Are there infinite combinations of partial pressures that give the same chemical equilibrium constant?

[zenterix]

Homework Statement

The chemical equilibrium constant $K_p$ is a constant that is specific to a balanced chemical reaction at a given temperature.

If we have a mixture of reactants and products from the chemical equation, then when equilibrium is reached each species has a partial pressure (or equivalently, a molar concentration) and the value of $K_p$ can be computed.

Relevant Equations

This is all fine for me in terms of doing calculations.

My question is about how to interpret $K_p$ from a different perspective.

Suppose we have the following balanced equation

$$\nu_A A+\nu_B B\rightleftharpoons \nu_C C+\nu_D D,\ \ \ \ \ \Delta G^\circ_{rxn}$$

At equilibrium we have

$$Q_{eq}=K_P=\frac{P_C^{\nu_C}P_D^{\nu_D}}{P_A^{\nu_A}P_B^{\nu_B}}=e^{-\Delta G_{rxn}^\circ/RT}$$

What I think I am confused about is how many "degrees of freedom" we have here?

Suppose I specify, say, $P_A$, and then I specify $P_B$.

Are there infinite combinations of $P_C$ and $P_D$ at which equilibrium occurs?

More generally, are there infinite combinations of $P_A,P_B,P_C,$ and $P_D$ that give the equilibrium constant?

 

[zenterix]

Another way of asking this is the following.

Consider pure water.

There is autoionization in pure water and experimentally, at 25 degrees Celsius we observe the concentration of $1.0\times 10^{-7}$M for each of hydronium and hydroxide ions.

The equilibrium constant is

$$\mathrm{K=\frac{[H_3O^+][OH^-]}{[H_2O]^2}}$$

I hadn't realized but we can actually compute $[H_2O]$.

Water has density 0.997g/mL at $25^\circ\text{C}$ and 1 mol of water has mass 18.02g.

Thus,

$$\mathrm{[H_2O]=997\frac{g}{L}\times \frac{1\ mol}{18.02\ g}}=55.3M$$

Since this is constant we rearrange

$$\mathrm{K_w=K[H_2O]^2=[H_3O^+][OH^-]=10^{-14}}$$

The way I understand this equation is as follows.

We approximate the concentration of water as constant.

If we were to add hydronium to pure water then $\mathrm{H_3O^+}$ would increase of course and there would be reactions between hydronium and some of the hydroxide present.

The new equilibrium would involve a hydronium concentration higher than $10^{-7}$ but lower than right after adding more hydronium, and a hydroxide concentration lower than $10^{-7}$.

The pure water would now be acidic. Would we still call it pure water?

At first I thought: well, won't the hydronium just disappear eventually? Well, the hydronium molecules are charged molecules and the solution now has excess charge that won't go away. It seems that this new acidic equilibrium is permanent.

But then I noticed this is incorrect, because the solution is always neutral.

When I said "add hydronium ions" I think I forgot about the question of how to do that. If we add it by way of an acid, then we need to add extra equations to the analysis to account for the need for charge balance.

Anyways, moving on and back to my original reason for this entire question, as I understand it, we can reach any concentration of hydronium that we wish in equilibrium by adding more hydronium or hydroxide.

Technically, we can't really reach any concentration because I think there are implicit assumptions being used when we consider that the concentration of $\mathrm{H_2O}$ is constant. If we keep adding ions forever this assumption seems like it would fail.

 

[zenterix]

So what happens when we add an acidic solute into pure water at pH 7?

Suppose we add HCl.

If we add a lot of acid such that the concentration of hydronium is relatively high, then we can use it directly to compute the resulting pH.

Otherwise, we need to take into account $K_w$.

Ignoring $K_w$ effectively means that the concentration of hydronium is so high that the production of ions due to autoprotolysis is irrelevant.

But technically, it seems to me that

- water continues to produce ions according to the water autoprotolysis equilibrium (which is now affected by the presence of a dang ton of hydronium)

- the need for charge balance means that we get a lot of positive hydronium and negative Cl from the HCl, and we also get positive hydronium and negative hydroxide from water autoprotolysis, and the sum of the charges needs to balance.

- material balance just means that all the concentration of negative Cl is the same as the concentration of acid initially.

- initially, the reaction quotient is not the same as $K_w$ because we have really high hydronium concentration.

