Problem $3\mathrm{.}4.$ Rate Equation for Ammonia Synthesis

Ammonia synthesis is thought to take place on an iron catalyst according to the

following sequence of elementary steps

N$2+2*2$N$*(1)$

H$2+2*2$H$*(2)$

N$*+$ H$*$NH$*+*(3)$

NH$*+$ H$*$ NH$2*+*(4)$

NH$2*+$ H$*$ NH$3+2*(5)$

Step $2$ must occur three times for every occurrence of step $1$ and steps $3$ to $5$ each

occur twice to give the overall reaction of N$2+3$ H$2=2$ NH$3.$ Experimental evidence

suggests that step $1$ is the rate$-$determining step, meaning that all of the other steps

can be represented by one pseudo$-$equilibrated overall reaction. Other

independent evidence shows that nitrogen is the only surface species with any

significant concentration on the surface $($most abundant reaction intermediate$).$

Thus, $[*]0=[*]+[$N$*].$

$($a$)$ Show that if the assumptions indicated above occurs, then the sum of the

individual steps leads to the overall reaction for ammonia synthesis

N$2+3$ H$22$ NH$3$

$($b$)$ Since step $(1)$ is assumed to be rate$-$determining, show that the sum of

steps $(2)(5)$ leads to the following equation for the equilibrium

relationship:

$2=[][2]3/2$

$[3][]$

$($c$)$ Next, show that the overall rate of reaction can be expressed as the rate of

dissociative adsorption of N$2$ from step $1$ so that

CHEN $5333$

Homework $3$

Page $6$

$=11=1[2][]2$

$[]0$

$-$

$1[]2$

$[]0$

where the subscripts denote the forward and reverse reactions, respectively.

$($d$)$ Solve the site balance equation $($above $($a$))$ along with the expression for K$2$

simultaneously to find expressions for $[*]$ and $[$N$*],$ respectively.

$($e$)$ Substitute the above expressions for $[*]$ and $[$N$*]$ into the expression derived

in step $($c$)$ to produce the following rate equation

$=$

$1[]0[2]12$

$2[]0$

$[3]2$

$[2]3$

$1+2$

$[3]2$

$[2]3/2$

$2$

Here, k$1$ and k$-1$ are the rate constants for the forward and reverse rates of

reaction, respectively, in step $1.$ The parameter K$2$ is the adsorption

equilibrium constant for step $2.$

$($f$)$ Using the result from step $($e$),$ derive the reduced form of the above rate

equation for the case where the reaction is at low conversion or far from

equilibrium.

$($g$)$ Now, assume that the rate$-$determining step is actually:

N$2+*$ N$2*($associative adsorption of N$2)$

with N$2*$ being the most abundant reaction intermediate. Derive the rate

expression for the reversible formation of ammonia following a procedure similar

to the one outlined above.

$($h$)$ Can the rate expression derived in parts $($e$)$ and the one from above be

discriminated through experimentation?