Thermodynamics S2

first law (general form)

second law (general form)

inequality applies when dQ is for an irreversible process

equations for a closed system where reversible PV work is involved

F

Fundamental equation for internal energy (U) for a closed system only involving PV work

the combined first and second law for a reversible PV work system (U, T, S, P and V are state functions)

what does the fundamental eqtn for U show

it shows that the internal energy of a closed system only involving PV work is a function of S and V: U(S,V)

dU as an exact differential

expressions for P and T by equation exact differential of dU to the fundamental equation

Why are S and V the natural variables of U

They are the basis for the fundamental equation of U, from which all the thermodynamic properties of the system (U, T, S, P, V) can be yielded. Considering U(T,P) or U(T,V) doesn’t let you calculate all these properties.

why consider different thermodynamic potentials

introducing new functions of state with intensive variables in them, such as P and T, allows us to explain thermodynamic questions when they are the independent variables. Also, the idea of thermodynamic potentials will show us when processes may occur.

functions of state table

thermodynamic potential

a scalar potential energy function. They don’t have an easy, insightful physical interpretation but are tools used to analyse situations depending on the variables we know.

internal energy and heat capacity

heat capacity at constant pressure

heat capacity and enthalpy

differential form of enthalpy (1)

heat needed for an isobaric process

heat absorbed in a process at a constant pressure is equal to the change in enthalpy IF the only work done is reversible PV work.

enthalpy definition

differential form of enthalpy (2)

natural variables of H are S and P

T and V using H

enthalpy exact differential

for a phase transition assuming constant pressure

change in enthalpy is the latent heat put into a system and measured for a phase transition

enthalpy of fusion

enthalpy change required to completely change on mole of a substance from solid to liquid

enthalpy of vaporisation

enthalpy change needed to completely change one mole of a substance from liquid to gas

enthalpy of sublimation

enthalpy change needed to change one mole of a substance completely from solid to gas

phase transitions and enthalpy of a system

condition for spontaneous change (S, V)

a change can occur spontaneously if the internal energy decreases when the change occurs at constant entropy and volume

differentlial of H inequality

condition for spontaneous change (S,P)

if an infinitesimal change takes place in a system of constant entropy and pressure, dH is negative if the change is spontaneous and 0 if the change occurs in the system at equilibrium.

Helmholtz function

differential of helmholtz function

natural variables are T and V

helmholtz function for an isothermal process

at constant temperature, a positive ΔF represents the reversible work done on a system by the surroundings.

condition for spontaneous change (T,V)

for any spontaneous or irreversible process at constant temperature and volume, there is a decrease in F, and for a reversible process it is constant. When this system experiences a spontaneous change, the helmholtz function decreases until equilibrium is reached.

derivatives of the helmholtz function

Gibbs Function

differential of gibbs function

(for a reversible process) natural variables are T and P

gibbs function for isothermal, isobaric process

derivatives of gibbs function

spontaneous change (T,P)

for any irreversible process in which only PV work is done, G decreases and for a reversible process it is constant. when a system undergoes spontaneous change at constant T and P, G will decrease until equilibrium is reached.

alternate expression for G

significance of natural variables

if the function of state can be determined as a function of its natural variables, then the function will yield all the thermodynamic properties of the system.

Gibbs-Helmholtz equation

equation of state with H

equation of state with G

equations of state using G(T,P)

maxwell relations

[the variables that are kept constant are the natural variables for the function each relation comes from]

internal energy of an ideal gas is not a function of volume

internal energy of an ideal gas is not a function of pressure

isobaric cubic expansivity (volumetric coefficient of thermal expansion)

linear expansion coefficient

the fractional change in length per degree of temperature change

isothermal compressibility

the (-) sign because materials reduce in volume when pressure increases

heat from a compressed object

general result for heat flow from a material that is being compressed

heat capacity at constant pressure (directly observables)

heat capacity at constant volume (directly observables)

difference between heat capacities

This is a general relationship.

