[Thermodynamics Note] The First Law

calorie: heat required to increase 1g water from 14.5°C to 15.5°C

1J = 0.239 calorie

First Law

dU = δq - δＷ

q: - heat flows out of a body (exothermic process)

+ heat flows into a body (endothermic process)

w: - work done on a body

+ work done by a body

"d": state function

"δ": not a state function

∫ dU = 0 -> a property of a state function

V = constant -> isochore, isometric processes -> △U = qv

P = constant -> isobaric processes -> △H = qp

T = constant -> isothermal processes -> △H = 0, △U = 0 (for ideal gas)

q = 0 -> adiabatic processes -> dU = - δW

enthalpy H = U + PV

Cv = (δq/dT)v = (dU/dT)v -> dU = Cv*dT

Cp = (δq/dT)p = (dH/dT)p -> dH = Cp*dT

(1.temperature change. 2.no phase change)

(processes not always V=constant or P=constant)

The difference between Cp and Cv:

Cp = (∂H/∂T)p = (∂U/∂T)p + P(∂V/∂T)p

Cv = (∂U/∂T)v

Cp - Cv = (∂U/∂T)p + P(∂V/∂T)p - (∂U/∂T)v

dU = (∂U/∂V)t * dV + (∂U/∂T)v * dT

(∂U/∂T)p = (∂U/∂V)t *  (∂V/∂T)p + (∂U/∂T)v

Cp-Cv = (∂U/∂V)t(∂V/∂T)p + (∂U/∂T)v + P *  (∂V/∂T)p - (∂U/∂T)v

= (∂V/∂T)p [P+(∂U/∂V)t]

P(∂V/∂T)p: work done by the system per degree rise in expanding against the constant external pressure P

for ideal gas: PV=RT Cp-Cv = P(∂V/∂T)p = P * (R/P) = R (for one mole)

(∂U/∂V)t(∂V/∂T)p: work done per degree rise in temperature in expanding against the internal cohesive forces

(∂U/∂V)t ideal gas: equal zero

liquids: very large

solids: very large

Reversible adiabatic processes

dU = - δW (∵ δq =0)

Ｃv * dT = - P * dV

one mole ideal gas

Cv * dT = - (RT * dV) / V

Cv * ln (T2/T1) = R * ln (V1/V2)

Let Cp/Cv = γ

Cp - Cv = R

(Cp/Cv) - 1 = R/Cv

γ - 1 = R/Cv

Cv ln (T2/T1) = R ln (V1/V2)

(T2/T1)^Cv = (V1/V2)^R

(T2/T1) = (V1/V2)^(R/Cv)

(T2/T1) = (V1/V2)^(γ - 1)

(P2V2/P1V1) = (T2/T1) = (V1/V2)^(γ - 1)

(P2/P1) = (V1/V2)^γ

P2V2^γ = P1V1^γ = PV^γ = constant

Reversible isothermal pressure or volume

dU = 0

δW = δq = P dV = RT/V dV

W = q = RT ln(V2/V1) = RT ln(P1/P2)