Derivations Flashcards
derive the work during an isothermal process
W = nRTln[V(1)/V(2)]
derive the work done during an isobaric
W = P(1)[V(1)-V(2)]
derive the work done during an adiabatic process
W = [P(1)V(1) - P(2)V(2)]/[1-γ]
derive enthalpy H
dU = TdS - PdV
dU = dU/dS)(V) dS + dU/dV)(S) dV
dV = dV/dT)(P) dT + dV/dP)(T) dP
δQ = dU+PdV
substitute dU and dV
H = U + PV
derive C(P)
from the differential form of H
gives on the formula sheet
enthalpy and Bernoulli’s equation
ΔH = Q - W(d)
ΔH + Δ(E(k) + E(p))(bulk) = Q - W
derive Clausius inequality
draw a diagram for a composite system
total work = ΣδW(i)
total heat = Σ(i) T(0)/T(i) δQ(i)
apply the first law
gives clausius inequality
derive the entropy showing it is a function of the state
start from Clausius inequality and integrate from f to i and i to f . it is path independent hence a function of the state. same can be said for the reverse.
derive the expaned form of the joule-kelvin coefficient
µ = (∂T/∂P)(H) from formula sheet
apply cyclical rule
identify C(P) = (∂H/∂T)(P)
use maxwell’s equation
to yield on formula sheet
determine the functional form for the inversion temperature of a van der Waals gas
van dar waals equation
take the partial derivative
equate to joule kelvin
eliminate pressure
substitute the derivative expression
and insert into joule-kelvin
lead to a lim on T = 2a/Rb
derive the Clausius Clapeyron equation
start from
(∂P/∂T)(V)
g1(A) = g2(A)
dg1 = dg2
dg = dP/ρ - set
dP/dT = [S2 - S1]/[V2 - V1]
leads yo clausius clapeyron equation
derive the density of states for a particle in a 3d box
ε(i) = ℏ^2/2m (kx^2+ky^2+kz^2)
1/8 4πk^2 dk = π/2 k^2 dk
1/2πk^2dk/[π^3/L^2] = Vk^2dk/ 2π^2
gives g(k)dk on formula sheet
derive the Maxwell-Boltzmann distribution
f(k) = N exp[{ε(k)/k(B)T)}/Z] g(k)dk
where Z = L^3 (mk(B)T/2πℏ^2)^3/2
gives the maxwell-boltzmann distribution
