# General Chemistry Pressure and Rate laws

## Homework Statement

The kinetics of the decomposition of phosphine at 950 K
4PH3 (g) -> P4 (g) + 6H2 (g)
was studied by injecting PH3(g) into a reaction vessel and measuring the total pressure at constant volume.

P total (Torr) Time (s)
100 0
150 40
167 80
172 120

What is the rate constant of this reaction?

PV = nRT
rate = k[PH3] ^n

## The Attempt at a Solution

Do I need to find the reaction order with respect to PH3 before finding the rate constant? I am lost.

## The Attempt at a Solution

Gokul43201
Staff Emeritus
Gold Member
Yes, you need to find the reaction order. But to do that, you need to relate the change in total pressure to the change in the number of mol/L of PH3. If at some time, t, there are some n moles of PH3 consumed, then what is the net change in the total number of moles of gas in the container?

Also, you've written the differential form of the n'th order rate law. Can you collect the terms, integrate it and write the expression for the concentration of PH3 as a function of time (and the initial concentration)?

Thanks for your help. This is what I have done so far, but it is still wrong.

ln[(PPH3)t / (PPH3)o] = -kt

Ptotal = PPH3 + PP4 + PH2

PPH3 = Po - PP4
PH2 = 6PP4

Ptotal = (Po - PP4) + 6PP4 + PP4
Ptotal= Po + 6PP4
PP4 = (Ptotal - Po) / 6

PPH3 = Po -PP4 = Po - [(Ptotal-Po) / 6]

Po = 100 torr

Pt= 100 - [(150-100) / 6] = 91.666 torr

ln0.91666 = -k * 40s
k = 0.0022

The correct answer is k=0.027 s-1

Gokul43201
Staff Emeritus
Gold Member
Thanks for your help. This is what I have done so far, but it is still wrong.

ln[(PPH3)t / (PPH3)o] = -kt
This is only true for n=1, and is not the correct equation for n != 1. You need to figure out the order of the reaction rate; you can't assume it is 1.

For n not 1, you have:

$$dA/dt = -kA^n \implies A^{-n}dA = -kdt \implies A^{1-n} - A_0^{1-n} = -k(t - t_0)$$

Ptotal = PPH3 + PP4 + PH2

PPH3 = Po - PP4
PH2 = 6PP4

Ptotal = (Po - PP4) + 6PP4 + PP4
Ptotal= Po + 6PP4
PP4 = (Ptotal - Po) / 6

PPH3 = Po -PP4 = Po - [(Ptotal-Po) / 6]
Not sure I follow what you're doing here. For every 4 moles of PH3 consumed there are 7 moles of products produced, resulting in a net change of +3 moles (for every 4 moles of PH3 consumed). So the decrease in partial pressure of PH3 is 4/3 of the increase in total pressure over any interval of time.

So, for instance, during the first 40s, since the p(tot) increases by 50 torr, p(PH3) must decrease by 50*4/3 torr. Using this method, you can translate the table for p(tot) into a similar table for p(PH3).

Write down the values in this new table and plug them into the general equation for the n'th order rate law (above) to find k and n.

Note that the above rate equation only applies if n is not 1. It is actually prudent to first check if n=1 works.

PS: I just checked. It is first order; so you can indeed use the rate equation you used in post #3, only you need to use it on the new table of numbers.

Last edited:
dialmformartian
I think this problem can be done by substituting the pressure and time variables in the concentration-rate constant equations for various ordered reaction The equation are founsd as :

r = -d[A]/dt
r = k[A]^n n - order of reaction.

equating

kdt = d[A]/[A]^n. Integrate (with limits c0 - c initial and final concentration, and 0 - t) to find equation for 1st, 2nd, 3rd... order reactions.

Substitute the given data in equations 1 by 1 till constant value of k for all pair of pressure values is obtained. This constant value gives the real k and n the order.

Im not surwe of this method but i have seen similar problems that can be done this way.

Gokul43201
Staff Emeritus