[trinitrotoluene]

Hopefully this is in the right forum, if not my apologies.

I am currently studying the Eyring equation. I am caught up on what seems to be a seemingly easy conversion going to d.r. to relative rates.

ka/kb     -     d.r.
2:1              66.6:33.3
10:1             91:9
100:1           99:1
10000:1        99.95:0.05

The first one makes perfect sense on going to d.r. to relative rates but I am not sure how the others were determined. Does anyone have a link on how to make this conversion or can explain it? I think I may just be getting caught up with rounding.

[ziqquratu]

I'd say that you're right - focus on the exact number is confusing you.

Try thinking in fractions:
A 2:1 ratio of k_a:k_b leads to ²/₃ of a and ¹/₃ of b - hence 66.6% a and 33.3% b or 66.6:33.3 a:b
10:1 leads to ¹⁰/₁₁ of a and ¹/₁₁ of b - 90.90:9.09, which is approximately 91:9
100:1 leads to ¹⁰⁰/₁₀₁ of a and ¹/₁₀₁ of b - 99.01:0.99, which is approximately 99:1

And so on.

Hope that helps!

[trinitrotoluene]

I'd say that you're right - focus on the exact number is confusing you.

Try thinking in fractions:
A 2:1 ratio of k_a:k_b leads to ²/₃ of a and ¹/₃ of b - hence 66.6% a and 33.3% b or 66.6:33.3 a:b
10:1 leads to ¹⁰/₁₁ of a and ¹/₁₁ of b - 90.90:9.09, which is approximately 91:9
100:1 leads to ¹⁰⁰/₁₀₁ of a and ¹/₁₀₁ of b - 99.01:0.99, which is approximately 99:1

This is exactly what I needed. thank you.

And so on.

Hope that helps!

[trinitrotoluene]

I thought I had this figured out, and I kind of do. Given the relative rates, I can go to d.r using the fraction method previously described. But given the d.r. how could u find the relative rates?

If given d.r. = 93:7, how would you determine the relative rates?

[ziqquratu]

I'd simply divide the larger by the smaller number, giving you the relative rates (or k_a relative to k_b).

So, 93 / 7 = 13.3:1 (call it 13:1)

As a check, convert that back to d.r. by our previous method:
¹³/₁₄ x 100 = 92.8 and ¹/₁₄ x 100 = 7.1, which rounds off to 93:7!