(a) What is the mass, in grams, of a mole of \({ }^{12} \mathrm{C} ?\) (b) How many carbon atoms are present in a mole of \({ }^{12} \mathrm{C} ?\)

Mole (symbol: mol) is a unit of measurement in chemistry that defines the amount of any substance. A mole of any substance has the same number of constituent particles (like atoms, ions, or molecules) as there are in 12 grams of pure carbon-12 isotope. This number of particles is known as Avogadro's number (N_A), which is approximately \(6.022 \times 10^{23}\) particles per mole. The atomic mass of an element is the average mass of the atoms of an isotope, scaled by the atomic mass unit (u), which is defined as one twelfth of the mass of one atom of carbon-12. For carbon, the atomic mass of carbon-12 isotope is \(12 u\), where \(1 u = 1.66 \times 10^{-24} g\).

To find the mass of one mole of carbon-12, we will convert the atomic mass from atomic mass units (u) to grams. Given that the atomic mass of carbon-12 is \(12 u\), we can write: \(mass_{C-12} = 12 u * 1.66 \times 10^{-24} \frac{g}{u}\) Using the conversion factor, we get: \(mass_{C-12} = 12 * 1.66 \times 10^{-24} g\) Now, multiply the values: \(mass_{C-12} = 1.99 \times 10^{-23} g\) Since we are looking for the mass of one mole of carbon-12, we need to multiply the mass of one carbon-12 atom by Avogadro's number: \(mass_{1 mol \ C-12} = N_A * mass_{C-12}\) So, we have: \(mass_{1 mol \ C-12} = (6.022 \times 10^{23}) * (1.99 \times 10^{-23} g)\) \(mass_{1 mol \ C-12} = 12 g\) (a) Therefore, the mass of one mole of carbon-12 is 12 grams.

To find the number of carbon atoms in one mole of carbon-12, we will use Avogadro's number. Since one mole of any substance has \(6.022 \times 10^{23}\) constituent particles, one mole of carbon-12 has: \(N_{atoms \ in \ 1 \ mol \ C-12} = N_A\) \(N_{atoms \ in \ 1 \ mol \ C-12} = 6.022 \times 10^{23}\) (b) Therefore, there are \(6.022 \times 10^{23}\) carbon atoms in one mole of carbon-12.