Thermodynamics
1. Properties of Gases
1.A The perfect gas
Definition
Gas: consists of a collection of molecules that are in ceaseless motion and which interact
significantly with one another only when they collide
Variables of state
Physical state: depends on the pressure (p), the amount of substance it contains (n) the
volume it occupies (V) and the temperature (T)
Thermodynamic equilibrium: the state variables of the system do not change spontaneously
Pressure: mechanical equilibrium is the condition of equality in pressure of two gasses →
standard pressure (pᶿ) is 1 bar) is 1 bar
Temperature: a perfect-gas/thermodynamic temperature scale (in Kelvins) is independent of
the identity of the gas → K = °C + 273.15
Perfect gas
Equation of state: interrelates the variables that define the state of a substance → the value
depends only on the current state of the system and is independent of how that state has
been prepared
Limiting law: strictly true only in a certain limit
pV = nRT = NkT → perfect gas law
Perfect gas: obeys the perfect gas law exactly under all conditions
Real gas: is described exactly by the perfect gas law in the limit of p→0
R = NAk → molar gas constant
Processes
Reversible process: the system is in thermodynamic equilibrium at every moment
Irreversible process: is not necessarily reversible
Isothermal process: at constant temperature
Isobaric process: at constant pressure
Isochoric process: at constant volume
Adiabatic process: without heat exchange with the environment (dQ = 0)
Mixture of gases
pJ = xJp → partial pressure (xJ = mole fraction)
pA + pB + … = (xA + xB + …)p = p → for real and perfect gases
Dalton’s law: the pressure exerted by a mixture of gases is the sum of the pressures that each
one would exert if it occupied the container alone
1.C Real gases
Intermolecular forces
Intermolecular forces: important when distance between the molecules is very small and the
temperature is very low
Repulsions: short-range interactions that assist expansion
Attractions: long-range interactions that assist compression
Vapour pressure: the pressure at which both liquid and vapour are present in equilibrium
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, Compression factor
Z = Vm / V°m → compression factor (ratio measured molar volume/perfect gas molar volume)
Z > 1: due to repulsive forces the molar volume is larger than that of a perfect gas
Z < 1: due to attractive forces the molar volume is smaller than that of a perfect gas
Virial coefficients
pVm = RT(1 + B/Vm + C/V2m + …) → virial equation of state (summarizes the behaviour of real
gases over a range of conditions)
Virial coefficients: depend on the temperature → second (B), third (C), … in the equation
Boyle temperature (TB): the properties of a real gas coincide with those of a perfect gas as
p→0
At TB: B = 0 & pVm = RTB
Critical constants
Critical point: compression takes place at TC and a surface separating two phases does not
appear and the volumes at each end of the horizontal part of the isotherm have merged to a
single point
Critical constants: critical temperature Tc, critical pressure pc and critical molar volume Vc
Tc = 8a / 27Rb , pc = a / 27b2 , Vc = 3b
Zc = pcVc / RTc = 3/8 → critical compression factor (for all gases that are described by the van
der Waals equation near the critical point)
Supercritical fluid: gas that occupies the entire volume (container) at T ≥ T C → much denser
than normally considered typical of gases
Van der Waals equation
p = (nRT / V – nb) – a(n2 / V2) → Van der Waals equation of state (real gases)
Van der Waals coefficients: represent the strength of attractive (a) and the repulsive
interactions between the molecules (b) → characteristic for each gas but independent of the
temperature
Van der Waals’ loops: the unrealistic oscillations below Tc → they suggest that under some
conditions an increase of pressure results in an increase of volume
Maxwell construction: the oscillations are replaced by horizontal lines drawn so the loops
define equal areas above and below the lines
Principles van der Waals equation
Perfect gas isotherms are obtained at high temperatures and large molar volumes → then the
perfect gas law applies (p = RT / Vm)
Liquids and gases coexist when the attractive and repulsive effects are in balance
The critical constants are related to the van der Waals coefficients → the extrema of the
oscillations converge as T→Tc and coincide at T=Tc (1st and 2nd derivatives are zero)
Corresponding states
Reduced variables: dimensionless variables of a gas used to compare the properties of
objects
Vr = Vm / Vc , pr = pm / pc , Tr = Tm / Tc → reduced variables
Principle of corresponding states: real gases at the same reduced volume and reduced
temperature exert the same reduced pressure → approximation that works best for spherical
(globular) molecules
Pr = (8Tr / 3Vr – 1) – (3 / V2r) → same equation but Van der Waals coefficients disappeared
2. The First Law
