How many molecules are in \(0.400\,\mathrm{mol}\) of a substance? Use \(N_A=6.02\times10^{23}\,\mathrm{mol^{-1}}\).

A substance has molar mass \(5.0\times10^{-2}\,\mathrm{kg\,mol^{-1}}\). Find the mass of one molecule.

A sample contains \(3.01\times10^{23}\) molecules. Find the amount in moles.

Find the mass of \(2.5\,\mathrm{mol}\) of a substance with molar mass \(1.8\times10^{-2}\,\mathrm{kg\,mol^{-1}}\).

A gas sample has \(n=0.80\,\mathrm{mol}\), molar mass \(2.8\times10^{-2}\,\mathrm{kg\,mol^{-1}}\), and volume \(0.020\,\mathrm{m^3}\). Find its density.

A container of volume \(1.0\times10^{-3}\,\mathrm{m^3}\) contains \(2.0\times10^{22}\) molecules. Estimate the average molecular spacing using \(r\sim(V/N)^{1/3}\).

A material has density \(900\,\mathrm{kg\,m^{-3}}\) and molar mass \(0.060\,\mathrm{kg\,mol^{-1}}\). Estimate its molecular number density \(N/V\).

Explain, in molecular terms, why a gas is much more compressible than a liquid.

A substance has molecular mass \(m_0\) and number density \(N/V\). Derive its mass density \(\rho\) in terms of these microscopic quantities, then rewrite it using molar mass \(M\) and moles per volume \(n/V\).

Two samples have the same mass density but molar masses \(M_A\) and \(M_B=4M_A\). Derive the ratio of their number densities and interpret what it implies about average molecular spacing.

Estimate the average molecular spacing in liquid water using \(\rho=1000\,\mathrm{kg\,m^{-3}}\), \(M=1.8\times10^{-2}\,\mathrm{kg\,mol^{-1}}\), and \(N_A=6.02\times10^{23}\,\mathrm{mol^{-1}}\).

A liquid and its vapor have the same molecular mass. If \(\rho_{\mathrm{liquid}}=800\,\mathrm{kg\,m^{-3}}\) and \(\rho_{\mathrm{vapor}}=1.25\,\mathrm{kg\,m^{-3}}\), estimate the ratio of average molecular spacing in the vapor to that in the liquid.

A material has density \(1200\,\mathrm{kg\,m^{-3}}\) and molar mass \(0.030\,\mathrm{kg\,mol^{-1}}\). Estimate the number of molecules in \(2.0\times10^{-6}\,\mathrm{m^3}\).

A gas has number density \(2.5\times10^{25}\,\mathrm{m^{-3}}\). Estimate the average molecular spacing.

A liquid has molecular spacing about \(3.0\times10^{-10}\,\mathrm m\). Estimate its molecular number density.

Derive \(m_0=M/N_A\) and explain why molar mass must be in \(\mathrm{kg\,mol^{-1}}\) for SI calculations.

Use number density to explain why gases have low density compared with liquids made of similar molecules.

Two materials have the same number density, but molecule B has twice the molecular mass of molecule A. Compare their mass densities.

Explain why molecular spacing estimates are order-of-magnitude models, not exact nearest-neighbor distances.

Connect molecular kinetic energy and intermolecular forces to the qualitative difference between solid, liquid, and gas.