- some of the hydronium will react with some of the hydroxide and we will reach an equilibrium with very low hydroxide concentration and still very high hydronium concentration but lower than right after the acid was added, and the reaction quotient will be $K_w$

- So, $K_w$ dictates what happens, but there are constraints: we have to have charge balance and material balance.

So, it seems we can't just have any combination of hydronium and hydroxide.

Does this answer the title question I asked?

 

[zenterix]

I think one snippet from a book that confuses me is the following

In pure water, the concentrations of the hydronium ion and the hydroxide ion are equal, and the solution is therefore neutral. If [H3O+] > [OH−], however, the solution is acidic, whereas if [H3O+] < [OH−], the solution is basic.

How can it be possible to have differing concentrations of hydroxide and hydronium in pure water?

Won't the pure water have charge?

 

[Borek]

If [H3O+] > [OH−], however, the solution is acidic, whereas if [H3O+] < [OH−], the solution is basic.

Where does it say this is a statement about pure water, and not a general one?

 

[Mayhem]

I think one snippet from a book that confuses me is the following



How can it be possible to have differing concentrations of hydroxide and hydronium in pure water?

Won't the pure water have charge?

The point is that [H3O⁺] = [OH^-] for pure water, but when you add acid or base to pure water, you disturb that balance. I commend your dedication to understanding acid base theory.

 

[zenterix]

Where does it say this is a statement about pure water, and not a general one?

The entire quote starts with "In pure water, the concentrations of the hydronium...".

By the way, here is the book itself and you can search for the quote above to see the context around it.

 

[zenterix]

The point is that [H3O⁺] = [OH^-] for pure water, but when you add acid or base to pure water, you disturb that balance. I commend your dedication to understanding acid base theory.

Yes, this is the general point which I understand. I'm trying to understand the details, however.

 

[Borek]

The entire quote starts with "In pure water, the concentrations of the hydronium...".

It still doesn't mean it applies to everything that is written in following phrases.

 

[zenterix]

It still doesn't mean it applies to everything that is written in following phrases.

I think the use of the word "however" shows that the second sentence is very much related to the first sentence.

But I won't debate the writing abilities of the author because what I want is to understand the equation for K_w.

 

[DrJohn]

For pure water, the concentration of H+ = that of OH-
This isn't necessarily true if the water is not pure. For example in your questions, after adding HCl.
But the Value of Kw is the same.
The If and However implies different conditions. Not pure water but water containing an acid which will ionise.
Same as in a computer program where If and Else applies to different conditions, which leads to two different results.

 

[zenterix]

Computer programs are unambiguous.

Written language need not be and is frequently and obviously open to interpretations.

Each person has a slightly different compiler or interpreter, unfortunately for some goals, fortunately for others.

In terms of a computer program, I understood the snippet (which I reproduce below)

In pure water, the concentrations of the hydronium ion and the hydroxide ion are equal, and the solution is therefore neutral. If [H3O+] > [OH−], however, the solution is acidic, whereas if [H3O+] < [OH−], the solution is basic.

to be

with (pure water) do
  if [hydronium] = [hydroxide] then
    solution = neutral
  else if [hydronium] > [hydroxide] then
    solution = acidic
  else
    solution = basic
end do

At this point, let's not debate whether the author wanted to express this or not. The chemistry question that I reiterate here is

1) in pure water at 25C, there is only one possible combination of [hydronium] and [hydroxide], correct?

If we add a little bit of acid, then [hydronium] increases immediately.

Why can we still describe the ensuing equilibrium using $K_w$?

I think the answer is that the solution is so dilute that we can approximate the solution as being pure water.

2) Suppose we could somehow inject hydronium ions into pure water (is this possible without an acid?).

Would it be possible to have an equilibrium in pure water with [hydronium] higher than [hydroxide]?

 

[hutchphd]

Would it be possible to have an equilibrium in pure water with [hydronium] higher than [hydroxide]?

Aren't you describing a solution of hydrogen peroxide in pure water (I am no chemist)??

?

 

[zenterix]

Aren't you describing a solution of hydrogen peroxide in pure water (I am no chemist)??

I am not sure but it seems that hydrogen peroxide decomposes into water and molecular oxygen in water, so it seems that no, I am not referring to something like that reaction.

In addition, I have no idea what the oxygen does in the water, ie if it ionizes the water molecules.

Let me try to reiterate my question, so I can try to make it clear to myself as well.

Experimentally, in pure water we have $\mathrm{[H_3O^+]=[OH^-]=10^{-7}M}$.

