Question

A blood sample is found to contain \(8.6 \mathrm{mg} / \mathrm{dL}\) of Ca. How many atoms of \(\mathrm{Ca}\) are present in \(8.6 \mathrm{mg}\) ? The atomic mass of Ca is 40.08 amu.

Short answer

Approximately \(1.291 \times 10^{20}\) calcium atoms are present.

Step-by-step solution

Convert mg to grams

First, convert the mass of calcium from milligrams to grams, as the molar mass is usually in grams per mole. Since 1 milligram is equal to 0.001 grams:\[8.6 \, \text{mg} = 8.6 \, \text{mg} \times 0.001 \, \frac{\text{g}}{\text{mg}} = 0.0086 \, \text{g} \text{ of } \text{Ca}\]

Calculate moles of Calcium

Next, use the atomic mass of calcium to find the number of moles. The atomic mass of calcium is 40.08 amu, which is equivalent to grams per mole:\[\text{Moles of Ca} = \frac{0.0086 \, \text{g}}{40.08 \, \text{g/mol}} \approx 0.0002146 \, \text{mol of Ca}\]

Calculate atoms of Calcium

Finally, calculate the number of calcium atoms using Avogadro's number (approximately \(6.022 \times 10^{23} \, \text{atoms/mol}\)):\[\text{Atoms of Ca} = 0.0002146 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 1.291 \times 10^{20} \, \text{atoms}\]

Key concepts

Mole Concept

The mole concept is a fundamental principle in chemistry that bridges the microscopic world of atoms and molecules with the macroscopic world we can observe. Imagine counting something as small as atoms—it's no easy task. That's where the mole comes in handy.

• One mole of any substance contains exactly Avogadro's number of particles (atoms, molecules, ions, etc.).
• This number, Avogadro's number, is approximately \(6.022 \times 10^{23}\).
• A mole helps chemists measure amounts of a substance in a meaningful way, similar to how a dozen refers to twelve items.

When you are given a mass of an element, like in our exercise, converting this mass to moles allows you to determine how many atoms are present. It's a key step that acts as a bridge.Understanding the mole concept effectively makes it possible to quantify the atoms in a given sample by first translating the mass to moles.

Atomic Mass Conversion

Converting atomic mass units (amu) to practical quantities like grams per mole is vital in stoichiometry. This conversion facilitates the link between the mass of an element and the number of atoms it contains.

• Atomic mass, given in amu, effectively tells us the relative weight of an atom compared to a standard, typically carbon-12.
• In practice, when you see an atomic mass, you can interpret it as grams per mole, allowing for seamless mass-to-moles conversion.
• For calcium, with an atomic mass of 40.08 amu, this translates directly into 40.08 grams per mole.

This conversion is essential for performing calculations like in our exercise. By changing the mass from milligrams to grams and using the atomic mass, you can easily find how many moles—and thus how many atoms—are present in a sample.

Avogadro's Number

Avogadro's number is a pillar of chemistry, essential for connecting theoretical calculations with real-world quantifications. Named after Amedeo Avogadro, its approximately \(6.022 \times 10^{23}\) number represents a fundamental constant in the molecular and atomic scale world.

• Avogadro's number defines the quantity of particles, whether atoms or molecules, in one mole of a substance.
• This constant allows chemists to convert moles to number of atoms or molecules, an otherwise intangible quantity.
• With Avogadro's number, the step from moles to individual atoms becomes straightforward, utilizing multiplication.

In problem-solving, such as our exercise on calcium atoms, once you know the amount in moles, Avogadro's number provides the exact count of atoms present. This powerful tool transforms abstract chemical data into practical understanding.