Free Energy and Phase & Chemical Equilibria Flashcards
(43 cards)
How do you derive the equation where du can be used to determine spontaneity of a reaction? Why is this not the most helpful?
ds>=dq/T
dq<=TdS
du= dq + dw
du <= TdS -Pex dV (from inequality above)
if constant S and V
du<=0 to be spontaneous, or =0 at equilibrium
Hard to maintain a constant entropy in a lab
What is the equation for Helmholtz free energy? Derive how the condition for equilibrium can obtained
A= u- TS
dA=du - TdS - SdT
dA<= TdS -P ex dV - TdS -SdT
dA<= -Pex dV - SdT
for a process at constant V and T
dA<= 0, at equilibrium=0
What is the equation for Gibbs free energy? Show how it can be used for a reaction at equilibrium?
G= A + pV
dG= dA + pdV + Vdp
dG<= -Pex dV - sdT + pdV + Vdp
at mechanical equilibrium where p=p ex
dG<= Vdp - sdT
for a process at constant p and t, dG<0
and dG=0 at equilibrium
e.g phase transitions, chemical reactions…
How can the Gibbs free energy incorporate enthalpy?
G= A + pV
G= U - TS + pV
G= U + pV - TS
G= H - TS
How can a systems ability to do non pV work be used within the free energy equations? What is the maximum available work of a system?
dw= -P ex dV + dwe, other work
dG<= dwe or equal at equilibrium
The maximum work from a process at constant T and P is equal to the decrease gibbs free energy
What are the terms involved in the fundamental equations?
They show how the 4 energy state functions, U, H, A, G, vary with P, V, S, and T
Can all be derived from definitions of the the energy state functions, and using the first and second law and pv work
How can the fundamental equation for H be derived? and A?
H= U + pV
dH= du + pdV + Vdp
dH= TdS - pdV + pdV + Vdp
dH= TdS + Vdp
A= u -TS
dA= du - Tds - SdT
dA= TdS- pdV - TdS - SdT
dA= pdV - SdT
Why can we remove the inequality when deriving the fundamental equations but not for spontaneity arguements?
Replacing du rev with du
As valid for changes between equilibrium states
How can the maxwell relations be derived?
Start with any fundamental equation
e.g dH= TdS + Vdp
implies H= H(S, p )
dH= (∂H/∂S)p dS + (∂H/∂p)s dP
implies (∂H/∂S)p = T and (∂H/∂p)s = V
and as ∂²H / ∂s∂p = ∂²H / ∂p∂s
so taking the derivative with respect to p for t as already done for s
(∂T/∂P)s = (∂V/∂s)p
How do you derive the Gibbs-Helmholtz equation?
G= H -TS
-S= (G-H) / T
From the maxwell equations,
-S=(∂G/∂T)p
(∂G/∂T)p - G/T= -H/T from subbing in and rearranging
but if we take G/T and differentiate implicitly with respect to T, using the product rule
= 1/T (∂G/∂T)p - G/T²
=1/T ((∂G/∂T)p - G/T)
which is 1/T a factor of -H/T
so
∂/∂T (G/T) = - H / T²
How do you derive the van’t Hoft equation from the Gibbs Helmholtz?
ΔrG⦵ = -RTlnkp
ΔrG/T⦵=-Rlnkp
∂/∂T (ΔrG/T⦵) = - ΔH / T²
∂/∂T (Rlnkp) = ∂/∂T (ΔrG/T⦵)
∂/∂T (lnkp) = - ΔH / RT²
How do you derive an equation to show how G varies with pressure for ideal gases and solids?
dG= Vdp - SdT
using the maxwell relation, (∂G/∂p)T= V
moving the p over and integrate
G2-G1= ∫ v dp
of a perfect gas, using v= nrt/p
G2-G1= nRTln(p2/p1)
using p1=1 atm
G2- G⦵=nRTln(p2/p⦵)
G2= G⦵ + nRTln(p2/p⦵)
for solids
G2-G1=v ∫ dp, as V independent of p
G2= G⦵ + V⦵(p-p⦵)
How can the idea of isothermal compressibility used for an equation of how G of solids varies with pressure?
