# What kind of 'product' is the 'product' in 'ionic product of water'?

My language has different words for product (=the result of a process) and product (=the sum of multiplied quantities), so I need to know the right meaning to choose the right word.

Which of the meanings above is present in the English term 'ionic product of water' ($K_\mathrm{w}$)?

This definition from http://www.chemguide.co.uk/physical/acidbaseeqia/kw.html seems to suggest it is a sum of multiplied quantities:

The units of $K_\mathrm{w}$: $K_\mathrm{w}$ is found by multiplying two concentration terms together. Each of these has the units of $\mathrm{mol~dm^{-3}}$. Multiplying $\mathrm{mol~dm^{-3}} \times \mathrm{mol~dm^{-3}}$ gives you the units above.

However, the use of the verb produce in this sentence from https://en.wikipedia.org/wiki/Self-ionization_of_water suggests something that is produced as a result of a process:

According to the theories of Svante Arrhenius, this must be due to the presence of ions. The ions are produced by the self-ionization reaction $\ce{H2O + H2O <=> H3O+ + OH-}$

Alternatively, could both meanings apply depending on your perspective?

Thanks in advance for any clarification.

• It refers to multiplication. – Zhe Jan 24 '17 at 17:59

Per Zhe's comment, in the specific context of the term ionic product of water, the word product is used in the sense of the result of the mathematical operation of multiplication:

The ionic product of water equals the concentration of $\ce{H+}$ times the concentration of $\ce{OH-}$:
$$K_\mathrm w = \ce{[H+][OH-]}$$

(Strictly, per MaxReid's comment, the ionic product water is of course the product of the activities of $\ce{H+}$ and $\ce{OH-}$, but concentrations are commonly used in routine/non-rigorous calculations as they provide good accuracy for dilute solutions.)

This sense of the word product applies any time one is referencing the mathematical description of the equilibrium among the ionic forms of a given species. This usage also includes the solubility product (again with the above caveat about activities):

The solubility product of silver chloride equals the concentration of $\ce{Ag+}$ times the concentration of $\ce{Cl-}$:
$$K_\mathrm{sp} = \ce{[Ag+][Cl-]}$$

As an aside, note that the main exception to this terminology is when the expression describing the distribution of species requires division by one or more quantities. In this case, the term to use is quotient, for which I expect little ambiguity exists:

The reaction quotient of the reaction
$$\ce{A + B->C + D}$$ is
$$Q = {\ce{[C][D]}\over \ce{[A][B]}}$$

However, to be complete: chemistry does also use the word product in the other sense you describe, as in something being produced by a reaction. Due to this, the word product in the phrase the ionic product(s) of X could also bear this meaning. For example:

The ionic products of the aqueous decomposition of dinitrogen tetroxide at neutral $\mathrm{pH}$ are nitrite and nitrate ion:
$$\ce{N2O4(aq) + 2H2O -> NO2-(aq) + NO3-(aq) + 2H3O+}$$

In general, I think the difference boils down to whether one is talking about the ionic distribution of a specific species of interest, as opposed to the transformation that occurs in a particular reaction of interest.

As I noted above, though: In the specific case you are asking about, product is used in the sense of the result obtained from multiplication.

• Thank you for the clear and unambiguous answer. It is much appreciated. – PrettyHands Jan 24 '17 at 18:34
• @PrettyHands Well, now that I think about it -- the answer isn't so cut and dried. See edit. – hBy2Py Jan 24 '17 at 18:48
• @PrettyHands And, well, now that I think about it some more: for the specific case you're asking about, it is cut and dried. See further edit. – hBy2Py Jan 24 '17 at 19:06
• @hBy2Py Isn't it defined as the product of the thermodynamic activities? In this case it should be equivalent, but for completeness sake I'd like to mention this. – Max Ried Jan 25 '17 at 5:51
• @MaxRied Strictly, you are of course correct. I'll edit. – hBy2Py Jan 25 '17 at 12:00

The product $K_w$ is a mathematical construct.

The ion species produced from ionization of water are not mathematical constructs. They are resulting entities (actual, tangible "things").

The word does have two meanings, but the essence in both cases is "result". There isn't really a case when both meanings would apply to the same "result".

• Thank you. So in the term 'ionic product of water' specifically, it refers to multiplication? – PrettyHands Jan 24 '17 at 18:22
• I think your last paragraph doesn't have the right focus. The gist to me is that just the phrase "ionic product of water" is ambiguous. You have to understand the context in which the phrase is being used to be sure if the mathematical formula, or a chemical species is being discussed. // Needing context is fairly common in English. Look at definition for shoe. merriam-webster.com/dictionary/shoe – MaxW Jan 24 '17 at 18:25
• I agree with Max. Yet the phrase 'ionic product of water' is reserved as the well-known $K_w$. – khaverim Jan 24 '17 at 18:29