Linearize each of the following \"common\" constitutive relations of chemical en

Question

Linearize each of the following "common" constitutive relations of chemical engineering, expressing your results in terms of deviation variables, which you should clearly dene.

(a) Chemical kinetics { temperature dependence of reaction rate constant": the Arrhenius

expression, with k0, E, and R constant:

k(T(t)) = k0e^(E/RT(t))

(b) Equations of state { (Empirical) temperature dependence of enthalpy, with bi constant:

H(T(t)) = b0 + b1T(t) + b2T^2(t) + b3T^3(t) + b4T^4(t)

(c) Equations of state { Antoine's equation for the temperature dependence of vapor pressure, with A, B, and C constant:

p0(T(t)) = e^[A-(B/(T(t)+C))]

(d) Phase equilibrium { Vapor-liquid equilibrium relation for binary mixture, alpha with , the

relative volatility, constant:

y(x(t)) = alpha* x(t)/(1+(alpha-1)*x(t))

(e) Heat transfer { radiation heat transfer, with epsilon ,sigma , and A constant:

q(T(t)) = epsilon*sigma*A*T^4(t)

Explanation / Answer

The Arrhenius equation is
k = A e^-Ea/RT
ln k = ln A - Ea/RT
we can rewrite it so it fits the equation of a straight line
y = mx + b
ln k = -Ea/R x 1/T + ln A
Now we can see that -Ea/R is the slope of the line when ln k is plotted against 1/T. We can also see that ln A is the y-intercept.

So to get A from your data: Plot ln k vs 1/T. Get the y-intercept. Since it is ln A, then A = e^(y-intercept).

The graphical method would be the best way, but I suppose you could solve the Arrhenius equation for A and then plug in each set of data, get a value for A and then average them.
k = A e^-Ea/RT
A = k / e^-Ea/RT

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