If we add a small amount of acid, then the water is not pure any more. However, if the solution is very diluted, then it seems we can keep using the equilibrium constant $K_w$ that we use for autoionization in pure water for this new acidic solution.

So my question is: is it possible to make pure water acidic by somehow increasing the concentration of hydronium relative to hydroxide without adding anything else into the water (ie, no other ions)?

In other words, at a specific temperature, does the equation $\mathrm{K_w=[H_3O^+][OH^-]}$ denote only one possible combination of concentrations or can we actually get an infinite number of combinations?

 

[hutchphd]

I am not sure but it seems that hydrogen peroxide decomposes into water and molecular oxygen in water, so it seems that no, I am not referring to something like that reaction.

One can obtain 3% hydrogen peroxide and 15% hydrogen peroxide aqueous solutions., so your paradigm seems logically incorrect. Again I am not a chemist

 

[Borek]

In other words, at a specific temperature, does the equation $\mathrm{K_w=[H_3O^+][OH^-]}$ denote only one possible combination of concentrations or can we actually get an infinite number of combinations?

What is the question here, really?

Choose any reasonable value of [H⁺], calculate corresponding value of [OH^-]. Yes, there are infinitely many pairs that satisfy the equation.

Typically there are other restrictions (charge balance, mass balances) that make only one pair valid. For example for pure water charge balance limits possible solutions to [H⁺]=[OH^-].

 

[zenterix]

What is the question here, really?

Choose any reasonable value of [H⁺], calculate corresponding value of [OH^-]. Yes, there are infinitely many pairs that satisfy the equation.

Typically there are other restrictions (charge balance, mass balances) that make only one pair valid. For example for pure water charge balance limits possible solutions to [H⁺]=[OH^-].

Well, I asked two in my previous post

Is it possible to make pure water acidic by somehow increasing the concentration of hydronium relative to hydroxide without adding anything else into the water (ie, no other ions)?

In other words, at a specific temperature, does the equation Kw=[H3O+][OH−] denote only one possible combination of concentrations or can we actually get an infinite number of combinations?

Here are two more

Why is charge balance required?

Why can we not put just hydronium ions in pure water?

 

[DrJohn]

"Is it possible to make pure water acidic by somehow increasing the concentration of hydronium relative to hydroxide without adding anything else into the water (ie, no other ions)?"
No.

 

[DrJohn]

"Why can we not put just hydronium ions in pure water?"
Because you can't pick up a bucket of just one type of ion.

 

[Borek]

Why is charge balance required?

Calculate forces involved when 1 mole of unbalanced H⁺ in one bottle attracts 1 mole of unbalanced OH^- in another bottle. Assume the distance between the bottles to be 1 meter. Compare that to the force keeping Moon on the Earth's orbit.

In other words, at a specific temperature, does the equation Kw=[H3O+][OH−] denote only one possible combination of concentrations or can we actually get an infinite number of combinations?

I explained to you it is infinite number of combinations. Looks like you are not even reading answers given. EOT

 

[zenterix]

Calculate forces involved when 1 mole of unbalanced H⁺ in one bottle attracts 1 mole of unbalanced OH^- in another bottle. Assume the distance between the bottles to be 1 meter. Compare that to the force keeping Moon on the Earth's orbit.



I explained to you it is infinite number of combinations. Looks like you are not even reading answers given. EOT

I read all of the answers.

I am also very, very careful with the specific words I use when I write.

You wrote that "Yes, there are infinitely many pairs that satisfy the equation." and also that "Typically there are other restrictions (charge balance, mass balances) that make only one pair valid. For example for pure water charge balance limits possible solutions to [H+]=[OH-]."

I asked "does the equation Kw=[H3O+][OH−] denote only one possible combination of concentrations or can we actually get an infinite number of combinations?".

First of all, obviously there are infinitely many pairs that satisfy the mathematical equation.

I am asking about the real-world possibilities that arise from this conceptual framework.
You seem to have given the answer that charge balance limits the mathematical possibilities to just one real-world possibility.

I did not understand why charge balance is such a fundamental law, which is why I asked the additional questions

Why is charge balance required?

Why can we not put just hydronium ions in pure water?

You and DrJohn seem to have answered the first by saying that Coulomb forces would be unreasonably large.

The answer to the second question is not clear to me though.

