How do we know when x is negligible when calculating equilibrium concentrations?
Actually you don't REALLY know if x is small enough to neglect. My advice has always been to work the problem neglecting x. When you get the answer you decide if x can be neglected; i.e., you see if you should have done it the long way. Here is an example:
Calculate the (H^+) and (Ac^-) in a 0.1 M solution of acetic acid(HAc)
......................HAc --> H^+ + Ac^-
I.......................0.1.........0..........0
C.......................-x............x..........x
E...................0.1-x...........x..........x
Ka = 1.8E-5 = (x)(x)/1.8E-5
x^2 = 1.8E-6. See I've neglected the x and done it the easy way.
x = 1.3E-3 or 0.0013
Now you decide if this is OK or if you should have done it the long way.
There are two ideas out there. One is the 5% rule; the other is the 10% rule. Take your pick. I prefer the 10% rule but that's what I used when I was a student which was a LONG TIME AGO when Ka values were not much better than 10%. Now days Ka values are better than that and 5% may be justified. Anyway, using the 10% rule you do this.
The answer is 0.0013 M for (H^+)
So 10% of the 0.1 is 0.1 x 0.1 = 0.01. The answer when neglecting x is 0.0013 which is less than the 10% so the shortcut is OK and no need to redo it the long way with the quadratic. Hope this is what you needed.
oops. BIG typo.
Here is what I typed.
Ka = 1.8E-5 = (x)(x)/1.8E-5
It should have been this.
Ka = 1.8E-5 = (x)(x)/(0.1-x)
Then neglecting x we get x^2 = 1.8E-6 etc.
Do you divide the concentration given in the question by the Kb or Ka value or is that incorrect?
Ah, the eternal battle between x and equilibrium. Determining when x is negligible depends on the context, my friend. But fear not, I shall attempt to enlighten you with some laughs along the way!
1. The "Blink and You'll Miss It" Rule: If x is so small that you can blink and miss it, then it's safe to say it's negligible. Just like that ice cream cone that mysteriously vanished when you blinked!
2. The "Tiny Tim" Rule: If x is as tiny as a flea riding a grain of sand on a sunny beach, it's probably negligible. I mean, who would even notice Tiny Tim the flea? Poor fella.
3. The "David vs. Goliath" Rule: If x is so small compared to the initial concentrations that it's like David trying to take down Goliath with a toothpick, then x can be considered negligible. Sorry, David. Toothpicks won't cut it.
4. The "Whisper in a Hurricane" Rule: If x is a mere whisper of a quantity amidst the roaring hurricane of initial concentrations, then it's safe to deem it negligible. Whispering in a hurricane? Good luck with that!
Remember, these are just guidelines to get you started. Your specific situation and the values of x will determine if it's truly negligible. And if in doubt, consult your textbook or professor. Good luck, brave equilibrium warrior!
To determine when the concentration of a reactant or product can be considered negligible when calculating equilibrium concentrations, you need to compare it to the initial concentration or the equilibrium constant (K).
In general, a reactant or product is considered negligible if its concentration decreases by more than 5% of the initial concentration at equilibrium. This ensures that its contribution to the equilibrium expression is minimal and would not significantly affect the calculated equilibrium concentrations.
Here's how you can know when x is negligible:
1. Write the balanced chemical equation for the reaction and assign the stoichiometric coefficients (a, b, c, d) to each reactant and product.
2. Define the initial concentrations of the reactants and products.
3. Use an ICE table (Initial concentrations, Change, Equilibrium concentrations) to express the concentration changes in terms of the variable "x" (representing changes in concentration).
4. Substitute the equilibrium concentrations (expressed in terms of "x") into the equilibrium expression based on the balanced equation.
5. Solve the expression for "x" algebraically using mathematical tools (quadratic equation, approximation techniques) or graphical methods.
6. Once you find the value of "x," compare it to the initial concentrations or the equilibrium constant (K). If "x" is significantly smaller than the initial concentration or K, then you can consider it negligible.
7. If "x" is negligible, you can assume that the initial concentration minus "x" is approximately equal to the equilibrium concentration for that species.
8. Calculate the equilibrium concentrations using the values of "x" for each species in the balanced equation.
Remember that this determination of negligible concentrations is an approximation, and your calculations could be affected if the concentration change is significant. It's always important to check the validity of your approximation by comparing the equilibrium concentrations calculated using the approximation with the initial concentration. If there is a significant difference, you may need to reconsider your assumption and use a different approach, like solving the equilibrium equation iteratively.