because κ_T is positive for all substances, it’s always true that C_P > C_V

difference in heat capacities for ideal gas

for n moles of ideal gas

molar entropy of an ideal gal (P,T)

molar entropy of an ideal gas (P,V)

alternate expression for molar entropy of ideal gas (P,T)

internal energy of a gas

general relation

internal energy of a van der waals gas

joule expansion

no work is done and no heat enters, so U is constant

joule coefficient (for expansion under constant internal energy)

rate of change of temperature with a change in volume when no work is done.

joule coefficient for an ideal gas

from PV = nRT

joule coefficient for a van der waals gas

free expansion of a real gas…

always results in cooling.

chemical potential

N is the number of particles

this is an intensive variable

modified first law for a change dN

conjugate pairs of extensive/intensive variables

expression for dS

derivatives of S

particle exchange to gain equilibrium

equilibrium is found for T1 = T2 when μ1 = μ2.

for dN > 0, particles flow from system 1 to 2 when μ1 > μ2

chemical potential (2)

it’s the gibbs function per particle

gibbs function for many types of particle in a system

helmholtz function for many types of particle in a system

general case for heat capacities

latent heat (L)

For two phases in thermodynamic equilibrium at a critical trainsition temperature (T_c), to change an amount of material between these phases some extra heat must be supplied while the system stays at constant temperature

liquid-gas transition of water

[change in entropy at boiling point]

chemical potential and phase changes

for a system in equilibrium, gibbs free energy is constant, and also equal to ΣN_iμ_i. If the number of particles in phase 1 increases, then the number of particles in phase 2 decreases by the same amount. This require μ₁ = μ₂ (along a line of coexistence). The phase with the lowest chemical potential is the stable phase.

Clausius-Clapeyron equation

two phases in the P-T plane can coexist at the phase boundary. The equation describing this boundary line is the Clausius-Clapeyron equation. It shows that the gradient of the phase boundary is determined by the latent heat, and the volume difference between the phases.

Clausius-Clapeyron for an ideal gas

using the substitutions R = N_Ak and l = L / N_A, it can be shown that this exponential is a boltzmann factor for the energy step for a particle to move from liquid to gas

P-T boundary between a liquid and solid phase

rearranging the CC equation and ingoring any temperature dependence for L and ΔV, then integrating to get this expression.

T₀ and P₀ are known points on the phase boundary.

triple point

the point where solid, liquid and gas phases exist simultaneously.

critical point

when the phase boundary between a liquid and gas terminates (end of vapour-pressure curve). At this point, liquid-gas coexistence is reached

termination of solid-liquid boundary

does not exist. after triple point, the boundary is very steep but does not terminate

phase diagram for a hypothetical pure substance

what happens exactly at a phase boundary

change in gibbs free energy is 0 and the chemical potentials of the two phases are identical

what happens upon approaching a phase boundary

the specific volumes (inverse of density) before and after a phase boundary are different. The slope of the Gibbs function must change at the boundary - the system will try to minimise it energy and travel along the lower of the two curves

what drives phase transitions

when a system undergoes spontaneous change at constant T and P, G decreases until equilibrium is reached

for any irreversible process where only P-V work is done, G decreases

for a reversible process, it is constant

ehrenfest classification for order of phase changes

the order of a phase transition is defined as the the order of the lowest differential of the Gibbs function which shows a discontinuity at the transition

table of gibbs free energy and its first/second derivatives w.r.t. pressure

first derivative - volume

second derivative - isothermal compressibility

alternative gibbs free energy curves

by finding the derivatives w.r.t. temperature, similar curves can be found such that:

1st derivative - entropy

2nd derivative - heat capacity at constant pressure

examples of first order phase changes

solid-liquid

solid-vapour

liquid-vapour

superconducting transition in mag. field

some allotropic transitions in solids (e.g. iron)

examples of second order phase changes

superconducting transition in 0 field

superfluid transition in liquid helium

order-disorder transition in 𝛽-brass

examples of third order phase changes

curie point of many ferro-magnets (e.g. iron)

modern approach to classifying phase transitions

show a latent heat (which are still called ‘first-order phase transitions’)

vs

do not have a latent heat (continuous phase transition)