### Topic: Equilibrium kinetics  (Read 72015 times)

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##### Equilibrium kinetics
« on: April 29, 2013, 04:43:08 PM »
I don't propose to go into the maths in my case, unless someone wants to point me towards it, but I'd like to ask a question that's on my mind: I know that it is possible to write a precise differential equation for any 1 equilibrium which (after integration) will yield a result of being able to calculate the concentration of any species involved in the equilibrium at any time, given the forward and backward rate constants, equilibrium constant, and initial concentrations of all species.

Is it possible, in theory, to write a differential equation for any large system of equilibria, which can then be solved to calculate the concentration of any species involved in the system at a certain time, given the rate constants for every equilibrium, every equilibrium constant, and the initial concentrations of all species involved?

#### Corribus

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##### Re: Equilibrium kinetics
« Reply #1 on: April 29, 2013, 06:09:09 PM »
This question is very abstract.  Can you ... er... deabstractify it?
What men are poets who can speak of Jupiter if he were like a man, but if he is an immense spinning sphere of methane and ammonia must be silent?  - Richard P. Feynman

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##### Re: Equilibrium kinetics
« Reply #2 on: April 29, 2013, 06:32:19 PM »
I'll try. http://en.wikipedia.org/wiki/Rate_equation#Equilibrium_reactions_or_opposed_reactions shows for the most simple possible equilibrium A  B, how a differential equation can be obtained in terms of the forward rate constant k1 and backward rate constant k2, as well as the equilibrium constant K, which can be solved for the concentration at any time t of any species present in the reaction. This is covered through to the end by the Wikipedia link, for the incredibly basic case of A  B.

Now let's say that instead of one equilibrium we have many. They do not necessarily "start" at the same time, e.g. I might place HCl in a solution and then at some time t later add K2HPO4, each of which will start more equilibria off. Is it in theory always possible to summarize all the equilibria in a differential equation which, when solved, calculates the concentration of every species that we now have in our system (e.g. for the above example, it would be H+, OH-, H2O, K+, and all the phosphate forms), given all the initial concentrations of these species, the time at which we put them in, and all needed constants?

#### Corribus

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##### Re: Equilibrium kinetics
« Reply #3 on: April 29, 2013, 09:25:14 PM »
You are asking if rate for every mechanism you could think of is exactly solvable?  I don't see why not.  At least, I can't think of a case off the top of my head where it isn't.
What men are poets who can speak of Jupiter if he were like a man, but if he is an immense spinning sphere of methane and ammonia must be silent?  - Richard P. Feynman

#### curiouscat

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##### Re: Equilibrium kinetics
« Reply #4 on: April 30, 2013, 01:54:28 AM »
Is it possible, in theory, to write a differential equation for any large system of equilibria, which can then be solved to calculate the concentration of any species involved in the system at a certain time, given the rate constants for every equilibrium, every equilibrium constant, and the initial concentrations of all species involved?

Yes.

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##### Re: Equilibrium kinetics
« Reply #5 on: April 30, 2013, 12:14:45 PM »
Is it possible, in theory, to write a differential equation for any large system of equilibria, which can then be solved to calculate the concentration of any species involved in the system at a certain time, given the rate constants for every equilibrium, every equilibrium constant, and the initial concentrations of all species involved?

Yes.

At what stage would I begin to get into the maths involved?

#### curiouscat

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##### Re: Equilibrium kinetics
« Reply #6 on: April 30, 2013, 12:46:09 PM »
Is it possible, in theory, to write a differential equation for any large system of equilibria, which can then be solved to calculate the concentration of any species involved in the system at a certain time, given the rate constants for every equilibrium, every equilibrium constant, and the initial concentrations of all species involved?

Yes.

At what stage would I begin to get into the maths involved?

Didn't understand your question. You could start right away if you were so inclined.

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##### Re: Equilibrium kinetics
« Reply #7 on: April 30, 2013, 01:33:02 PM »
Is it possible, in theory, to write a differential equation for any large system of equilibria, which can then be solved to calculate the concentration of any species involved in the system at a certain time, given the rate constants for every equilibrium, every equilibrium constant, and the initial concentrations of all species involved?

Yes.

At what stage would I begin to get into the maths involved?

Didn't understand your question. You could start right away if you were so inclined.

