Question

Which has more atoms: a 1-gram sample of carbon-12 or a 1-gram sample of carbon-13? Explain.

Short answer

A 1-gram sample of carbon-12 has more atoms than a 1-gram sample of carbon-13.

Step-by-step solution

Determine the Number of Moles in Each Sample

To find out how many moles of carbon are in each sample, we use the formula for moles: \( n = \frac{m}{M} \), where \( m \) is the mass of the substance and \( M \) is the molar mass.* For carbon-12: \( M = 12 \, \text{g/mol} \) \[ n = \frac{1 \, \text{g}}{12 \, \text{g/mol}} = 0.0833 \, \text{mol} \]* For carbon-13: \( M = 13 \, \text{g/mol} \) \[ n = \frac{1 \, \text{g}}{13 \, \text{g/mol}} = 0.0769 \, \text{mol} \]

Calculate the Number of Atoms in Each Sample Using Avogadro's Number

To find the number of atoms, we multiply the number of moles by Avogadro's number \( N_A = 6.022 \times 10^{23} \, \text{atoms/mol} \).* For carbon-12: \[ \text{Number of atoms} = 0.0833 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 5.02 \times 10^{22} \, \text{atoms} \]* For carbon-13: \[ \text{Number of atoms} = 0.0769 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 4.63 \times 10^{22} \, \text{atoms} \]

Compare the Number of Atoms

After calculating the number of atoms in each sample, compare the results:* Carbon-12 has approximately \( 5.02 \times 10^{22} \) atoms.* Carbon-13 has approximately \( 4.63 \times 10^{22} \) atoms.The sample of carbon-12 has more atoms than the sample of carbon-13.

Key concepts

Moles Calculation

The concept of moles calculation is crucial in chemistry for quantifying the amount of substance. Moles serve as a bridge between the atomic scale and the real world that we can measure. When calculating moles, you use the formula:

• \( n = \frac{m}{M} \)

where \( n \) is the number of moles, \( m \) is the mass of the substance in grams, and \( M \) is the molar mass of the substance, which is usually expressed in grams per mole (g/mol). This calculation is foundational because it allows conversion from the mass of a sample, like carbon-12 or carbon-13, into moles which provide a direct connection to the number of particles.

For example, when you have a 1-gram sample of carbon-12, with a molar mass of 12 g/mol, the number of moles \( n \) is calculated as follows:

• \( n = \frac{1 \, \text{g}}{12 \, \text{g/mol}} = 0.0833 \, \text{mol} \)

This same calculation applies to any element or compound you're dealing with, adapting \( M \) to match the specific substance.

Molar Mass Comparison

Comparing molar masses is important when assessing different isotopes or different compounds because it influences the number of moles present in a sample.

• Carbon-12 has a molar mass of 12 g/mol.
• Carbon-13 has a molar mass of 13 g/mol.

This slight difference in molar mass between the isotopes of carbon means that for the same mass of each (e.g., 1 gram), the number of moles will differ.

A higher molar mass, like that of carbon-13 compared to carbon-12, results in fewer moles when you have the same mass. This is because more mass per mole (higher molar mass) reduces the number of moles \( n = \frac{m}{M} \).

When comparing the calculations:

• For carbon-12: \( n = 0.0833 \) mol
• For carbon-13: \( n = 0.0769 \) mol

Thus, carbon-13 has fewer moles than carbon-12 when each has a mass of 1 gram.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry that defines the number of particles (usually atoms or molecules) present in one mole of a substance. This value, \( N_A = 6.022 \times 10^{23} \) particles per mole, creates a crucial relationship between moles and the actual number of atoms or molecules.

In solving the original problem, once the number of moles is known, Avogadro's number can be used to find the exact number of atoms. For carbon-12, with 0.0833 moles:

• \( \text{Number of atoms} = 0.0833 \, \text{moles} \times 6.022 \times 10^{23} \, \text{atoms/mole} \approx 5.02 \times 10^{22} \) atoms

Similarly, for carbon-13, with 0.0769 moles:

• \( \text{Number of atoms} = 0.0769 \, \text{moles} \times 6.022 \times 10^{23} \, \text{atoms/mole} \approx 4.63 \times 10^{22} \) atoms

By multiplying the moles by Avogadro’s number, we convert a relatively small, abstract number into a count of tangible particles, facilitating easy comparison between different samples, like carbon-12 and carbon-13 in this case.