In fact, if you look at the OP, it is still not clear to me, in the context of a general chemical reaction, how many combinations of partial pressures are real-world possibilities in chemical equilibrium.

The case of water was just to try to understand in a simpler case, but it has not really elucidated the question of a more general reaction for me.

 

[hutchphd]

Why can we not put just hydronium ions in pure water?

Because it would not then be "pure water". Also we are not God (How do you propose to do this? Please do the calculation as requested. )

What if pigs could fly??

 

[zenterix]

Because it would not then be "pure water". Also we are not God (How do you propose to do this? Please do the calculation as requested. )

What if pigs could fly??

I don't have a proposal, which is why I am asking.

It is not at all obvious to me that we can or can't put hydronium in water.

 

[hutchphd]

Your question then becomes easy to ask but impossible to answer.

 

[DrJohn]

You seem to be forgetting that if you add some acid to pure water, the acid dissociates and increases the amount of H+ ions. This disturbs the equilibrium and some of the H+ ions react with the OH- ions to form some more H2O, until the equilibrium is restored, and satisfies the Kw value. Whether you add a tiny amount of acid, or a large amount of acid, the position of equilibrium alters until the {H+].[OH-] = Kw.
Remember, this is a chemical reaction where water dissociates into ions, and when you add more of one particular ion, you are affecting the dissociation process momentarily, the ions combine and the equilibrium is restored.

 

[zenterix]

If we place a bar of solid zinc in water, does the water not begin to contain $\mathrm{Zn^{2+}}$ ions?

Is it not possible to do something like this with hydrogen?

Here is a standard hydrogen electrode:

Now, one question I have about this setup is why is it necessary to have that sulphuric acid in the water?

Apparently, there is a dynamic equilibrium between hydrogen gas molecules and hydrogen ions in solution.

$$\mathrm{H^+(aq)+2e^-\rightleftharpoons H_2(g)}$$

Why can't this equilibrium exist without the sulphuric acid?

From what I have read, the platinum catalyzes the reaction.

Anyway, if we could have just pure water (and no sulphuric acid), would not the water become more acidic?

 

[mjc123]

The equilibrium would exist without the sulphuric acid. But [H⁺] would be much smaller than 1M, so it wouldn't be a standard hydrogen electrode.

To return to your general point, you are possibly confusing infinite combinations with unlimited variability. Consider a reaction A + B → C + D with
K = [C][D]/[A][B ]
If you fix [A] and [B ], then [C][D] = K[A][B ] = constant, call it q.
If you plot a graph of [C] vs. [D], then the line [C][D] = q is (within reasonable physical limits) the locus of all combinations of [C] and [D] that satisfy the equilibrium. These are infinite because there are infinite points in a line, however short. But that does not mean that any point on the plane represents an equilibrium. [C] and [D] are not independently variable. There is 1 degree of freedom - you can move along the line, but not off it.
In the case of the water equilibrium, the same is true. Any combination of [H₃O⁺] and [OH^-] that satisfies [H₃O⁺][OH^-] = Kw is possible. But if [H₃O⁺] ≠ [OH^-], there must be other ions present to maintain charge balance.

 

[hutchphd]

we place a bar of solid zinc in water, does the water not begin to contain Zn2+ ions?

These are equilibrium constants. So wait a few hours (or years) and then ask what your system looks like. I am no chemist. .

 

[Mayhem]

The equilibrium would exist without the sulphuric acid. But [H⁺] would be much smaller than 1M, so it wouldn't be a standard hydrogen electrode.

To return to your general point, you are possibly confusing infinite combinations with unlimited variability. Consider a reaction A + B → C + D with
K = [C][D]/[A][B ]
If you fix [A] and [B ], then [C][D] = K[A][B ] = constant, call it q.
If you plot a graph of [C] vs. [D], then the line [C][D] = q is (within reasonable physical limits) the locus of all combinations of [C] and [D] that satisfy the equilibrium. These are infinite because there are infinite points in a line, however short. But that does not mean that any point on the plane represents an equilibrium. [C] and [D] are not independently variable. There is 1 degree of freedom - you can move along the line, but not off it.
In the case of the water equilibrium, the same is true. Any combination of [H₃O⁺] and [OH^-] that satisfies [H₃O⁺][OH^-] = Kw is possible. But if [H₃O⁺] ≠ [OH^-], there must be other ions present to maintain charge balance.

NHE requires pH 1 but SHE doesn't.