Reef Chemistry Question of the Day #241 How Many Water Molecules Between the Ions in Seawater?

[Randy Holmes-Farley]

Reef Chemistry Question of the Day #241

On average, how many water molecules are directly between one ion in seawater, and its next nearest neighbor?

For example, for a given magnesium ion, how far away is the nearest other ion (sodium, calcium, chloride, sulfate, etc.) in terms of the number of water molecules directly between them.

A. 3.66 water molecules
B. 13.4 water molecules
C. 49.1 water molecules
D. 61.6 water molecules
E. 137.1 water molecules

For purposes of answering this question, we will assume that:

1. The water concentration in seawater is 55 M (molar)
2. The total concentration of ions in seawater is 1.12 M (molar)

If you are familiar with ion pairing, ignore that for purposes of this question. I'm looking for the neighbor distance for an ion not attached to the first ion. I'll discuss this detail in the answer.

 

[Randy Holmes-Farley]

Bump

 

[JimWelsh]

From the choices given, I choose "A. 3.66 water molecules", but my answer was 2.69.

I calculated the average distance between ions:
(1000 cm^3 / 1.12M * 6.022E+23 / (4/3*3.14159)) ^ (1/3) * 2 = 1.415 nm.

I then calculated the average diameter of the space occupied by the sum of 55M of H2O + 1.12M other ions:
(1000 cm^3 / 56.12M * 6.022E+23 / (4/3*3.14159)) ^ (1/3) * 2 = 0.384 nm.

Now, since I assume that the ions are not points of absolute zero size, but instead that they occupy this average space occupied by 56.12M particles per liter, the average distance between the outer edges of the space occupied by the ions is 1.415 nm - 0.384 nm = 1.031 nm, which when divided by this same average diameter of 0.384 nm = 2.69 water molecules. My model assumes that each salt ion/water molecule occupies an equal proportion of the volume, and also accounts for the space occupied by the ions.

I find that if I calculate the average diameter of the space occupied by only the 55M of water in a liter:
(1000 cm^3 / 55M * 6.022E+23 / (4/3*3.14159)) ^ (1/3) * 2 = 0.386 nm,

...and divide the 1.415 nm distance between ions by that 0.386 nm average diameter of a water molecule without accounting for the space occupied by the other ions, I get 3.66 (which is what I presume you did).

EDIT: Of course, since all of the terms except the molarity are the same in each calculation, then my answer can be simplified to (56.12^(1/3) / 1.12^(1/3))-1, and Randy's answer becomes just 55^(1/3) / 1.12^(1/3).

 

[Randy Holmes-Farley]

From the choices given, I choose "A. 3.66 water molecules", but my answer was 2.69.

I calculated the average distance between ions:
(1000 cm^3 / 1.12M * 6.022E+23 / 3.14159) ^ (1/3) * 2 = 1.557 nm.

I then calculated the average diameter of the space occupied by the sum of 55M of H2O + 1.12M other ions:
(1000 cm^3 / 56.12M * 6.022E+23 / 3.14159) ^ (1/3) * 2 = 0.422 nm.

Now, since I assume that the ions are not points of absolute zero size, but instead that they occupy this average space occupied by 56.12M particles per liter, the average distance between the outer edges of the space occupied by the ions is 1.557 nm - 0.422 nm = 1.135 nm, which when divided by this same average diameter of 0.422 nm = 2.69 water molecules. My model assumes that each salt ion/water molecule occupies an equal proportion of the volume, and also accounts for the space occupied by the ions.

I agree that there are multiple ways to interpret molecule volumes. I assumed a cube of volume to occupy all space. You assumed spheres.

If you pack them all in as spheres, what is in the gaps between the edges of the spheres?

 

[JimWelsh]

Fine. Let's use cubes. That doesn't change the result. I stand by my answer (edited again to fix the "4/3" part I left out).

 

[Randy Holmes-Farley]

Fine. Let's use cubes. That doesn't change the result. I stand by my answer (edited again to fix the "4/3" part I left out).

lol Glad you are here to check up on me.

It's the assumptions that make the difference. Neither of us has a perfect set of assumptions to get a perfect answer, but your assumptions were definitely better.

I assumed the ions had no size, and you assumed they are the same as water molecules. Your assumption is clearly more accurate. In my weak defense, I initially thought the answer was going to be large enough that such a difference wouldn't matter, but it clearly does because the ions are actually a lot closer together than I had thought.


Here's a slightly different way to describe how to get both of these answers:

In a liter we have 55 moles of water molecules and 1.12 moles of ions. So there are 55/1,12 = 49.1 water molecules for each ion.

Consequently, each ion is surrounded by 49.1 water molecules.

If we imagine the ion as the center of a cube, and no size for the ion (come back to this later) each cube will contain 49.1 water molecules, and so has a length of a side of the cube of dimension 49.1^(1/3) = 3.66 water molecules. That is, each cube is a stack of 3.66 x 3.66 x 3.66 water molecules.

In this configuration, the "distance" between the centers of adjacent cubes is the same as the side of a cube. One half to get from the center of one ion to the edge of the cube, and another half to get from the edge to the center of the next ion over. Answer: 3.66 water molecules of distance between them

Jim assumed the ions have size, which, of course, they do. He assumed they are similar sized to water molecules. This gets into weird chemistry of the relative "size" of something that is round (a free sodium ion) compared to something that is not (a water molecule), but let's assume they are the same.

So the cube now holds 49.1 water molecules plus one ion, for a total of 50.1. The cube would now be 50.1^(1/3) = 3.69 entities on a side. That gives a "total distance" between the centers of the ions of 3.69 water molecules.

BUT, part of that total is the ion itself, and not water molecules between them. So we subtract one water molecule of distance to get 2.69. Jim's answer, which is (I think) about as good as we can get in this sort of simplistic analysis.

Happy Reefing!

 

[MnFish1]

Well - part of it would depend on the temperature of the water. And I thought that water has a relative charge (positive on the hydrogen side, negative on Oxygen side). which means ions of different charge may not be equally distributed in the water (at least not enough to calculate a distance). And I would also assume that a magnesium ion would tend to be closer to a chloride ion than a sodium ion. . Probably overthinking the question though

 

[JimWelsh]

I gotta say I too am surprised at the small number of water molecules between ions. It sheds a whole new light (for me) on the relatively high ionic strength of seawater.