CHM2104 Lecture 15 -The Clausius inequality, Gibbs and Helmholtz Function , Maxwell Relation

Clausius Inequality

the total entropy of system + surroundings must increase for a spontaneous process (dS_sys + dS_surr ≥ 0)

Entropy (S)

measure of disorder or spreading of energy; higher S means more randomness

Spontaneous Process

occurs naturally without external work; total entropy increases

Heat (q)

energy transfer due to temperature difference

Work (w)

energy used to move or push matter; expansion work often involves volume change

Internal Energy (U)

total energy inside a system (motion + bonds + vibrations)

dU = dq + dw

first law of thermodynamics; change in internal energy = heat + work

Constant Volume Condition

no expansion work occurs, so w = 0

At constant volume

dU = dq (all heat changes internal energy)

Clausius system form

dS ≥ dq/T (entropy change is linked to heat and temperature)

Constant Pressure Heat

dq_p = dH (heat at constant pressure equals enthalpy change)

Enthalpy (H)

heat content of a system at constant pressure

Free Energy Purpose a

llows spontaneity to be predicted using system-only variables

Helmholtz Free Energy

(A) A = U - TS, useful at constant T and V

Helmholtz spontaneity rule

a process is spontaneous at constant T and V when dA ≤ 0

Gibbs Free Energy (G)

G = H - TS, useful at constant T and P

Gibbs spontaneity rule

spontaneous at constant T and P when dG ≤ 0

Why Gibbs is common

most reactions happen at constant pressure, so G is most useful

Meaning of TS term

energy unavailable for work due to disorder (entropy)

∆G and work meaning

∆G is maximum non-expansion work obtainable at constant T and P

Non-expansion Work Examples

electrical work in batteries, moving charges, lifting mass

∆A and work ∆A gives

maximum work at constant volume and temperature

Fundamental Equation

dU = T dS - P dV (from first + second laws)

Reversible Conditions

derived for reversible steps but valid since U is a state function

Maxwell Relations Purpose

link properties that are hard to measure to ones easier to measure

Maxwell Relation example

(∂T/∂V)_S = - (∂P/∂S)_V

Spontaneity Summary

spontaneous when free energy decreases (G or A goes down)

Signs of ∆G

negative = spontaneous, zero = equilibrium, positive = not spontaneous