises from the intermolecular forces due to the overlap of electron clouds.
• The constant b takes into account the finite volume occupied by the molecules.
• Multiplication of Equation (2.3) by yields the equation (2.4)
• Equation (2.4) is a cubic equation in with three roots, only one of which needs to be real.
• In Figure 2.2 some isotherms calculated from the van der Waals equation have been drawn.
Figure 2.2 Isotherms for a Van Der Waals’ gas.
2.3 Van Der Waals’ Equation for a Real Gas
2.4 Surfaces for Real Substances
• Figure 2.3 is a schematic diagram of the surface for a substance that contracts on freezing
Figure 2.3 𝑃 − 𝑣 − 𝑇 surface for a substance that contracts on freezing
2.4 Surfaces for Real Substances
• Notice the regions(solid, liquid, gas or vapor) in which the substance can exist in a single phase only.
• Elsewhere two phases can exist simultaneously in equilibrium, and along the so-called triple line, all three phases can coexist.
Figure 2.4 𝑃 − 𝑇 diagrams for (a) a substance that contracts on freezing; and (b) a substances that expands on freezing
CP: Critical point TP: Triple point
2.5 Expansivity and Compressibility
• Suppose that the equation of state of a given substance is written in the form (2.5)
• Taking the differential, we obtain
(2.6)
(2.7)
• The expansivity, or coefficient of volume expansion, is given by
• This is the fractional change of volume resulting from a change in temperature, at constant pressure.
2.5 Expansivity and Compressibility
• Similarly, the isothermal compressibility is defined as (2.8)
• This is the fractional change in volume as the pressure changes, with the temperature held constant.
• The negative sign is used since the volume always decreases with increasing pressure (at constant temperature)
• For an ideal gas,
(2.9) (2.10)
2.7 Specific heat
• Specific heat: the amount of heat required to change a unit mass of a substance by one degree in temperature
,
→ The properties we can measure(2.11) (2.12)
If there is no volume change (constant v)
(2.13)
2.7 Specific heat
→ →
V constant V constant
Easier to measure heat and internal energy in difference
can be measured
2.7 Specific heat
(2.14)
(2.15)
(2.16) If there is no pressure change
(constant P)
2.7 Specific heat
→
Constant pressure
Amount of heat added =
can be measured
2.8 Maxwell equation
(2.17) (2.18) (2.19)
(2.20)
2.8 Maxwell equation
P
Wnet
v
T
s Wnet
Jacobian
2.9 Various equations of state Cubic equation of state (EOS)
• Van der Waals EOS (1873)
• Redlich-Kwong (RK) EOS (1949)
• Soave-Redlich-Kwong (SRK) EOS (1972)
• Peng-Robinson (PR) EOS (1976)
• Patel-Teja EOS (1982)
2.9 Various equations of state Virial equation of state (EOS)
• BWR (Benedict–Webb–Rubin) Starling EOS (1970)
• Lee-Kesler EOS (1975)
• MBWR EOS (1973, 1987)
𝑍 = 𝑝𝑣
𝑅𝑇 = 1 + 𝜌 𝐸 −𝐸 𝑇 −𝐸
𝑇 + 𝐸 𝑇 −𝐸
𝑇 + 𝜌 𝐸 −𝐸 𝑇 −𝐸
𝑇 + 𝜌 𝐸
𝑇 +𝐸
𝑇 +𝐸 𝜌
𝑇 1 + 𝐸 𝜌 exp −𝐸 𝜌
𝐸 = 𝑎 + 𝛾 𝑏
𝑍 = 𝑝𝑣
𝑅𝑇 = 1 + 𝐵 𝑣 + 𝐶
𝑣 + 𝐷
𝑣 + 𝑐
𝑇 𝑣 𝛽 + 𝛾
𝑣 exp − 𝛾 𝑣
𝐵 = 𝑏 −𝑏 𝑇 − 𝑏
𝑇 − 𝑏 𝑇 𝐶 = 𝑐 −𝑐
𝑇 + 𝑐
𝑇 𝐷 = 𝑑 +𝑑 𝑇
𝑍 = 𝑍 + 𝜔
𝜔 𝑍 − 𝑍
𝑝 = 𝑎 𝑇 𝜌 + exp −𝛿 𝑎 𝑇 𝜌 𝛿 = 𝜌
𝜌
𝑎 = 𝑅𝑇
𝑎 = 𝑏 𝑇 + 𝑏 𝑇 ⁄ + 𝑏 +𝑏 𝑇 +𝑏
𝑇 𝑎 = 𝑏 𝑇 + 𝑏 +𝑏
𝑇 +𝑏 𝑇 𝑎 = 𝑏 𝑇 + 𝑏 +𝑏 𝑎 = 𝑏 𝑇
𝑎 =𝑏 𝑇 +𝑏
𝑇 𝑎 =𝑏
𝑇 𝑎 =𝑏
𝑇 +𝑏 𝑇
𝑎 =𝑏 𝑇 𝑎 =𝑏
𝑇 +𝑏 𝑇 𝑎 =𝑏
𝑇 +𝑏 𝑇 𝑎 =𝑏
𝑇 +𝑏 𝑇 𝑎 =𝑏
𝑇 +𝑏 𝑇 𝑎 =𝑏
𝑇 +𝑏 𝑇 𝑎 =𝑏
𝑇 +𝑏 𝑇 +𝑏
𝑇
2.9 Various equations of state
• Helmholtz energy equation of state (EOS) for pure fluid
Helmholtz energy
𝛼 𝛿, 𝜏 = 𝑎 𝜌, 𝑇
𝑅𝑇 = 𝑎 𝜌, 𝑇 + 𝑎 𝜌, 𝑇
𝑅𝑇 = 𝛼 𝛿, 𝜏 + 𝛼 𝛿, 𝜏
𝛼 𝛿, 𝜏 =𝑎 𝜌, 𝑇
𝑅𝑇 =ℎ 𝜏 𝑅𝑇 −𝑠
𝑅 − 1 + ln𝛿𝜏 𝛿 𝜏−𝜏
𝑅 𝑐
𝜏 𝑑𝜏 +1 𝑅
𝑐 𝜏 𝑑𝜏
𝛼 𝛿, 𝜏 = 𝑎 + 𝑎 𝜏 + ln𝛿 + 𝑐 − 1 ln𝜏 − 𝑛 𝜏
𝑡 (𝑡 − 1)+ 𝑣 ln 1 − exp −𝑢 𝜏 𝑇⁄
𝛼 𝛿, 𝜏 = 𝑛 𝛿 𝜏 + 𝑛 𝛿 𝜏 exp −𝛿
+ 𝑛 𝛿 𝜏 exp −𝜂 𝛿 − 𝜖 − 𝛽 𝜏 − 𝛾 + 𝑛 𝛿𝛥 𝜓
Departure function
𝛿 = 𝜌
𝜌 𝜏 =𝑇
𝑇
