1. No category

One Page Thermodynamics Alex Yanjun Hao At thermodynamic equilibrium 𝜐! 𝜇! = 0 Perfect gas !
𝜇 = 𝜇 ∘∗ + 𝑅𝑇𝑙𝑛(𝑃) 𝜇 is chemical potential for the pure gas (denoted by *), at unit pressure P=1 atm (denoted by °), (it’s written as 𝜇 ∘ in the textbook); 𝜇 ∘∗ is a function of T, 𝜇 ∘∗ = 𝑓(𝑇). Perfect gas mixture 𝜇! = 𝜇!∘∗ + 𝑅𝑇𝑙𝑛(𝑃! ) 𝜇!∘∗ pure i, unit P; f(T). Imperfect gas 𝜇 = 𝜇 ∘∗ + 𝑅𝑇𝑙𝑛(𝑓) 𝜇 ∘∗ pure, unit P; f(T); fugacity coefficient 𝜒 = 𝑓/𝑝 → 1 𝑎𝑠 𝑝 → 0 . Imperfect gas mixture 𝜇! = 𝜇!∘∗ + 𝑅𝑇𝑙𝑛(𝑓! ) 𝜇!∘∗ pure i, unit P; f(T); 𝑓! /𝑝! → 1 𝑎𝑠 𝑝 → 0 Ideal solution (gas, liquid, solid) 𝜇! = 𝜇!∗ + 𝑅𝑇𝑙𝑛(𝑋! ) 𝜇!∗ pure i, arbitrary P; f(T,P). Non-­‐ideal solution (gas, liquid, solid) 𝜇! = 𝜇!∗ + 𝑅𝑇𝑙𝑛(𝑎! ) 𝜇!∗ pure i, arbitrary P; f(T,P); activity 𝑎! = 𝛾! 𝑋! Aqueous solution (electrolyte) 𝜇! = 𝜇!☐ + 𝑅𝑇𝑙𝑛(𝑎! ) ☐
𝜇! chemical potential of hypothetical ideal solution of unit molality, arbitrary P; f(T,P); activity 𝑎! = 𝛾! 𝑚! ; activity coefficient of j: log!! 𝛾! = −𝛼𝑍!! 𝐼 ; ionic strength 𝐼 =
∘∗
!
−
𝑚! 𝑍!! (sum of all ions); equilibrium !
constant 𝐾!" = 𝑎! !! . Explain: 𝜇! = 𝜇!∘∗ + 𝑅𝑇𝑙𝑛(𝑋! 𝑃) = 𝜇!∘∗ + 𝑅𝑇𝑙𝑛(𝑃) + 𝑅𝑇𝑙𝑛(𝑋! ) = 𝜇!∗ + 𝑅𝑇𝑙𝑛(𝑋! ) For standard state, 𝜇!∗ = 𝜇!∘∗ , pure i, unit P. For other state, 𝜇!∗ (𝑝) need to be converted to unit pressure value, 𝜇!∘∗ , via 𝜇 ∗ = 𝜇 ∘∗ + 𝑅𝑇𝑙𝑛(𝑃) for gas (or use f for imperfect gas); or 𝜇 ∗ = 𝜇 ∘∗ + 𝑉! (𝑃 − 1) for liquid and solid, if V is assumed constant. If V is not constant, use compressibility 1 𝜕𝑉
𝜅 = −
𝑉 𝜕𝑃 ! !" !
(which gives 𝑉 = 𝑉! 𝑒 !!(!!!) in relation !"
= 𝑉) when deriving the P-­‐V relation. !" !
Now 𝜇 ∗ = 𝜇 ∘∗ −
𝑉! !!(!!!)
𝑒
− 1 𝜅
All the 𝜇 ∘∗ (pure, unit pressure) are functions of T, to get the chemical potential at reference temperature 𝑇! , use 𝑇
𝑇
°
°
°
∆𝐺!"#
(𝑇) = ∆𝐺!"#
(𝑇! ) + ∆𝐻!"#
(𝑇! ) 1 −
𝑇!
𝑇!
°
where ∆𝐻!"#
(𝑇! ) is assumed to be constant in °
°
range [𝑇! , 𝑇]: ∆𝐻!"#
(𝑇! ) = ∆𝐻!"#
(𝑇) Derive: 𝐺
𝜕
𝑇 = − 𝐻 𝜕𝑇
𝑇!
!
!
𝐺
1
𝑑
= −𝐻
𝑑𝑇 𝑇
!!
!! 𝑇
𝐺(𝑇) 𝐺(𝑇! )
1 1
−
=𝐻
−
𝑇
𝑇!
𝑇 𝑇!
𝑇
𝑇
𝐺(𝑇) = 𝐺(𝑇! ) + 𝐻 1 −
𝑇!
𝑇!