Formulas Flashcards
differential form of the first law
dU = δQ + δW
all infinitesimal changes
U - internal energy (J)
Q - heat or thermal energy (J)
W - work (J)
second law
ΔS ≥ 0
reciprocal theorem
(∂x/∂y)(z) = [(∂y/∂x)(z)]^-1
reciprocity theorem
(∂x/∂y)(z) (∂y/∂z)(x) (∂z/∂x)(y) = -1
work during reversible processes
δW = -PdV
W = -(2 ∫ 1) PdV
heat capacity at constant volume
C(V) =lim(ΔT->0) (∂Q/∂T)(V) = (∂U/∂T)(V)
enthalpy
H = U + PV
enthalpy in differential form
dH = d(U+PV) = dU +PdV + VdP = ∂Q + VdP
heat capacity at constant pressure
C(P) =lim(ΔT->0) (∂Q/∂T)(P) = (∂H/∂T)(P)
efficiency of an engine
η = 1 - Q(2)/Q(1)
efficiency of a carnot engine
η(C) = 1 - T(2)/T(1)
triple point of water
T(K) = 273.16K Q/[Q(TP) H(2)O]
efficiency of a carnot refrigirator
η(C)^(R) = T(2)/T(1)-T(2)
efficiency of a carnot heat pump
η(C)^(HP) = Q(1)/Q(1)-Q(2) = T(1)/T(1)-T(2)
Clausius inequality
∮ δQ/T(0) ≤ 0
central equation of thermodynamics
dU = TdS - pdV
δQ
= TdS
δW
= -PdV
entropy change for heating a body
ΔS = cmln(Tf/Ti)
entropy change for adding heat to a reservoir at constant T
ΔS = Q/T
entropy change for phase changes
phase changes are isothermal processes
ΔS = mL/T
absolute entropy of an ideal gas
ΔS = n[c(V) ln(T2/T1) + Rln(V2/V1)]
molar specific entropy of an ideal gas
Δs = c(V) ln(T2/T1) + R ln(V2/V1)
c(P) - C(V) =
R
isobaric coefficient of thermal cubic expansion
β(P) on the formula sheet
adiabatic coefficient of thermal cubic expansion
β(S) on the formula sheet
isothermal bulk compressibility
κ(T) on the formula sheet
adiabatic bulk compressibility
κ(S) on the formula sheet
isothermal bulk modulus
K(T) on the formula sheet
latent heat of expansion change
L(V) on the formula sheet
latent heat of pressure change
L(P) on the formula sheet
absolute entropy
s = s(0) + c(v) ln[T/T(0)] + R ln[V/v(0)]
enthalpy in a chemical reaction under constant pressure
Q = ΔH for ΔP = 0
entropy in a chemical reaction under constant pressure and enclosed in an adiabatic wall
ΔS + ΔS(0) ≥ 0
ΔG ≤ 0 for ΔP = ΔT = 0
entropy in a chemical reaction under constant volume and enclosed in an adiabatic wall
ΔS + ΔS(0) ≥ 0
ΔF ≤ 0
molar specific enthalpy of an ideal gas
h = u + pV(m)
= u + RT
monoatomic
3/2 RT
diatomic
5/2 RT
enthalpy for a phase change at constant pressure
ΔH = ΔQ + VΔP = mL(p)
enthalpy for a incompressible fluid
ΔH = CΔT + VΔP
or
Δh = cΔT + ΔP/p
where p is density
van der waals gas
see formula sheet
a, accounts for attractive forces between molecules
b, accounts for the finite particle volume
dieterici equation
Pexp(an/RTV)(V-nb) = nRT
joule-kelvin coefficient
µ on formula sheet
inversion temperature
= joule-kelvin coefficient
where µ is the gradient of the curve
overall enthalpy is conserved in the liquification and hence
h(i) = αh(f,l) + (1-α)h(f,v)
α = [h(i)-h(f,v)]/[h(f,l)-h(f,v)]
phase change equilibria
g(1) = g(2)
Clausius-Clapeyron equation for first-order phase changes
dP/dT = mL/[T(V(2) - V(1))]
where P is the saturation vapour pressure
L is the specific latent heat of vaporisation in J/kg
V(1) and V(2) are the specific volumes of the vapour and liquid phases
partition function
Z = ( Σ i) exp[-ε(i)/k(b)T ]
ε(i) is the microstates of energy
boltzmann’s hypothesis
S = k(b) ln Ω
Ω is the number of ways a system can be configured
three dimensional density of states
g(k)dk on formula sheet
maxwell-boltzmann distribution
f(k)dk on formula sheet
rayleigh-jeans law
u(BB) (λ) on formula sheet
the entropy for a single particle
S = k(b) [lnV + 3/2ln[mk(B)T/2πℏ^2 + 3/2]
probability or fraction of particles in a state
P(i) = 1/Z exp[-ε(i)/k(B)T]
Z = 1/N! [V (mk(B)T/2πℏ^2) ]^N
the partition function given derives from Z(gas) = 1/N! Z^N(particle) for Z^N(particle) the one-particle partition function
N! derives from the indistinguishability of particles
entropy
dS = δQ/T
where entropy is measured in JK^(-1)kg^(-1)
