Question

Suppose that the atomic mass of \(\mathrm{C}-12\) is taken to be 5.000 amu and that a mole is defined as the number of atoms in \(5.000 \mathrm{~kg}\) of carbon-12. How many atoms would there be in one mole under these conditions? (Hint: There are \(6.022 \times 10^{23} \mathrm{C}\) atoms in \(12.00 \mathrm{~g}\) of \(\left.\mathrm{C}-12 .\right)\)

Short answer

Answer: Approximately \(2.508 \times 10^{24}\) Carbon atoms.

Step-by-step solution

Convert kg to grams

First, we need to convert the 5.000 kg of Carbon-12 to grams. As 1 kg = 1000 g, we will multiply the given value of kg by 1000. 5.000 kg * 1000 = 5000 g

Calculate the number of moles

We will now use the hint provided, which says that there are \(6.022 \times 10^{23}\) Carbon atoms in 12.00 g of Carbon-12. We can set up a proportion to find the number of moles in 5000 g of Carbon-12: \(\frac{6.022 \times 10^{23} \ \mathrm{atoms}}{12.00 \ \mathrm{g}} = \frac{x \ \mathrm{atoms}}{5000 \ \mathrm{g}}\) Here, x represents the number of Carbon atoms in 5000 g of Carbon-12.

Solve for x

We will now cross-multiply and solve for x: \((6.022 \times 10^{23} \ \mathrm{atoms}) * (5000 \ \mathrm{g}) = (12.00 \ \mathrm{g}) * x \ \mathrm{atoms}\) Now, we will divide both sides of the equation by 12.00 g: \(x \ \mathrm{atoms} = \frac{(6.022 \times 10^{23} \ \mathrm{atoms}) * (5000 \ \mathrm{g})}{12.00 \ \mathrm{g}}\)

Calculate the value of x

Finally, we will find the value of x: \(x \ \mathrm{atoms} = \frac{(6.022 \times 10^{23})*(5000)}{12.00}\) \(x \ \mathrm{atoms} = 2.508 \times 10^{24}\) So there would be approximately \(2.508 \times 10^{24}\) Carbon atoms in one mole under the given conditions.