Question

(a) What is the mass, in grams, of a mole of \({ }^{12} \mathrm{C} ?\) (b) How many carbon atoms are present in a mole of \({ }^{12} \mathrm{C} ?\)

Short answer

(a) The mass of one mole of carbon-12 is \(12 g\). (b) There are \(6.022 \times 10^{23}\) carbon atoms in one mole of carbon-12.

Step-by-step solution

Understand the concept of a mole and atomic mass

Mole (symbol: mol) is a unit of measurement in chemistry that defines the amount of any substance. A mole of any substance has the same number of constituent particles (like atoms, ions, or molecules) as there are in 12 grams of pure carbon-12 isotope. This number of particles is known as Avogadro's number (N_A), which is approximately \(6.022 \times 10^{23}\) particles per mole. The atomic mass of an element is the average mass of the atoms of an isotope, scaled by the atomic mass unit (u), which is defined as one twelfth of the mass of one atom of carbon-12. For carbon, the atomic mass of carbon-12 isotope is \(12 u\), where \(1 u = 1.66 \times 10^{-24} g\).

Calculate the mass of one mole of carbon-12 in grams

To find the mass of one mole of carbon-12, we will convert the atomic mass from atomic mass units (u) to grams. Given that the atomic mass of carbon-12 is \(12 u\), we can write: \(mass_{C-12} = 12 u * 1.66 \times 10^{-24} \frac{g}{u}\) Using the conversion factor, we get: \(mass_{C-12} = 12 * 1.66 \times 10^{-24} g\) Now, multiply the values: \(mass_{C-12} = 1.99 \times 10^{-23} g\) Since we are looking for the mass of one mole of carbon-12, we need to multiply the mass of one carbon-12 atom by Avogadro's number: \(mass_{1 mol \ C-12} = N_A * mass_{C-12}\) So, we have: \(mass_{1 mol \ C-12} = (6.022 \times 10^{23}) * (1.99 \times 10^{-23} g)\) \(mass_{1 mol \ C-12} = 12 g\) (a) Therefore, the mass of one mole of carbon-12 is 12 grams.

Calculate the number of carbon atoms in one mole of carbon-12

To find the number of carbon atoms in one mole of carbon-12, we will use Avogadro's number. Since one mole of any substance has \(6.022 \times 10^{23}\) constituent particles, one mole of carbon-12 has: \(N_{atoms \ in \ 1 \ mol \ C-12} = N_A\) \(N_{atoms \ in \ 1 \ mol \ C-12} = 6.022 \times 10^{23}\) (b) Therefore, there are \(6.022 \times 10^{23}\) carbon atoms in one mole of carbon-12.

Key concepts

Avogadro's number

Avogadro's number is a fundamental constant in chemistry that represents the number of constituent particles, usually atoms or molecules, in one mole of a substance. It is denoted as \(N_A\), and its value is approximately \(6.022 \times 10^{23}\) particles per mole. This incredibly large number allows chemists to count particles even though they are extremely tiny and numerous.

Understanding Avogadro's number is crucial when dealing with the mole concept, which is essentially a bridge between the atomic world and the macroscopic world we can measure. It tells us how many atoms are in a specific amount of a substance that we can weigh or measure.

In practical terms, if you have a mole of something, such as a mole of carbon-12 atoms, you have \(6.022 \times 10^{23}\) atoms of carbon-12. This direct correspondence between mass and number of particles makes Avogadro's number a key player in chemical calculations.

Atomic mass unit

The atomic mass unit (amu), denoted as \(u\), is a standard unit of mass that quantifies the mass of individual atoms or molecules. It's used to express atomic and molecular weights in a way that avoids dealing with extremely small numbers. One atomic mass unit is defined as one twelfth of the mass of a neutral atom of carbon-12 isotope. This definition is internationally accepted and provides a convenient scale for measuring atomic masses.

Typically, the mass of an atom in amu is very close to the sum of the number of protons and neutrons in its nucleus. For example, the carbon-12 isotope has an atomic mass of 12 amu because it contains 6 protons and 6 neutrons, making it the standard for the atomic mass unit.

When converting to grams, as part of chemical calculations, 1 amu is equivalent to approximately \(1.66 \times 10^{-24}\) grams. This conversion is key in calculating the mass of a substance in macroscopic terms, as seen in converting the atomic mass of elements to grams per mole.

Carbon-12 isotope

Carbon-12, often represented as \({}^{12}\text{C}\), is one of the isotopes of carbon. Isotopes are different forms of the same element that contain equal numbers of protons but different numbers of neutrons in their nuclei. Carbon-12 is the most abundant carbon isotope, making up about 98.9% of naturally occurring carbon.

It is used as a standard in multiple scientific measurements, including the atomic mass unit and the mole concept. Because of its stability and abundance, one atom of carbon-12 has been assigned a mass of exactly 12 atomic mass units. This makes it an excellent standard reference, ensuring consistent and accurate scientific calculations.

Understanding carbon-12 is essential when discussing the concept of moles and atomic masses. When we say that the atomic mass of a carbon-12 atom is exactly 12 amu, we're using this isotope as a baseline for defining the scale of atomic masses and the concept of a mole as 12 grams per mole based on its atomic mass.