Question

\(1 \mathrm{~g}=\ldots \ldots \ldots \ldots \ldots\) amu (a) \(6.02 \times 10^{23}\) (b) \(6.02 \times 10^{-23}\) (c) \(1.66 \times 10^{-27}\) (d) \(1.66 \times 10^{27}\)

Short answer

None of the options provided are correct. The closest answer is (b) \(6.02 \times 10^{-23}\), but the actual conversion is approximately \(1.66 \times 10^{-24}\) amu. None of the given options match the calculated answer.

Step-by-step solution

Understanding the conversion between grams and amu

An atomic mass unit (amu) is defined as 1/12 of the mass of a carbon-12 atom. The number of amu in 1 gram can be calculated using Avogadro's number, which is approximately \(6.022 \times 10^{23}\) atoms/mol. Since 1 mole of any substance contains Avogadro's number of atoms or molecules, and the molar mass of that substance in grams, we can use the molar mass and Avogadro's number to convert grams to amu.

Converting 1 gram to amu

First, we need to find the molar mass of the substance in question. Since we were only given the mass (1 gram) and not the substance, we have to make use of the mass-to-amu conversion: \[1\, \text{amu} = \frac{1\, \text{g}}{N_A}\] Where \(N_A\) represents Avogadro's number, which is approximately \(6.022 \times 10^{23} \, \text{atoms/mol}\). Now, we need to solve for the amu equivalent of 1 gram:

Calculate the amu equivalent of 1 gram

\[\text{amu} = \frac{1\, \text{g}}{6.022 \times 10^{23}\, \text{atoms/mol}} = 1.66 \times 10^{-24} \, \text{amu}\] Comparing this result to the options given in the exercise: (a) \(6.02 \times 10^{23}\) : It's Avogadro's number, but not the answer here. (b) \(6.02 \times 10^{-23}\) : Close to the reciprocal of Avogadro's number, but not the answer. (c) \(1.66 \times 10^{-27}\) : Not the correct conversion factor. (d) \(1.66 \times 10^{27}\) : Incorrect, since it is the reciprocal of our answer (1.66 × 10^{-24} amu). None of the options provided in the exercise exactly match our calculated answer (\(1.66 \times 10^{-24}\) amu). However, it is noteworthy that none of them is correct and the closest option (b) differs by a factor of 100. Please note that the number "\(1.66 \times 10^{-24}\) amu" that we found is only an approximation, and it is assumed that our calculations were aimed at a more accurate conversion value. Since there might be rounding errors, it would be wise to double-check the given exercise and the expected answer's precision.

Key concepts

Avogadro's number

Avogadro's number, denoted as \(N_A\), is a central concept in chemistry. It is defined as the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. This number is approximately \(6.022 \times 10^{23}\). This handy figure allows chemists to count particles by weighing them, bridging atomic scale with the macroscopic world.

When given in the context of conversions, Avogadro's number enables calculations between grams and atomic mass units (amu). It allows for the transformation of quantities measured in moles, making it easier to understand and quantify reactions and substances on a macro level.

• Helps in determining the number of atoms or molecules in a given mole.
• Forms the basis for the link between atomic mass units and grams.
• Essential for accurate stoichiometric calculations in chemistry.

Avogadro's number is a foundational bridge in chemistry, making abstract atomic-scale concepts accessible and measurable.

atomic mass unit

The atomic mass unit, abbreviated as amu, is a standard unit of mass that quantifies the weight of atoms and molecules. It is defined as one twelfth of the mass of a carbon-12 atom, approximately equivalent to \(1.66 \times 10^{-27}\) kilograms. This unit helps in expressing extremely small masses typical of subatomic particles in a comprehensible way.

Atomic mass units make it easier to compare the masses of different atoms and molecules. Since atoms are incredibly small, using kilograms or grams directly would result in cumbersome numbers that are difficult to manipulate. Using amu provides:

• A common scale for assessing the mass of atoms and molecules.
• A unit that aligns with Avogadro's number to enable conversions to grams.
• Simplicity in communicating small-scale mass values.

In essence, the amu allows chemists to describe, compare, and understand masses at the atomic level without losing precision.

molar mass

Molar mass is the mass of one mole of a chemical element or compound, expressed in grams per mole (g/mol). It connects the mass of a substance to the amount of substance present in moles. Molar mass is crucial for converting between mass and moles, enabling chemists to determine how many atoms or molecules are present in a sample.

A substance's molar mass allows it to be weighed and compared, bridging the gap between atomic theory and real-world measurements. Here's what to remember about molar mass:

• It is calculated by summing the atomic masses of the constituent atoms in a molecule.
• Used alongside Avogadro's number to find the number of particles in a given mass.
• Essential for calculating reagents and yields in chemical reactions.

Molar mass provides a practical connection between atomic-scale properties and laboratory-scale quantities, forming a necessary tool for accurate chemical analysis.

conversion factors

Conversion factors are mathematical tools used to convert one unit of measurement to another. In chemistry, conversion factors are pivotal for calculations involving quantities like mass, moles, and volume. They allow chemists to use measurements that are more practical and understandable.

When converting between grams and amu, or moles and molecules, conversion factors are essential. Here's why they're important:

• Facilitate the translation between different kinds of measurements.
• Enable the transformation of abstract scientific values into usable data.
• Include constants like Avogadro's number or derived formulas for specific conversions.

In essence, conversion factors help bridge the theoretical and practical aspects of chemistry, ensuring that scientific calculations are rooted in real-world applications and units.