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1. Properties of Gases
1.A The perfect gas
Definition
Gas: consists of a collection of molecules that are in ceaseless motion and which interact
significantly with one another only when they collide
Variables of state
Physical state: depends on the pressure (p), the amount of substance it contains (n) the
volume it occupies (V) and the temperature (T)
Thermodynamic equilibrium: the state variables of the system do not change spontaneously
Pressure: mechanical equilibrium is the condition of equality in pressure of two gasses →
standard pressure (pᶿ) is 1 bar) is 1 bar
Temperature: a perfect-gas/thermodynamic temperature scale (in Kelvins) is independent of
the identity of the gas → K = °C + 273.15
Perfect gas
Equation of state: interrelates the variables that define the state of a substance → the value
depends only on the current state of the system and is independent of how that state has
been prepared
Limiting law: strictly true only in a certain limit
pV = nRT = NkT → perfect gas law
Perfect gas: obeys the perfect gas law exactly under all conditions
Real gas: is described exactly by the perfect gas law in the limit of p→0
R = NAk → molar gas constant
Processes
Reversible process: the system is in thermodynamic equilibrium at every moment
Irreversible process: is not necessarily reversible
Isothermal process: at constant temperature
Isobaric process: at constant pressure
Isochoric process: at constant volume
Adiabatic process: without heat exchange with the environment (dQ = 0)
Mixture of gases
pJ = xJp → partial pressure (xJ = mole fraction)
pA + pB + … = (xA + xB + …)p = p → for real and perfect gases
Dalton’s law: the pressure exerted by a mixture of gases is the sum of the pressures that each
one would exert if it occupied the container alone
1.C Real gases
Intermolecular forces
Intermolecular forces: important when distance between the molecules is very small and the
temperature is very low
Repulsions: short-range interactions that assist expansion
Attractions: long-range interactions that assist compression
Vapour pressure: the pressure at which both liquid and vapour are present in equilibrium
1
, Compression factor
Z = Vm / V°m → compression factor (ratio measured molar volume/perfect gas molar volume)
Z > 1: due to repulsive forces the molar volume is larger than that of a perfect gas
Z < 1: due to attractive forces the molar volume is smaller than that of a perfect gas
Virial coefficients
pVm = RT(1 + B/Vm + C/V2m + …) → virial equation of state (summarizes the behaviour of real
gases over a range of conditions)
Virial coefficients: depend on the temperature → second (B), third (C), … in the equation
Boyle temperature (TB): the properties of a real gas coincide with those of a perfect gas as
p→0
At TB: B = 0 & pVm = RTB
Critical constants
Critical point: compression takes place at TC and a surface separating two phases does not
appear and the volumes at each end of the horizontal part of the isotherm have merged to a
single point
Critical constants: critical temperature Tc, critical pressure pc and critical molar volume Vc
Tc = 8a / 27Rb , pc = a / 27b2 , Vc = 3b
Zc = pcVc / RTc = 3/8 → critical compression factor (for all gases that are described by the van
der Waals equation near the critical point)
Supercritical fluid: gas that occupies the entire volume (container) at T ≥ T C → much denser
than normally considered typical of gases
Van der Waals equation
p = (nRT / V – nb) – a(n2 / V2) → Van der Waals equation of state (real gases)
Van der Waals coefficients: represent the strength of attractive (a) and the repulsive
interactions between the molecules (b) → characteristic for each gas but independent of the
temperature
Van der Waals’ loops: the unrealistic oscillations below Tc → they suggest that under some
conditions an increase of pressure results in an increase of volume
Maxwell construction: the oscillations are replaced by horizontal lines drawn so the loops
define equal areas above and below the lines
Principles van der Waals equation
Perfect gas isotherms are obtained at high temperatures and large molar volumes → then the
perfect gas law applies (p = RT / Vm)
Liquids and gases coexist when the attractive and repulsive effects are in balance
The critical constants are related to the van der Waals coefficients → the extrema of the
oscillations converge as T→Tc and coincide at T=Tc (1st and 2nd derivatives are zero)
Corresponding states
Reduced variables: dimensionless variables of a gas used to compare the properties of
objects
Vr = Vm / Vc , pr = pm / pc , Tr = Tm / Tc → reduced variables
Principle of corresponding states: real gases at the same reduced volume and reduced
temperature exert the same reduced pressure → approximation that works best for spherical
(globular) molecules
Pr = (8Tr / 3Vr – 1) – (3 / V2r) → same equation but Van der Waals coefficients disappeared
2. The First Law
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