I am so inclined! What should I look into?

#### curiouscat

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##### Re: Equilibrium kinetics
« Reply #8 on: April 30, 2013, 01:55:13 PM »
Is it possible, in theory, to write a differential equation for any large system of equilibria, which can then be solved to calculate the concentration of any species involved in the system at a certain time, given the rate constants for every equilibrium, every equilibrium constant, and the initial concentrations of all species involved?

Yes.

At what stage would I begin to get into the maths involved?

Didn't understand your question. You could start right away if you were so inclined.

I am so inclined! What should I look into?

It's really not that hard. Good bookkeeping followed by a reliable simultaneous ODE solver.

If you only want equilibrium concentrations its just a bunch of simultaneous non-linear equations.

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##### Re: Equilibrium kinetics
« Reply #9 on: April 30, 2013, 02:07:37 PM »
Is it possible, in theory, to write a differential equation for any large system of equilibria, which can then be solved to calculate the concentration of any species involved in the system at a certain time, given the rate constants for every equilibrium, every equilibrium constant, and the initial concentrations of all species involved?

Yes.

At what stage would I begin to get into the maths involved?

Didn't understand your question. You could start right away if you were so inclined.

I am so inclined! What should I look into?

It's really not that hard. Good bookkeeping followed by a reliable simultaneous ODE solver.

If you only want equilibrium concentrations its just a bunch of simultaneous non-linear equations.

Ok that video makes sense on the whole. But it seems to apply to reactions going to completion. Will the equilibrium balances be written in the same way?

#### curiouscat

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##### Re: Equilibrium kinetics
« Reply #10 on: April 30, 2013, 03:09:37 PM »

Ok that video makes sense on the whole. But it seems to apply to reactions going to completion.

Nope.

Quote

Will the equilibrium balances be written in the same way?

Yes.

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##### Re: Equilibrium kinetics
« Reply #11 on: April 30, 2013, 03:24:14 PM »

Ok that video makes sense on the whole. But it seems to apply to reactions going to completion.

Nope.

Quote

Will the equilibrium balances be written in the same way?

Yes.

If it's an equilibrium shouldn't we be concerned with a forward rate constant and a backward rate constant for each reaction? There only appears to be a forward rate constant. And no term for the equilibrium constants either ...

#### curiouscat

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##### Re: Equilibrium kinetics
« Reply #12 on: April 30, 2013, 03:27:31 PM »

Ok that video makes sense on the whole. But it seems to apply to reactions going to completion.

Nope.

Quote

Will the equilibrium balances be written in the same way?

Yes.

If it's an equilibrium shouldn't we be concerned with a forward rate constant and a backward rate constant for each reaction? There only appears to be a forward rate constant. And no term for the equilibrium constants either ...

B  with k1 & k2

is the same as

B with k1
and
A with k2

No equilibrium constant is needed. If k1 and k2 are correct then Keq=k1 / k2 automatically.

#### curiouscat

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##### Re: Equilibrium kinetics
« Reply #13 on: April 30, 2013, 03:30:25 PM »

If it's an equilibrium shouldn't we be concerned with a forward rate constant and a backward rate constant for each reaction? There only appears to be a forward rate constant. And no term for the equilibrium constants either ...

Out of pure curiosity, what is your current level? HS? College? etc.

Again, pure curiosity. Have you exposure to how to solve ODE's simultaneously numerically?

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##### Re: Equilibrium kinetics
« Reply #14 on: April 30, 2013, 04:37:52 PM »

Ok that video makes sense on the whole. But it seems to apply to reactions going to completion.

Nope.

Quote

Will the equilibrium balances be written in the same way?

Yes.

If it's an equilibrium shouldn't we be concerned with a forward rate constant and a backward rate constant for each reaction? There only appears to be a forward rate constant. And no term for the equilibrium constants either ...

B  with k1 & k2

is the same as

B with k1
and
A with k2

No equilibrium constant is needed. If k1 and k2 are correct then Keq=k1 / k2 automatically.

Hmmm, but in the video I don't see where the extra terms are for the backward reaction ... to clarify:

A+B  C with k1
2E with k2

But nowhere does there appear to be a k1,rev or k2,rev. And how can we arrive at final ODEs to solve, for equilibrium, which do not include the equilibrium constants?