What is the edge length (in cm) of a titanium cube that contains
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Question

What is the edge length (in cm) of a titanium cube that contains
2.55 * 1024 titanium atoms? The density of titanium is
4.50 g/cm3.

Accepted Answer

Welcome back everyone. A copper cube with a density of 8.96 g per cubic centimeter contains 1.58 times 10 to the 24th power copper atoms. We need to calculate the edge length in centimeters of this cube recall that edge length is represented by the variable s where we would find our volume of our cube by taking our edge length cubed. And that means that therefore because our edge length should be our final answer, our edge length will be found by taking the cube root of our volume. We don't know our volume of this cube. So we're going to have to solve for it. And we would find the volume of the cube by taking our formula where rho for density is related to the mass divided by our volume of our atom. And so in this case we want to isolate for volume. So we would multiply both sides by our denominator volume where we would have pressure or sorry, rho times volume equal to mass. And that means we would have to divide both sides by row so that we have volume equal to mass over rho for density. And so now we just need to find our mass because we also don't know the mass of our copper atom. And so we do know the number of atoms within copper given in the prompt as 1.58 times 10 to the 24th power copper atoms. And so to find our mass of copper we're going to multiply this by our stock geometry conversion factor to go from atoms of copper in the denominator, two moles of copper in the numerator. And we'll just move this over to make more room. We would recall avocados number which tells us that in our denominator we have 6.22 times 10 to the 23rd power atoms of copper equivalent to one mole. So canceling out atoms of copper. We have moles of copper but we need mass. So we're going to multiply by our next conversion factor to go from moles of copper in the denominator. two g of copper in the numerator. Recall that we can utilize our molar mass of copper. We're on our product table. We see that for one mole of copper we have a mass of 63.5 46 g of copper in our numerator. So canceling out moles of copper. We have grams of copper as our final unit and we'll find that copper's mass is equal to 166.7 26 g of copper making up our cube. Now with this mass, this is going to be step one, we're going to go into step two which utilizes our formula for volume. Where we stated that our volume is equal to our mass which the numerator will plug in as 166.7 26 g of copper divided by rho for density where we're given density as 8.96 g per cubic centimeter in the denominator. And so canceling out our unit of grams. We're left with cubic centimeters as our final unit, which is what we want for volume. And this will simplify in our calculators to a volume of our copper cube as 18.64 cubic centimeters. Now that we have this volume, we can get to step three which is our final step of this prompt in which we would take or we would find our edge length by taking the cube root of our volume, which we just found as 18.64 cubic centimeters. So taking the cube root will cancel out cubic centimeters, leaving us with centimeters as our final unit for edge length. And this will simplify to an edge length equal to 2.65 centimeters. Sorry, this says centimeters. And this would be our final answer as the edge length of our copper cube which corresponds to choice a in the multiple choice as the correct answer. So I hope this made sense. And let us know if you have any questions

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