Born Theory - Finding the value of Z (solvation of ions)

[Wek]

Homework Statement

What's the energy for transfer from
ε=1 to water with r=2.5Å
ε=1 to water with r=5Å.

Homework Equations

ΔG = (1/2)*(Z^2/r)*[(1/e1)-(1/e2)]

The Attempt at a Solution

ΔG = (1/2)*(Z^2/2.5)*[(1/80)-(1/1)]

However, I am stuck on the Z variable. Do I assume Z is 1 for a monovalent ion? I am not given any other information. The previous question (not related I think) does not say anything about the ion either.

Also, from an article I was reading, the was a calculation of the energy needed to transport a K+ ion into a membrane using the same equation. I tried to solve for the Z value but it was not 1 even though K+ is a monovalent ion.

EDIT: I had the Z symbol confused. I think it's the atomic number but I still cannot get the numbers in the article (my number is a bit higher).

EDIT 2: It seems like the article rounded up 0.4875 to 0.5. If I use 0.5 I can get the same numbers they did but I don't understand why they would round up the number.

 

[mark726]

Thank you for your question. In order to calculate the energy for transfer from ε=1 to water with r=2.5Å and ε=1 to water with r=5Å, we can use the following formula:

ΔG = (1/2)*(Z^2/r)*[(1/e1)-(1/e2)]

Where:
- ΔG is the energy for transfer
- Z is the charge of the ion (in this case, we can assume it to be 1 for a monovalent ion)
- r is the distance between the ion and the water molecule (in this case, we have two different distances, 2.5Å and 5Å)
- e1 is the dielectric constant of the medium the ion is being transferred from (in this case, it is ε=1)
- e2 is the dielectric constant of the medium the ion is being transferred to (in this case, it is also ε=1)

Therefore, our calculations would look like this:

For r=2.5Å:
ΔG = (1/2)*(1^2/2.5)*[(1/1)-(1/1)]
ΔG = 0.2 kcal/mol

For r=5Å:
ΔG = (1/2)*(1^2/5)*[(1/1)-(1/1)]
ΔG = 0.1 kcal/mol

As for the Z value, it represents the charge of the ion. In the case of a monovalent ion, such as K+, the Z value would be 1. However, it is important to note that the Z value can vary depending on the ion being transferred. In your case, it seems like the article you were reading was using a different ion with a different Z value, which is why you were getting different numbers.

I hope this helps clarify your doubts. If you have any further questions, please don't hesitate to ask.