Carbon adsorption linearization A graduate student conducts

Carbon adsorption linearization A graduate student conducts the carbon adsorption experiments and plots In(X/M) vs In(Ce) to obtain the following curves to estimate Freundlich contants Express these equations in the general Freundlich equation format as: 3 .2 0 2979-4 6473 09146 -3 y-0.3587x-5 2689 R -09681 In(Ce) sludge derived activated carbon commercial activated carbon Figure 6. Freundlich isothern for commercial and sludge derived activated carbons b.Another student performs the same experiments but plots log (X/M) vs log(Ce) (base 10) What would be his equations in linearized form? How would his general Freuindlich equation parameter differ (if any) in the general format? X/M-ar

Solution

a)

X/M = a C_f^b

Using natural logarithm on both sides,

ln(X/M) = ln(a C_f^b)

=> ln(X/M) = ln(a) + ln(C_f^b)

=> ln(X/M) = ln(a) + b ln(C_f) ............(eqn 1)

For commercial activated carbon,

y = 0.2979 x - 4.6473

=> ln(X/M) = 0.2979 ln(C_f) - 4.6473 .............. (eqn 2)

Comparing eqn1 and eqn 2,

ln(a) = -4.6473

=> a = e^-4.6473 = 9.587 x 10^-3

b = 0.2979

For sludge derived activated carbon,

y = 0.3587 x - 5.2689

=> ln(X/M) = 0.3587 ln(C_f) - 5.2689 .............. (eqn 3)

Comparing eqn1 and eqn 3,

ln(a) = -5.2689

=> a = e^-5.2689 = 5.149 x 10^-3

b = 0.3587

General Freundlich equation for commercial activated carbon : X/M = 9.587 x 10^-3 C_f^0.2979

General Freundlich equation for sludge derived activated carbon : X/M = 5.149 x 10^-3 C_f^0.3587

b)

For commercial activated carbon,

ln(X/M) = 0.2979 ln(C_f) - 4.6473

=> ln(X/M) / 2.303 = 0.2979 ln(C_f) / 2.303 - 4.6473 / 2.303 .....[Dividing both sides with 2.303]

=> log(X/M) = 0.2979 log(C_f) - 2.018

For sludge derived activated carbon,

ln(X/M) = 0.3587 ln(C_f) - 5.2689

=> ln(X/M) / 2.303 = 0.3587 ln(C_f) / 2.303 - 5.2689 / 2.303 ....[Dividing both sides with 2.303]

=> log(X/M) = 0.3587 log(C_f) - 2.288

There will be no change in the Freuindlich equation parameters a and b.

So,

General Freundlich equation for commercial activated carbon : X/M = 9.587 x 10^-3 C_f^0.2979

General Freundlich equation for sludge derived activated carbon : X/M = 5.149 x 10^-3 C_f^0.3587