University:
The Pennsylvania State University
Course:
CHEM 450 | Thermodynamics
Academic year:

2024
Views:

375

Pages:

3
Author:

Alexis H.

Properties of internal energy 1st law: dU = dq + dw When the system is closed and reversible (only expansion work) dw = -pdV dq = TdS Then: dU = dq + dw dU = TdS -pdV. I This is called the fundamental equation dU is a state function (independent of path). This means that l is true whether the change is reversible or irreversible When the change is irreversible: TdS > dq (Clausius inequality) -pdV > dw However the sum dq + dw will always equal the sum TdS-pdV (assuming constant composition) Using equation l , we can write U as a function of S and V: U(S,V) Then: µ ¶ µ ¶ @U @U dU = dS + dV @S V @V S Using the fundamental equation l dU = TdS - pdV, it follows that… µ @U @S ¶ µ =T V E @U @V ¶ = ¡p S This is the thermodynamic definition of temperature The relations above imply that there are also other interesting relations between the thermodynamic properties of a system Since dU is an exact differential, then from dU = TdS - pdV we get… µ @T @V ¶ =¡ S µ @p @S ¶ V This is called a Maxwell relation This particular Maxwell relation is fairly useless, but we can derive more Maxwell relations from the other thermodynamic state function State function U H A G Exact differential dU = T dS ¡ pdV dH = T dS + V dp dA = ¡pdV ¡ SdT dG = V dp ¡ SdT Maxwell Relation µ @T @V µ @p @T @V @T =¡ S ¶ @T @p µ µ ¶ = S ¶ ¶ = V µ µ Example: variation of internal energy with volume from chapter 2: Let us express pi(T) as a function of pressure and temperature ¶ @V @S ¶ µ =¡ p @P @S @S @V µ V p ¶ @S @p T ¶ T ¼T = µ @U @V ¶ T dU = µ @U @V µ ¶ @U @S = T ¶ µ dS + V @U @S ¶ µ V µ @S @V T It ¼T = T µ @P @T @U @V ¶ ¶ + T dV S µ @U @V ¶ S p ¶ ¡p V For a van der Waals gas: nRT n2 p= ¡a 2 V ¡ nb V µ ¶ µ ¶ @p @nRT = (V ¡ nb) = @T V @T V ¼T = T = nR V ¡ nb µ nR V ¡ nb ¶ nRT ¡p= ¡ V ¡ nb µ nRT n2 ¡a 2 V ¡ nb V Let us derive the exact differential for G: G = H ¡ TS dG = dH ¡ d (T S) = dH ¡ T dS ¡ SdT remember : H = U + pV dH = dU + pdV + V dp dU = T dS ¡ pdV dH = T dS ¡ pdV + pdV + V dp dH = T dS + V dp dG = T dS + V dp ¡ T dS ¡ SdT ¶ n2 =a 2 V dG = V dp ¡ SdT