Question

What is the number of carbon atoms in 0.5 nanomoles of carbon? One mole contains \(6.02 \cdot 10^{23}\) atoms. a) \(3.2 \cdot 10^{14}\) atoms e) \(3.19 \cdot 10^{17}\) atoms b) \(3.19 \cdot 10^{14}\) atoms f) \(3 . \cdot 10^{17}\) atoms c) \(3 .+10^{14}\) atoms d) \(3.2 \cdot 10^{17}\) atoms

Short answer

Answer: The approximate number of carbon atoms in 0.5 nanomoles of carbon is b) \(3.19 \times 10^{14}\) atoms.

Step-by-step solution

Understand the units of measurement

First, understand that "nanomole" means \(10^{-9}\) moles. So when we say 0.5 nanomoles, we should convert it to moles by multiplying 0.5 by \(10^{-9}\).

Convert 0.5 nanomoles to moles

To do the conversion, multiply 0.5 by \(10^{-9}\): $$ 0.5 \, \text{nanomoles} \times 10^{-9} \frac{\text{moles}}{\text{nanomole}} = 0.5 \times 10^{-9} \, \text{moles} $$

Use Avogadro's number to find the number of carbon atoms

Next, multiply the number of moles by Avogadro's number (\(6.02 \times 10^{23}\)) to find the number of carbon atoms: $$ (0.5 \times 10^{-9} \, \text{moles}) \times (6.02 \times 10^{23} \text{ atoms/mole}) $$

Perform the calculation

Now, perform the multiplication: $$ (0.5 \times 6.02 \times 10^{-9} \times 10^{23}) \, \text{atoms} $$

Simplify and find the final answer

After simplifying the expression, we get: $$ (3.01 \times 10^{14}) \, \text{atoms} $$ Looking at the options provided, the answer closest to \(3.01 \times 10^{14}\) atoms is option b) \(3.19 \times 10^{14}\) atoms.

Key concepts

Moles to Atoms Conversion

Converting moles to atoms is a fundamental task in chemistry that involves using Avogadro's number, a key concept in understanding the relationship between microscopic particles and measurable quantities. Avogadro's number, approximately \(6.02 \times 10^{23}\), represents the quantity of atoms or molecules in one mole of a substance.

To perform a moles to atoms conversion, simply multiply the number of moles by Avogadro's number. For instance, if you have 2 moles of helium, you would have \(2 \times 6.02 \times 10^{23}\) helium atoms. It's essential to handle exponentials correctly to ensure accuracy in your calculations, especially when dealing with non-standard units like nanomoles.

Let's not forget that a nanomole is \(10^{-9}\) moles, adding complexity to the task because it requires an understanding of exponential notation and the ability to work with very small numbers. Always convert nanomoles to moles before using Avogadro's number for a seamless and precise calculation.

Understanding Nanomoles

A nanomole is to a mole what a nanometer is to a meter - a billionth part of it. This tiny unit of measurement, nanomole (nmol), is particularly useful in biochemistry and molecular biology, where reactions often occur on a minuscule scale.

To conceptualize this, imagine taking a single mole (which is like a dozen, but instead contains \(6.02 \times 10^{23}\) entities) and slicing it into a billion equal parts; each part is one nanomole. So, when working with nanomoles, you're often dealing with tiny amounts of substances, which require precise calculations and conversions using scientific notation to handle the sheer difference in magnitude.

For example, to convert 500 nanomoles to moles, you would use the conversion factor of \(10^{-9}\) and multiply it by the amount in nanomoles: \(500 \times 10^{-9}\) moles. This conversion must be second nature to anyone frequently dealing with phenomena at the molecular level.

Molar Mass Calculation

Calculating molar mass is a critical skill in chemistry that allows us to translate between the weight of a substance and the number of moles. The molar mass is the weight of one mole (in grams) of a given substance and is numerically equivalent to the substance's atomic or molecular weight.

To calculate the molar mass of any element, you just look at the periodic table and note the atomic weight listed for that element. For example, carbon has a molar mass of approximately 12.01 g/mol. For compounds, you calculate the molar mass by summing the molar masses of all the individual atoms in the compound's formula.

As an example, to find the molar mass of water (H₂O), you would add together twice the molar mass of hydrogen (roughly 1.01 g/mol for each hydrogen atom) and the molar mass of oxygen (approximately 16.00 g/mol), giving you about 18.02 g/mol. With the molar mass known, conversions between mass and moles become straightforward, allowing for a clearer understanding of chemical reactions and stoichiometry.