Thermodynamics Flashcards
mole fraction
mole fraction of component J
amount of J (in moles) divided by the total moles, n of molecules in the mixture
partial molar volume
partial molar volume of component J in a mixture
change in volume per mole of J added to a large volume of the mixture
V𝐽 is the tangent (derivative) of the curve describing the total
volume, V of the mixture as a function of the amount of J (in moles)
at constant p, T and amount 𝑛𝑖≠𝐽 of any other component in the mixture
change in the total volume V of a mixture due to the addition of
small amounts of A and small amounts of B
V is a state function, only depends on the composition of the mixture.
equation can be used independently of how nA and nB are mixed
partial molar volume varies with mixture composition
when composition is altered, the environment experienced by each component (and its interaction with the other components) changes
also varies with p and T
extensive property
any property (mass, volume) that depends on the absolute amount of matter
intensive property
any property (partial molar quantity/volume, density) that depends on the amount of matter normalised to the sample size
partial molar Gibbs energy (chemical potential)
the tangent (derivative) of the curve describing the Gibbs energy as a function of the amount of J (in moles)
at constant p, T and amount 𝑛𝑖≠𝐽 of any other component in the mixture.
Gibbs energy for binary mixture
fundamental equation of chemical thermodynamics (modified to account for changes in composition)
chemical potential changes with changes to p, T and composition
new term accounts for any additional non-expansion work that the system can
perform due to changes of its composition (changes in 𝑛𝐽)
as a consequence of phase changes/chemical reactions
modified equation for binary mixtures
Gibbs energy of mixing: the case of ideas gases
gas A and gas B initially occupy different compartments of a container - separated by a partition
their chemical potentials is the molar Gibbs energy of the pure substances, μ = Gm.
μ(p) - chemical potential of an ideal gas
Gibbs energy before mixing
Gibbs energy after mixing
after the partition is removed
p = pA + pB
pA and pB = gases’ partial pressures in the mixture
Gibbs energy of mixing
Gibbs energy of mixing in terms of mole fraction
𝑥𝐽 is always <1
hence, ln 𝑥𝐽 < 0 so that ∆mixG < 0 (always negative) for all compositions
confirms that mixing ideal gases is always a spontaneous process at any composition
entropy of mixing
ln x < 0, ∆mixS > 0 (always positive) for all compositions
mixing gases increases disorder, greater entropy
enthalpy of mixing
Using ∆G = ∆H −T∆S,
we obtain that ∆mixH = 0 for all compositions
(i.e. no interaction exist between molecules of ideal gases).
enthalpy in the system constant, therefore the increase in entropy is what drives the mixing of two ideal gases.
Gibbs energy (left) and entropy of mixing (right) of two ideal gases
When the liquid and gas phases of pure A are at equilibrium, the chemical potentials of the two phases are identical
When the liquid phase also contains solute B, the criterion of phase equilibrium still holds for A
combine equations for 𝝁𝑨* and 𝝁𝑨
Raoult’s law
partial vapour pressure, pA
mole fraction, 𝑥A
vapour pressure of pure A, pA*
physical interpretation of Raoult’s law
when there is solute B, the rate at which the solvent A evaporates decreases without altering the rate at which it condenses to the liquid phase from the gas phase
solute molecules (small, black) prevent solvent molecules (large, blue) at the surface of a solution to pass into the gas phase (dashed arrow) but not the opposite exchange (solid arrows).
rates of vaporization and
condensation for A
Vapourisation rate = kxA
Condensation rate = k′pA
k and k′ are proportionality constants
vapourisation and condensation rates must coincide at equilibrium
for a pure liquid, xA = 1
therefore, k/k′ = pA*, thus giving Raoult’s law.
combine Raoult’s law for A and B with Dalton’s law of partial pressures to construct vapour-pressure diagram for the mixture
ideal solutions (or ideal mixtures)
mixtures that follow Raoult’s law at all compositions
substituting Raoult’s law in the expression of the liquid chemical potential:
expression shows that: in a solution, the solvent (A) chemical potential is always reduced by the solute B.
Gibbs energy before mixing of two liquids
Gibbs energy after mixing of two liquids
Gibbs energy of mixing of two liquids
entropy of mixing of two liquids
enthalpy of mixing of the two liquids
the rise in the system’s entropy is the reason
for the mixing, as ∆mixH = 0
volume variation due to mixing is null for ideal solutions
ideal gas
molecules do not interact
ideal solution
molecules interact
A-B, A-A and B-B interactions are all equivalent energetically
phase-diagrams of binary mixtures
a: single phase (liquid)
b: bubble-point line
c: dew-point line, single phase
(vapour of same composition as the liquid in a)
vapour pressure (fixed temperature)
Using Raoult’s law, express the total vapour pressure p of an ideal binary mixture as:
Hence, at a given temperature, p depends linearly on composition.
