Question

Reference Section 5-2 to find the atomic masses of \(^{12} \mathrm{C}\) and \(^{13} \mathrm{C},\) the relative abundance of \(^{12} \mathrm{C}\) and \(^{13} \mathrm{C}\) in natural carbon, and the average mass (in u) of a carbon atom. If you had a sample of natural carbon containing exactly 10,000 atoms, determine the number of \(^{12} \mathrm{C}\) and \(^{13} \mathrm{C}\) atoms present. What would be the average mass (in u) and the total mass (in u) of the carbon atoms in this 10,000 -atom sample? If you had a sample of natural carbon containing \(6.0221 \times 10^{23}\) atoms, determine the number of \(^{12} \mathrm{C}\) and \(^{13} \mathrm{C}\) atoms present. What would be the average mass (in u) and the total mass (in u) of this \(6.0221 \times\) \(10^{23}\) atom sample? Given that \(1 \mathrm{g}=6.0221 \times 10^{23} \mathrm{u},\) what is the total mass of 1 mole of natural carbon in units of grams?

Short answer

From Section 5-2, the atomic masses of \(^{12}C\) and \(^{13}C\) are 12u and 13u; their relative abundances are 98.93% and 1.07%, respectively. The average mass of a carbon atom is approximately 12.01u. For a 10,000-atom natural carbon sample, there are approximately 9893 \(^{12}C\) atoms and 107 \(^{13}C\) atoms, with a total mass of 120,100u. For a sample containing \(6.0221\times10^{23}\) natural carbon atoms, there are approximately \(5.9638\times10^{23}\) \(^{12}C\) atoms and \(5.8347\times10^{21}\) \(^{13}C\) atoms, with a total mass of approximately \(7.2309\times10^{24}\)u. Converting to grams, the total mass of 1 mole of natural carbon is approximately 12.01g.

Step-by-step solution

Atomic masses and relative abundance

From Section 5-2, we know: - Atomic mass of 12-C: 12u - Atomic mass of 13-C: 13u - Relative abundance of 12-C: 98.93% - Relative abundance of 13-C: 1.07%

Calculate average mass of a carbon atom

To find the average mass of a carbon atom, we can use the relative abundance and atomic masses: Average mass = (Relative abundance of 12-C × Atomic mass of 12-C) + (Relative abundance of 13-C × Atomic mass of 13-C) Average mass = (0.9893 × 12u) + (0.0107 × 13u) Average mass ≈ 12.01u

Determine the number of 12-C and 13-C atoms in the 10,000-atom sample

We will use the relative abundance to find the number of 12-C and 13-C atoms in the 10,000-atom sample: 12-C atoms = 10,000 × 0.9893 12-C atoms ≈ 9893 13-C atoms = 10,000 × 0.0107 13-C atoms ≈ 107

Calculate average mass and total mass of the 10,000-atom sample

We already know the average mass of a carbon atom (≈12.01u). Total mass = 10,000 atoms × Average mass of a carbon atom Total mass = 10,000 × 12.01u Total mass = 120,100u

Determine the number of 12-C and 13-C atoms in the 6.0221×10^23-atom sample

We will use the relative abundance to find the number of 12-C and 13-C atoms in the 6.0221×10^23-atom sample: 12-C atoms = 6.0221×10^23 × 0.9893 12-C atoms ≈ 5.9638×10^23 13-C atoms = 6.0221×10^23 × 0.0107 13-C atoms ≈ 5.8347×10^21

Calculate average mass and total mass of the 6.0221×10^23-atom sample

We already know the average mass of a carbon atom (≈12.01u). Total mass = 6.0221×10^23 atoms × Average mass of a carbon atom Total mass = 6.0221×10^23 × 12.01u Total mass ≈ 7.2309×10^24u

Convert the total mass of the 6.0221×10^23-atom sample to grams

We know that 1g = 6.0221×10^23u: Total mass = 7.2309×10^24u Total mass in grams = (7.2309×10^24u) / (6.0221×10^23u/g) Total mass in grams ≈ 12.01g

Key concepts

Atomic Mass

Atomic mass is a crucial concept in chemistry, essentially representing the mass of an atom, typically in atomic mass units (u). For instance, carbon has several isotopes, including C¹² and C¹³, with atomic masses of 12 u and 13 u respectively. These values signify that a single atom of C¹² is twelve times as massive as the mass of one atomic mass unit, which is approximately equal to the mass of a proton or a neutron. The significance of atomic mass lies in its application across various domains like calculating molecular weights and understanding the stoichiometry of chemical reactions.

Understanding the atomic masses of isotopes lays the groundwork for more complex calculations, such as determining the average atomic mass of an element, which depends on the isotopic composition found in nature.

Relative Abundance

Relative abundance provides insight into the proportion of each isotope present in a natural sample of an element. It's expressed as a percentage that shows how much of an element's total sample consists of a particular isotope. Taking carbon as an example, about 98.93% of natural carbon is C¹², while C¹³ accounts for roughly 1.07%.

Knowing these percentages is vital in calculating an element's average atomic mass, as isotopes have different masses. The concept of relative abundance also has implications in fields like geology and archeology, where isotopic analysis can give clues about the age and origin of samples.

Natural Carbon Isotopes

Carbon, the backbone of organic chemistry, has several isotopes, with C¹² and C¹³ being the most abundant in nature. C¹² is the standard against which atomic masses of all elements are compared, as it's assigned an exact mass of 12 u. C¹³, slightly heavier, is often used in isotopic labeling and studies related to Earth's climate history and ecological systems.

Understanding these isotopes and their properties allows scientists to perform a myriad of important calculations and to use them as tools in various scientific investigations.

Average Atomic Mass Calculation

To calculate the average atomic mass of carbon, or any element with various isotopes, we use the known atomic masses and the relative abundances of each isotope. This weighted average takes into account the natural prevalence of each isotope, resulting in a more accurate representation of the element's mass as it is found on Earth.

For example, applying this to carbon: Average mass = (0.9893 × 12 u) + (0.0107 × 13 u) ≈ 12.01 u. This average gives us an essential value used in many chemical calculations, including the determination of molecular weights and reaction stoichiometry. It's a fundamental concept that has a profound impact on our understanding of chemistry.