(a) How is the law of combining volumes explained by Avogadro's hypothesis? (b) Consider a 1.0 - \(\mathrm{L}\) flask containing neon gas and a 1.5-L flask containing xenon gas. Both gases are at the same pressure and temperature. According to Avogadro's law, what can be said about the ratio of the number of atoms in the two flasks? (c) Will 1 mol of an ideal gas always occupy the same volume at a given temperature and pressure? Explain.

Avogadro's hypothesis states that equal volumes of all gases, at the same temperature and pressure, contain the same number of molecules. The law of combining volumes states that the ratio of the volumes of gaseous reactants and products in a chemical reaction can be expressed as a simple ratio of small whole numbers, provided that all measurements are made at the same temperature and pressure. Using Avogadro's hypothesis, we can explain the law of combining volumes since equal volumes of gases contain the same number of molecules, and thus, the volumes of reacting gases are proportional to the number of molecules involved in the reaction. That's why the volume ratios of gaseous reactants and products come out to be in simple whole numbers.

Since both the neon and xenon gases are at the same pressure and temperature, we can apply Avogadro's law to find the ratio of the number of atoms in the two flasks. Avogadro's law states: \[\frac{n_1}{V_1} = \frac{n_2}{V_2}\] Where \(n_1\) and \(n_2\) are the number of atoms (or moles) of two gases and \(V_1\) and \(V_2\) are the volumes they occupy. Given that the volume of the neon gas flask is 1.0 L and the volume of the xenon gas flask is 1.5 L, we can find the ratio of atoms as: \[\frac{n_\mathrm{Ne}}{1.0\, \mathrm{L}} = \frac{n_\mathrm{Xe}}{1.5\, \mathrm{L}}\] To find the ratio of the number of atoms, we can solve for \(n_\mathrm{Ne}/n_\mathrm{Xe}\): \[\frac{n_\mathrm{Ne}}{n_\mathrm{Xe}} = \frac{1.0\, \mathrm{L}}{1.5\, \mathrm{L}} = \frac{2}{3}\] Thus, the ratio of the number of atoms of neon gas to xenon gas is 2:3.

According to the ideal gas law, the relationship between the number of moles of a gas (n), pressure (P), volume (V), and temperature (T) is given by the equation: \[PV = nRT\] Where R is the ideal gas constant. Given a constant temperature and pressure, the product of pressure and volume will remain constant. That means, \[PV = \mathrm{constant}\] Since the number of moles (n) is also constant (1 mol), and R is always a constant value, the product of pressure and volume will remain constant, regardless of the type of gas. Therefore, 1 mol of an ideal gas will always occupy the same volume at a given temperature and pressure.