vapour pressure diagrams (fixed temperature)
zA - overall composition
xA - mole fraction in liquid
yA - mole fraction in vapour
total vapour pressure, p of an ideal binary mixture as a function of its overall composition, zA
on LHS, points of compositions a and b have the same vapour pressure - represent a mixture with liquid of composition a=xA in equilibrium with gas of composition b=yA
all points above the upper line (i.e. high pressure) correspond to points where only the liquid phase is stable (zA = xA)
all points below the curve (i.e. low pressure) correspond to points where only the vapour phase is stable (zA = yA)
for the points contained between the two curves, the two phases coexist and zA quantifies how much A is present in total in the two phases
isopleth
vertical line on vapour pressure diagram
- it is a line of constant composition
tie line
horizontal line on vapour pressure diagram
- connects the compositions of the 2 phases in equilibrium
lever rule: focusing on the points in the area between the curves where we have the coexistence of liquid and vapour
- a point in this region (green point) gives quantitative information about their relative amounts
α and β are the two phases
- nα and nβ are their amounts (in moles)
temperature-composition diagrams (fixed pressure)
important for distillation where pressure rather than temperature is kept constant
colligative properties (physical properties)
due to presence of solute B, dilute solutions show colligative properties
- these are only determined by how much B there is rather than
what B is
colligative properties examples
- elevation of the boiling point
- depression of the freezing point
- osmotic pressure
these properties are discussed assuming that solute B is:
* not volatile - it is not present in the vapour phase
* not soluble in the solid solvent
this means that the solute only influences the chemical potential of the liquid phase (and not the solid or vapour phases)
origin of all colligative properties
attributed to the fact that, in a solution, a solute B always reduces the chemical potential of the liquid solvent A
both freezing point and boiling point shift.
elevation of boiling point - consider eqm between solvent’s liquid and vapour phases
A = solvent
B = solute - only contributes to the chemical potential of the liquid phase as it is non-volatile
T = the mixture’s boiling temperature (increases)
Tb = boiling temperature of pure A
∆vapH = vaporisation enthalpy of pure A.
equation shows that
solute B induces an increase in the boiling point from Tb to Tb + ∆T, proportional to its mole fraction xB
depression of freezing point - consider eqm between solvent’s liquid and solid phases
A = solvent
B = solute - only contributes to the chemical potential of the liquid phase due to its insolubility in the solid phase
T = the mixtures freezing temperature (decreases)
Tf = freezing temperature of pure A
∆fusH = fusion enthalpy of pure A
osmosis
the net unprompted movement of solvent molecules through a semipermeable membrane into a region of higher solute concentration
semipermeable membrane
a membrane that allows only solvent and not solute
molecules to pass
osmotic pressure Π
the pressure that, if applied to region of higher solute concentration, will oppose osmosis
Π stems from the equilibrium between pure solvent A (at pressure p) and A in an ideal solution (at higher pressure p + Π) at both sides of semipermeable membrane
importance of osmosis
- measurements based on osmosis (osmometry) - used to measure the molar masses of macromolecules.
- osmosis helps cells keep their shape.
- osmosis is used in dialysis - the purification of solutions.
- reverse osmosis is applied to desalinate seawater.
osmosis figure
solvent A diffuses from the left of the semipermeable membrane (low solute concentration) to its right (high solute concentration) to dilute the solution and establish equilibrium.
this movement can be halted by a pressure Π being applied to the compartment on the right
∴ at equilibrium, there is no net mass transport
- the concentrations on both sides of the membrane do not change
chemical potential of A at equilibrium
must coincide at both sides of the membrane
- although presence of B would lower chemical potential of A, this effect is counterbalanced by the increased pressure
Vm = the molar volume of A
assume solution is dilute - amount of solute B is very small compared to solvent A
for a dilute solution, since ln xA = ln(1 − xB) ≈ −xB and assuming Vm to be constant in the range of integration (which holds for small values of Π), we obtain
van ’t Hoff equation
𝜫 = [B]RT
[B] = nB/V
B molar concentration.
osmotic pressure can be measured from hydrostatic pressure
Π = ρgh
ρ = density of solution at equilibrium
g = gravitational acceleration (9.81 m s-2)
h = height of solution in column at equilibrium
the spontaneous passage of A from the bottom container to the solution in the upper container generates a difference in hydrostatic pressure.
- the osmotic pressure can then be measured from this hydrostatic pressure
criterion of spontaneous change
at constant T and p, a system naturally evolves towards a minimum of Gibbs free energy G - this also applies to chemical reactions
extent of reaction, ξ - measures by how much the reaction proceeds (units: moles)
equilibrium A ⇌ B
If a very small amount ∆ξ of A becomes B, the amount of A will change
by ∆nA = −∆ξ
the amount of B will change by ∆nB = +∆ξ
___
A variation of ξ by ∆ξ will cause the amount of any reactant/product J to vary by ∆nJ = νJ∆ξ
reaction Gibbs energy ∆rG
Since chemical potentials depend on composition, ∆rG also varies with the stage of the reaction (as the composition changes)
Gibbs energy vs Extent of reaction
(1) Spontaneous reaction in the forward case for ∆rG < 0
(2) Spontaneous reaction in the the reverse case for ∆rG > 0
(3) equilibrium is reached at the point of zero slope (when ∆rG = 0), where the Gibbs energy has its minimum
perfect gas (ideal gas law)
𝑝𝑉 = 𝑛𝑅𝑇 with R = 8.314 J K-1 mol-1
law represents an approximate equation of state for all gases (real gases)
- works progressively better to describe a real gas behaviour the lower the pressure (p → 0), i.e. when gas molecules are far apart
departures from this law become significant when gas molescules are packed together - high pressure and low temperature
- V → 0 when p → ∞ or T → 0;
- molecules do not interact;
- a gas does not become a liquid or a solid upon cooling.
