Question

Which has the greater volume, \(100.0 \mathrm{~g}\) of hydrogen gas \(\left(d=0.0000899 \mathrm{~g} / \mathrm{cm}^{3}\right)\) or \(25.0 \mathrm{~g}\) of argon gas \(\left(d=0.00178 \mathrm{~g} / \mathrm{cm}^{3}\right) ?\)

Short answer

100 g of hydrogen has a greater volume than 25 g of argon.

Step-by-step solution

Understand the Problem

We need to compare the volume of two gases: hydrogen and argon. To do this, we'll use the relation between mass, volume, and density which is given by the formula: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \] Using this, we will calculate the volume of 100.0 g of hydrogen gas and 25.0 g of argon gas.

Calculate the Volume of Hydrogen Gas

We have the mass of hydrogen as 100.0 g and its density as 0.0000899 g/cm³. Substitute these values into the volume formula: \[ \text{Volume of Hydrogen} = \frac{100.0 \text{ g}}{0.0000899 \text{ g/cm}^3} \] This gives us:\[ \text{Volume of Hydrogen} \approx 1,112,347.05 \text{ cm}^3 \]

Calculate the Volume of Argon Gas

For Argon, the mass is 25.0 g and the density is 0.00178 g/cm³. Substitute these values into the volume formula:\[ \text{Volume of Argon} = \frac{25.0 \text{ g}}{0.00178 \text{ g/cm}^3} \] This gives us:\[ \text{Volume of Argon} \approx 14,044.94 \text{ cm}^3 \]

Compare the Volumes

Now that we have both volumes, we can compare them. The volume of hydrogen gas is approximately 1,112,347.05 cm³, and the volume of argon gas is approximately 14,044.94 cm³. Clearly, the volume of hydrogen is much greater than the volume of argon.

Key concepts

Understanding Density

Density is a fundamental concept in physics and chemistry that helps us understand how much mass a substance has per unit of volume. It's often represented by the symbol \(d\) and calculated using the formula:

• \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)

In simple terms, density tells us how tightly packed the molecules of a substance are. For instance, if you have two identical sized containers, one filled with feathers and the other with iron, the container with iron will have a higher density because iron molecules are packed more tightly.
For gases, density can be quite different from solids and liquids. Gases have molecules that are spread out and move freely compared to solids and liquids. As a result, gases generally have much lower densities. When working with gases like hydrogen and argon, understanding density is crucial because it allows us to compare their volumes based on their masses. This is key to solving problems where volume comparison is necessary.

Mass to Volume Conversion

Converting mass to volume is a useful technique in many scientific situations. To find the volume from the mass and density, you use the formula:

• \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \)

This formula is straightforward yet powerful. It allows you to rearrange the basic density formula to solve for volume, which is often a necessary step in many physical chemistry problems.
Let's break it down with our gases example. If you have 100 grams of hydrogen gas with a density of \(0.0000899 \text{ g/cm}^3\), you can find the volume by dividing the mass by the density. Similarly, if you have 25 grams of argon gas with a density of \(0.00178 \text{ g/cm}^3\), this formula will also help you determine its volume. By calculating these volumes, you are then able to compare the extent of space each gas occupies.

Properties of Gases

Gases have unique properties that set them apart from solids and liquids. Understanding these properties is important when working with exercises like volume comparison. Firstly, gases are highly compressible. This means their volumes can change significantly under different pressure conditions. Secondly, gases expand to fill the shape and volume of their containers. This expansion is due to the large amount of space between gas molecules, allowing them to move freely. Gas properties are influenced by temperature and pressure. As you increase the temperature, the energy and movement of gas molecules increase, which can affect the volume they occupy. Conversely, increasing pressure can decrease the volume of a gas as the molecules are forced closer together. In the context of our hydrogen and argon gas problem, knowing that hydrogen is much lighter than argon—even though it occupies a much larger volume—is due to these intrinsic gas properties. The low density of hydrogen allows it to take up more space for the same mass when compared to a heavier gas like argon.