graph lnV against p
slope=-k = (∂lnV/∂p)T= -1/v (∂V/∂p)T
rearrange for V
then same G integral as before, replace V with -1/k (∂V/∂p)T
G2-G1= (v2-v1) / k
-k = (∂lnV/∂p)T, rearrange for v2 in terms of v and e
end up with
G2= G⦵ + V/k(1-e^-k(p2-p⦵)
and as k approaches 0,
G2= G⦵ + v⦵(p2-p⦵)
What are the conditions for phase equilibrium and how have these been derived?
Suppose an equilibrium between a and b, where dn moles of a is converted to b, constant p,t
dG= G(b) - G (a) dn
at equilibrium dG=0
so G(b) = G(a)
can be 2 or 3 states. but 3 states have isolated points in the p,t plane
What does the p/t graph for water look like?
Y=mx line from origin between s and g
almost backwards line from triple point between solid and liquid
almost quadratic from triple point up ish between liquid and gas
How do you calculate the degrees of freedom for a substance?
F= 3 - p, the number of phase in equilibrium
Derive the Clapeyron equation
The slope of a pt graph is dp/dT
Consider the points p,t and p+dp, T + dT
at p+dp, T + dT, G(a) + dG(a) = G(b) + dG(b)
at p,T G(a)=G(b)
subtracting the equations
dG(a)= dG(b)
using the fundamental equation dG= Vdp - SdT, which applies to both
V(a)dp - S(a)dT= V(b)dp - S(b)dT
V(b)dp - V(a)dp= S(b)dT- S(a)dT
ΔV(b-a)dp= ΔS(b-a)dT
dp/dT=ΔS/ΔV
where ΔV is the volume change between the phases and ΔS the entropy change between the phases
How can the Clapeyron equation be used to incorporate enthalpy?
ΔG=ΔH - TΔS
at equilibrium, ΔG=0
ΔS= ΔH/T
dp/dT=ΔS/ΔV (Clapeyron equation)
dp/dT= ΔH/TΔV
How can the Clausius-Clapeyron equation be derived?
Clapeyron= dp/dT= ΔS/ΔT = ΔH/TΔV
For sublimations and vaporisations, we can assume the volume change= volume of the gas, and there is such a large increase in volume, volume solid/liquid negligible
And we can use the ideal gas law for the volume of gas
pV=nrt with 1 mol
V=RT/p
substituting into the Clapeyron equation
dp/dT= ΔH/T/(RT/p) = pΔH/RT²
1/p dp/dT = ΔH/RT²
dlnp/dT=ΔH/RT²
an approximation
If given the vapour pressures of a gas at different temperatures, how can the enthalpy changes of vaporisation and sublimation be calculated?
The Clapeyron equation can be written in the form
ln(p)= C -ΔH/RT When integrating out
Sketch a graph of this and it will give two opposing best fits
The first one is vaporisation, with higher temperatures but lower pressures and the second sublimation
Then you can calculate the slopes
Or you can use the equation with suitable data points
What is the chemical potential of a pure substance?
The amount of Gibbs energy of one mole of the pure substance
G(m)=μ
G=nμ
What is the fundamental equation for dG in an open system?
How does this vary for a mixture?
dG= Vdp - SdT + sum of μdn
for a mixture, μ1dn1 + μ2dn2 …
from exact differential from G=G(P,T,n)
What is the Gibbs energy of a system using chemical potential? What is the differential for μ?
G= sum of n x μ for all components of the system
G=nμ
∂G/∂n = μ
What is the Gibbs- Duhem equation and how is it derived?
G= sum of nμ
dG= sum of ndμ + sum of μdn
dG=Vdp - SdT + sum of μdn
at constant p and t
dG= sum of μdn
subtracting the equations
0= sum of ndμ