gas expansion
driven by repulsive interactions between molecules
gas compression
driven by attractive forces between molecules
real gas (virial equation of state)
temperature-dependent coefficients B, C, etc. are known as the second, third, etc. virial coefficients
- account for non-ideal behaviour.
- the first virial coefficient is always 1
real liquid mixtures - positive deviation from Raoult’s Law
A-B interactions less favourable than A-A and B-B interactions
- causing an increased vapour pressure compared to ideal case
A and B do not like each other - go from liquid to vapour phase earlier as they prefer to be further apart in vapour phase.
in liquid phase A and B would be forced to be closer together
real liquid mixtures - negative deviation from Raoult’s Law
A-B interactions more favourable than A-A and B-B interactions
- causing a decreased vapour pressure compared to ideal case
A and B like each other - go from liquid to vapour phase later as they prefer to be closer together in liquid phase
in vapour phase A and B would be forced to be further apart
using Raoult’s Law for real solutions
can still use Raoult’s Law to describe behaviour of solvent A in real solutions (when xA is close to 1)
Henry’s Law
describes the behaviour of the solute B at low concentrations (i.e. when xB is close to 0)
KB, is the slope of the tangent to the experimental vapour pressure curve as a function of mole fraction of B at xB = 0.
KB is experimentally determined and has the dimensions of pressure
ideal-dilute solutions
solutions where the solvent and the solution follow Raoult’s law and Henry’s law, respectively
ideal-dilute solution diagram
small amount of solute (B)
- B is mainly surrounded by solvent (A)
solvent molecules are mainly surrounded by other molecules of the same nature, whilst solute molecules are not
- small amount of B in a lot of A
Henry Law: the vapour pressure of an ideal-dilute solution is proportional to the mole fraction xB
azeotropes
a max/min can occur in temperature-composition diagram
max can appear in presence of favourable A-B interactions (HCl/H2O mixture)
- due to favourable interactions, molecules prefer to stay in liquid phase rather than escaping to the gas phase more than for an ideal mixture
min can appear in presence of unfavourable A-B interactions (ethanol/water mixture)
- due to unfavourable interactions, molecules escape to the gas phase faster than in the ideal case
azeotropes form at these points - vapour and liquid have the same composition
negative azeotrope (e.g. hydrochloric acid and water)
a high boiling point
c denotes azeotropic composition ~ vapour and liquid have the same composition
implication of azeotropes on distillation
the composition at max/min is reached during distillation -
evaporation of an azeotrope takes place without a change in composition - therefore, distillation can no longer separate the two components beyond this composition (point c)
positive azeotrope (e.g. ethanol and water)
a low boiling point
c denotes azeotropic composition ~ vapour and liquid have the same composition
activity - effective mole fraction
contains the information of how the liquid deviates from ideal behaviour
(assume ideality of liquid phase)
The activity of a liquid in a mixture can be obtained in experiments by measuring its vapour pressure at all compositions
activity coefficient, γ
γ<1 for favourable A-B interactions
γ>1 for unfavourable A-B interactions
activity coefficient changes with T, p and mole fraction (tending to 1 when 𝑥𝐴 tends to 1).
only for ideal mixtures, 𝛾𝐴 = 1 at all temperatures, pressures and compositions
equation relating acitivity, α to activity coefficient, γ
colligative properties with non-ideal solutions: osmosis
assume that the solute B is not volatile or soluble in the solid phase so that it only influences the liquid chemical potential
experimentally determined constant B = the osmotic virial coefficient
osmosis is often used to measure molar masses of macromolecules.
- solutions of these big molecules are hardly ideal so use 𝛱 = 𝜌𝑔ℎ
excess functions, X^E
for real mixtures
quantify the deviation of the measured value of a thermodynamic function of mixing from that of the corresponding function expected assuming ideal behaviour.
excess Gibbs free energy
excess Gibbs free energy of mixing
excess entropy
excess enthalpy and excess volume
directly defined by the measured enthalpy
and volume of mixing
excess free Gibbs energy can be measured by:
- Measuring the vapour pressures of A/B mixtures, e.g. for chloroform/acetone
- Calculating their activities
- Calculating the activity coefficients
- Calculating G^E
In this case, for example, mixing acetone and chloroform is more favourable than
mixing an ideal mixture.
