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 d) \(3.2 \cdot 10^{17}\) atoms b) \(3.19 \cdot 10^{14}\) atoms e) \(3.19 \cdot 10^{17}\) atoms c) \(3 . \cdot 10^{14}\) atoms f) \(3 . \cdot 10^{17}\) atoms

Short answer

a) \(6.02 \cdot 10^{23}\) atoms b) \(3.19 \cdot 10^{14}\) atoms c) \(5.0 \cdot 10^{-10}\) atoms d) \(2.50 \cdot 10^{9}\) atoms Answer: b) \(3.19 \cdot 10^{14}\) atoms

Step-by-step solution

Convert nanomoles to moles

To convert the amount of carbon from nanomoles (nmol) to moles (mol), we need to remember the relationship between these units: 1 mole = \(10^9\) nanomoles. So to convert 0.5 nmol of carbon to moles, simply divide by \(10^9\): $$ 0.5 \ nmol \cdot \frac{1 \ mol}{10^9 \ nmol} = 0.5 \times 10^{-9} \ mol $$

Calculate the number of carbon atoms

Now that we have the amount of carbon in moles, we can use Avogadro's number (\(6.02 \cdot 10^{23}\) atoms/mol) to calculate the number of carbon atoms. Multiply the amount in moles by Avogadro's number: $$ (0.5 \times 10^{-9} \ mol) \cdot (6.02 \cdot 10^{23} \ atoms/mol) = 3.01 \cdot 10^{14} \ atoms $$

Find the closest answer in the choices

Now that we have calculated the number of carbon atoms, we can look at the given options to find the closest answer. The closest answer to our calculated value (\(3.01 \cdot 10^{14}\) atoms) is option b) \(3.19 \cdot 10^{14}\) atoms.

Key concepts

Mole Conversions

Understanding mole conversions is essential in chemistry because it provides a bridge between the number of particles and the amount of substance measured in laboratory settings. A mole is an amount that refers to approximately \(6.02 \times 10^{23}\) entities of the substance, whether they are atoms, molecules, or ions. This large number is known as Avogadro's number.
To convert from nanomoles to moles, you have to acknowledge that 1 mole is equivalent to \(10^9\) nanomoles. Nanomoles are just one billionth of a mole, which allows chemists to work quantitatively with small amounts of substances. For instance, converting 0.5 nanomoles of a substance to moles involves this simple calculation:

• Divide the number of nanomoles by \(10^9\).
• Thus, \(0.5 \space nmol = 0.5 \times 10^{-9} \space mol\).

This conversion helps in further calculations, like determining the number of atoms or molecules and facilitates the application of Avogadro’s number.

Nanomoles to Moles

Converting nanomoles to moles is a common task in chemistry, especially when dealing with very small quantities of a substance. Nanomoles are used to measure extremely minute amounts, which are crucial in fields like pharmacology and biochemistry. The conversion factor is straightforward. Since 1 mole equals \(10^9\) nanomoles, the conversion from nanomoles (nmol) to moles (mol) is accomplished by dividing the given amount in nanomoles by \(10^9\).
To illustrate, for the conversion of 0.5 nanomoles of carbon to moles, we perform the following operation:

• Calculate \(0.5 \space nmol \divide 10^9 = 0.5 \times 10^{-9} \space mol\).

This precise conversion is crucial when calculating further properties like the number of atoms present, which can then be used to make meaningful scientific analyses and conclusions.

Carbon Atoms Calculation

Once you have the number of moles, you can calculate the number of atoms in a sample using Avogadro's Number. Avogadro's Number \(6.02 \times 10^{23}\) is a fundamental constant depicting the number of atoms in one mole of a pure substance.
To determine the number of carbon atoms in 0.5 nanomoles of carbon, which is equal to \(0.5 \times 10^{-9} \space mol\) as previously calculated, you use the following steps:

• Multiply the amount in moles by Avogadro’s Number.
• This results in \((0.5 \times 10^{-9} \space mol) \times (6.02 \times 10^{23} \space atoms/mol) = 3.01 \times 10^{14} \space atoms\).

Thus, using Avogadro’s Number in this conversion allows chemists to quantitatively interpret chemical quantities in terms of atoms and molecules, exactly as represented in a balanced chemical equation.