[sodium.dioxid]

Suppose we have 2NO₂ N₂O₄
K_eq=[N₂O₄]/[NO₂]²

But the bottom term is squared. How can there be a constant ratio for two terms with different powers? (it seems that the lower term would overshoot the upper term). Why isn't it just K_eq=[product]/[concentration] without the square.

[sodium.dioxid]

Suppose we have 2NO₂ N₂O₄
K_eq=[N₂O₄]/[NO₂]²

But the bottom term is squared. How can there be a constant ratio for two terms with different powers? (it seems that the lower term would overshoot the upper term). Why isn't it just K_eq=[product]/[concentration] without the square.

By the way, is this supposed to be intuitive? Or is it derived? If the latter, I will leave it alone.

[fledarmus]

How can there be a constant ratio for two terms with different powers?

If I understand your question correctly, the answer lies in the fact that the values for the two terms are not independent. Whatever concentration of starting material you begin the reaction with absolutely fixes the final equilibrium concentration of all the components in the reaction in accordance with the equilibrium constant for the reaction.

[sodium.dioxid]

How can there be a constant ratio for two terms with different powers?

If I understand your question correctly, the answer lies in the fact that the values for the two terms are not independent. Whatever concentration of starting material you begin the reaction with absolutely fixes the final equilibrium concentration of all the components in the reaction in accordance with the equilibrium constant for the reaction.

I understand what you are saying. You are saying that the reaction acts in such away so as to achieve concentrations that maintains the Keq. This is a mysterious conclusion. The next logical question would be why does it behave this way? How can this be physically described?

[fledarmus]

That is where the derived part comes in. An equilibrium constant is actually a ratio of two reaction rate constants; the rate constant of the forward reaction and the rate constant of the reverse reaction. We say that a reaction is in equilibrium when the rate of the forward reaction is equal to the rate of the reverse reaction. For every two molecules of NO₂ that react to form one molecule of N₂O₄, somewhere there is an N₂O₄ molecule falling apart to form two NO₂ molecules.

So let's suppose that it is much faster for NO₂ to react than for N₂O₄ to decompose - the forward rate constant is much higher than the reverse rate constant, and the K_eq is a large number. For the forward reaction and the reverse reaction to occur at the same rate, you have to slow down the forward rate by making the NO₂ molecules few and far between, and speed up the reverse rate by increasing the number of N₂O₄ molecules. At equilibrium, the concentration of product molecules is much higher than the concentration of starting material molecules.

If you try to increase the concentration of starting materials at this point by adding more NO₂, you will speed up the forward rate even more and form more product, decreasing the amount of NO₂ again until a new equilibrium is established, still in accord with the same K_eq

[Jorriss]

You can also derive equilibrium constants without appealing to kinetics from thermodynamics. A derivation of the equilibrium constant for ideal gases is very approachable if you take a look at Levine.

[sodium.dioxid]

Ok, one more question. For any equilibrium reaction, starting with only reactants, are these statements true:

In the forward direction, rate of reaction decreases at a decreasing rate.
In the reverse direction, rate of reaction increases at a decreasing rate.

[Jorriss]

Ok, one more question. For any equilibrium reaction, starting with only reactants, are these statements true:

In the forward direction, rate of reaction decreases at a decreasing rate.
In the reverse direction, rate of reaction increases at a decreasing rate.

You've taken calculus, think about this in terms of calculus. You want to know if the rate of the reaction is monotonically decreasing. How do you tell if a function is monotonically decreasing (increasing)?

[sodium.dioxid]

Ok, one more question. For any equilibrium reaction, starting with only reactants, are these statements true:

In the forward direction, rate of reaction decreases at a decreasing rate.
In the reverse direction, rate of reaction increases at a decreasing rate.

You've taken calculus, think about this in terms of calculus. You want to know if the rate of the reaction is monotonically decreasing. How do you tell if a function is monotonically decreasing (increasing)?

Got it. For this particular problem, rate is a function of concentration squared in the forward direction. Taking the derivative of this shows a linearly increasing deceleration. So that's a check for the first one. The reverse direction is a problem. While the function is linear, the concentrations do not build up linearly. The surge comes in the beginning and fades down with increasing concentration (because the gap between the rates are closing). Nevertheless, while the gap is closing, the concentration of product is increasing, just not as fast since the difference is becoming smaller. Thus, That's a check for the second one also.

[Jorriss]

Got it. For this particular problem, rate is a function of concentration squared in the forward direction. Taking the derivative of this shows a linearly increasing deceleration. So that's a check for the first one. The reverse direction is a problem. While the function is linear, the concentrations do not build up linearly.

Not quite, don't forget implicit diff. You differentiate with respect to time so you have to use implicit differentiation on A. y(t)=x(t)^2 => dy/dt = 2x (dx/dt)

[sodium.dioxid]

Got it. For this particular problem, rate is a function of concentration squared in the forward direction. Taking the derivative of this shows a linearly increasing deceleration. So that's a check for the first one. The reverse direction is a problem. While the function is linear, the concentrations do not build up linearly.

Not quite, don't forget implicit diff. You differentiate with respect to time so you have to use implicit differentiation on A. y(t)=x(t)^2 => dy/dt = 2x (dx/dt)

Oops. That makes sense.
And I completely understand now the relationship between equilibrium constant and rate law constants.
But the book just gave me another surprise. In my own words, it says that the powers come from the coefficients in the composite reaction, not the powers in the rate law. Why is that?

Is it because equilibrium reactions don't occur in a composite fashion? In other words, is the composite reaction itself an elementary reaction (for equilibrium reactions)?

[sodium.dioxid]

Please don't mind my last question. I got myself another textbook, which explained this particular topic more clearly.