Problem 27 - Chemistry: The Central Science

Question

(a) How is the law of combining volumes explained by Avogadro's hypothesis? (b) Consider a 1.0-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?

Step-by-Step Solution

Answer

(a) Avogadro's hypothesis explains the law of combining volumes because the whole number ratios of gas molecules in a chemical reaction are directly related to their volumes if the temperature and pressure remain the same. Since gas molecules occupy the same volume and exchange with each other in whole number ratios, their reaction volumes also follow these same whole number ratios. (b) According to Avogadro's law, the ratio of the number of neon atoms in the 1.0-L flask to the number of xenon atoms in the 1.5-L flask is \(\frac{N_{Ne}}{N_{Xe}} = \frac{1.0}{1.5}\), which simplifies to \(\frac{2}{3}\). So, there are 2 neon atoms in the 1.0-L flask for every 3 xenon atoms in the 1.5-L flask when the gases are at the same temperature and pressure.

Step 1: (Understanding Avogadro's hypothesis and the law of combining volumes)

Avogadro's hypothesis states that equal volumes of gases, under the same temperature and pressure conditions, contain the same number of molecules. The law of combining volumes states that the volumes of reacting gases and the volumes of the products are in simple whole number ratios when the volumes are measured under the same conditions of temperature and pressure. To see how Avogadro's hypothesis explains the law of combining volumes, we need to understand that the whole number ratios of volumes come from the whole number ratios of molecules in the gases.

Step 2: (Answering Part (a))

Avogadro's hypothesis explains the law of combining volumes because the whole number ratios of gas molecules in a chemical reaction are directly related to their volumes if the temperature and pressure remain the same. Since gas molecules occupy the same volume and exchange with each other in whole number ratios, their reaction volumes also follow these same whole number ratios.

Step 3: (Analyzing Part (b))

We have a 1.0-L flask containing neon gas and a 1.5-L flask containing xenon gas, both at the same pressure and temperature. According to Avogadro's law, the number of atoms in each flask is directly proportional to their volumes under the same conditions of temperature and pressure. We need to find the ratio of the number of neon atoms in the 1.0-L flask to the number of xenon atoms in the 1.5-L flask.

Step 4: (Calculating the ratio of atoms using Avogadro's law)

First, let's denote the number of Neon atoms as \(N_{Ne}\) and the number of Xenon atoms as \(N_{Xe}\). According to Avogadro's law, under the same conditions of temperature and pressure, the ratio between the number of atoms in each flask will be the same as the ratio between the volumes of the flasks. Thus, we can write the ratio as: \[\frac{N_{Ne}}{N_{Xe}} = \frac{V_{Ne}}{V_{Xe}}\] We are given the volumes of the flasks as \(V_{Ne} = 1.0\,\text{L}\) and \(V_{Xe} = 1.5\,\text{L}\). Plugging these values into the equation, we get: \[\frac{N_{Ne}}{N_{Xe}} = \frac{1.0}{1.5}\]

Step 5: (Simplifying the ratio of atoms)

Now, we can simplify the ratio and get: \[\frac{N_{Ne}}{N_{Xe}} = \frac{2}{3}\] So, according to Avogadro's law, there are 2 neon atoms in the 1.0-L flask for every 3 xenon atoms in the 1.5-L flask when the gases are at the same temperature and pressure.

Key Concepts

Law of Combining Volumes

In the world of gases, the Law of Combining Volumes is a fascinating principle. It states that when gases combine or react, their volumes (when measured at constant temperature and pressure) are in simple, whole number ratios. For example, if you combine hydrogen and oxygen gases to make water vapor, two volumes of hydrogen will combine with one volume of oxygen, resulting in two volumes of water vapor.
This law is deeply related to Avogadro's hypothesis, which explains why volumes relate to whole numbers. Since equal volumes under constant conditions contain an equal number of molecules, the simple ratios of volumes during reactions reflect the simple ratios of whole molecules reacting.

Gas Molecules

Gas molecules are tiny particles that make up gases. Under the same conditions of temperature and pressure, these molecules spread out to occupy the space available to them. This is a key aspect of Avogadro's hypothesis.
Imagine two flasks, each filled with a different gas, but each at the same temperature and pressure. If both flasks have the same volume, they will contain the same number of gas molecules, regardless of what kind of gas you have in each flask.
This universality makes it easier to predict and calculate the behavior of gases. It's also what gets translated into the consistent whole number ratios in the Law of Combining Volumes when gases react.

Volume Ratios

When it comes to gases, volume tells us a lot. Rather than tracking tiny individual molecules, it's often more convenient to measure gases by volume. That's because Avogadro's law assures us that volume is directly proportional to the number of molecules.
This means if you have a gas at a specific volume, by knowing other gases' volumes, you can infer the ratios of the number of molecules involved, as seen in chemical reactions. For example, if Gas A and Gas B are both at the same temperature and pressure, and you have twice the volume of Gas A compared to Gas B, there will be twice as many molecules of Gas A as there are of Gas B.

Neon and Xenon Gases

Let's consider an example with neon and xenon gases, both noble gases with unique properties. Imagine you have a 1.0-liter flask of neon gas and a 1.5-liter flask of xenon gas, both at the same temperature and pressure. According to Avogadro's law, the number of atoms in these flasks is proportional to the volume of the gas.
Thus, we can assume the number of neon atoms relates to xenon in the same way as their volumes. So if the volume of neon gas is 1.0 L and the volume of xenon gas is 1.5 L, then for every 2 neon atoms, there are 3 xenon atoms.
This ratio of 2:3 is derived directly from their volume ratio, showcasing Avogadro’s law in action with these gases.