A student has 54 grams of water (H₂O, molar mass 18 g/mol). How many water molecules are in this sample?
A54 molecules — one gram of water contains one molecule
B3 × 6.022 × 10²³ molecules — three moles each contain Avogadro's number of molecules
C54 × 10²³ molecules — multiply the mass in grams directly by Avogadro's number
D6.022 × 10²³ molecules — any macroscopic sample contains exactly one mole of molecules
54 g ÷ 18 g/mol = 3 moles. Each mole contains N_A = 6.022 × 10²³ molecules, so 3 moles = 3 × 6.022 × 10²³ molecules. Option C is the classic error: you cannot multiply grams directly by Avogadro's number — you must first convert to moles using molar mass. Option D confuses 'macroscopic sample' with 'exactly one mole,' which is only true if the sample happens to be one molar mass's worth.
Question 2 Multiple Choice
An element has an atomic mass of 32 amu. Which correctly explains why one mole of this element weighs 32 grams?
ABecause 32 is a round number conveniently close to the element's true mass in grams
BBecause the gram and the amu were independently derived from the same underlying mass standard
CBecause the mole is defined as the number of atoms in 12 grams of carbon-12, which makes the amu and g/mol scales numerically identical by construction
DBecause Avogadro's number happens to equal 6.022 × 10²³, which creates this numerical equality by coincidence
The equality is deliberate design, not coincidence. The mole was defined so that one mole of carbon-12 (atomic mass 12 amu) weighs exactly 12 grams, which fixes the conversion: 1 amu = 1 g/mol ÷ N_A. Every other element's molar mass in g/mol then equals its atomic mass in amu by the same construction. Option D gets causation exactly backward — Avogadro's number was chosen to produce this equality, not the other way around.
Question 3 True / False
The ideal gas law can be written as PV = nRT (with n in moles) or as PV = Nk_BT (with N as the number of molecules). These are the same equation expressed in different counting units.
TTrue
FFalse
Answer: True
R = k_B × N_A, so nRT = n × (k_B × N_A) × T = (n × N_A) × k_B × T = N × k_B × T, where N is total molecule count. Avogadro's number is the conversion factor between the per-molecule world (k_B) and the per-mole world (R). The physics is identical; only the bookkeeping unit changes.
Question 4 True / False
The numerical equality between atomic mass in amu and molar mass in g/mol is a remarkable physical coincidence that chemists discovered experimentally.
TTrue
FFalse
Answer: False
This equality is a deliberate definitional choice, not a coincidence. The mole was defined so that one mole of carbon-12 (atomic mass 12 amu) weighs exactly 12 grams. That definition fixes 1 amu = 1 g/mol ÷ N_A, ensuring the numerical equality for all elements. In the 2019 SI redefinition, Avogadro's number was fixed to an exact value precisely to preserve this relationship. It is engineered convenience, not discovered coincidence.
Question 5 Short Answer
What problem does Avogadro's number solve for scientists working with macroscopic quantities of matter?
Think about your answer, then reveal below.
Model answer: Avogadro's number bridges the microscopic scale of atoms (masses ~10⁻²⁷ kg, measured in amu) and the macroscopic scale of laboratory measurements (grams, liters). By grouping 6.022 × 10²³ atoms into one mole, scientists can apply atomic-scale properties — like atomic mass — using ordinary laboratory scales without counting individual atoms or working with absurdly small numbers. The mole converts between 'how many atoms' and 'how many grams' seamlessly.
The core utility is unit translation. Without this bridge, you could never connect a measurement on a scale (grams) to a prediction about atomic behavior (which operates in amu and particle counts). Avogadro's number is the conversion factor that makes atomic physics quantitatively useful in the lab.