Ksp Question

[sanguinee]

I was hoping someone could help me make sense of this -- I'm following all of it until the explanation talks about cubing 5.5.. why is 5.5 cubed?

What is the minimum number of moles of Be(OH)2(s) must be added to one liter of water in order to create a saturated solution if the Ksp of Be(OH)2(s) is 6.92 x 10-22?
Ksp = [Be2+][OH-]2. So the expresion for molar solubility and Ksp becomes Ksp = (x)(2x)2 = 4x3. The value 6.92 x 10-22 can be approximated to ~10-21, so the cube root is about 10-7 (eliminate choices B and D). 6.92/4 is about 7/4 or 1.75. Cubing 5.5 gives a number greater than 125 and less than 216 (5 and 6 cubed, respectively), making choice C the best answer

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[NextStepTutor_3]

I actually think the way they answer this question is really convoluted. Either they left out several steps, they're wrong, or I'm missing something. Here's a MUCH easier way to do it:

We have 4x^3 = 6.92 x 10^-22. Now, right away we can see that we aren't going to want to use -22 as our exponent, because it isn't easily divisible by 3. In other words, taking its cube root would give us a fractional exponent. For some reason, they change their scientific notation to give 6.92 x 10^21 instead. This isn't ideal - we shouldn't just randomly change the exponent while leaving the coefficient alone.

Do this instead: change it into 692 x 10^-24. This is exactly the same as 6.92 x 10^-22, the exponent is divisible by 3, AND it gives us a nice big number to work with in the front (people tend to prefer working with large whole numbers instead of decimals). Now, we have 4x^3 = 692 x 10^-24, or about x^3 = 175 x 10^-24. Well, the cube root of 10^-24 is 10^-8, but what about the cube root of 175?

NOW we can estimate in the way they did at the end. What's 5 cubed? 125. What's 6 cubed? 216. So "5.something" cubed must be 175. The final answer will be something very close to 5.5 x 10^-8.

 

[sanguinee]

@NextStepTutor_3 Makes a lot of sense -- thank you!!