Problem 29 - Chemistry The Central Science

Question

(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.

Step-by-Step Solution

Answer

(a) Avogadro's hypothesis explains the law of combining volumes by stating that equal volumes of all gases contain the same number of molecules at the same temperature and pressure. This results in volume ratios of gaseous reactants and products being in simple whole numbers. (b) According to Avogadro's law, the ratio of the number of neon and xenon gas atoms in the flasks is 2:3, since their volumes are 1.0 L and 1.5 L, respectively, and they are under the same pressure and temperature. (c) Yes, 1 mol of an ideal gas will always occupy the same volume at a given temperature and pressure, since the product of pressure and volume remains constant, according to the ideal gas law formula.

Step 1: (a) Avogadro's Hypothesis and the Law of Combining Volumes

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.

Step 2: (b) Finding the Ratio of Neon and Xenon Gas Atoms

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.

Step 3: (c) Volume Occupied by 1 Mol of an Ideal Gas

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.

Key Concepts

Law of Combining Volumes

The law of combining volumes is a fascinating principle in chemistry that simplifies how we view gaseous reactions. It tells us that gaseous reactants and products bind to one another in simple, whole number ratios by volume. This only holds true if the conditions of temperature and pressure remain unchanged.
The key idea behind this law is connected closely with Avogadro's hypothesis. Avogadro suggested that equal volumes of different gases, given the same temperature and pressure, will consist of the same number of molecules.
Hence, when two gases react, the law of combining volumes and Avogadro's insight together explain why the volume ratios are simple integers. Because equal volumes contain equal numbers of molecules, changes in volume directly reflect changes in the number of reacting molecules. Therefore, when gases react, they do so in whole number ratios, making calculations in chemical reactions much more predictable and straightforward.

Ideal Gas Law

The Ideal Gas Law is a cornerstone concept in the study of gases, providing a comprehensive relationship between the physical properties of gases. It is summarized by the equation \(PV = nRT\), where \(P\) denotes pressure, \(V\) denotes volume, \(n\) represents the number of moles of gas, \(R\) is the ideal gas constant, and \(T\) stands for temperature.
Essentially, this equation tells us that a gas's pressure fitting into a specific volume is proportional to the number of moles multiplied by the temperature and a constant.

• As pressure increases, assuming temperature and moles are constant, the volume must reduce.
• Meanwhile, at a fixed pressure, the volume expands with rising temperature, provided moles remain constant.

These relationships allow scientists to predict how a gas will behave under different conditions. This makes the Ideal Gas Law an indispensable tool when studying or working with gases in various scenarios.

Mole Concept

The mole concept is an essential idea in chemistry that helps scientists measure quantities of chemical substances. It introduces a way of counting atoms, molecules, or other entities in a substance by using a large, yet convenient number known as Avogadro's number, \(6.022 \times 10^{23}\).
One mole of any substance contains exactly this many atoms or molecules. Hence, whether you have a bucket of water or a flask of gases, 1 mole will present precisely the same number of molecules.
The concept is particularly important in gas calculations. For instance, at standard temperature and pressure (STP), 1 mole of any ideal gas occupies 22.4 L of space. This consistent volume provides a straightforward understanding, aiding calculations when determining reactant quantities for achieving desired reactions. Recognizing the mole concept simplifies chemical equations and enables chemists to predict how substances will react and in what proportions.