Question

If \(1.04 \mathrm{~g}\) of chlorine gas occupies a volume of \(872 \mathrm{~mL}\) at a particular temperature and pressure, what volume will \(2.08 \mathrm{~g}\) of chlorine gas occupy under the same conditions?

Short answer

The final volume of \(2.08\mathrm{~g}\) of chlorine gas under the same conditions is \(1744\mathrm{~mL}\).

Step-by-step solution

Find the ratio of the initial and final mass of chlorine gas

Let's first find the ratio between the initial mass (\(1.04\mathrm{~g}\)) and the final mass (\(2.08\mathrm{~g}\)). This can be done using division: \[\frac{2.08\mathrm{~g}}{1.04\mathrm{~g}} = 2\] This means the final mass is twice the initial mass.

Use the proportionality relationship to find the final volume

Since the initial mass and volume of the chlorine gas is given, we can use that to find the final volume with the proportionality relationship: \[V_2 = V_1 \times \frac{m_2}{m_1}\] where: - \(V_1\) is the initial volume (\(872 \mathrm{~mL}\)) - \(m_1\) is the initial mass (\(1.04\mathrm{~g}\)) - \(m_2\) is the final mass (\(2.08\mathrm{~g}\)) - \(V_2\) is the final volume that we want to find.

Calculate the final volume

Substitute the values into the proportionality equation and solve for \(V_2\): \[V_2 = 872 \mathrm{~mL} \times 2 = 1744 \mathrm{~mL}\] The final volume of \(2.08\mathrm{~g}\) of chlorine gas under the same conditions is \(1744\mathrm{~mL}\).

Key concepts

Chlorine Gas

Chlorine gas, often represented by the chemical symbol \( \text{Cl}_2 \), is a diatomic molecule composed of two chlorine atoms. It is a greenish-yellow gas at room temperature and is well-known for its pungent odor. Chlorine is widely used in the manufacture of everyday products, including disinfectants and bleach, as well as in water treatment.

As a gas, chlorine follows the fundamental principles of gas laws, which describe how gases behave under different conditions of temperature, pressure, and volume. One of the key characteristics of gases, including chlorine, is that they fill the shape of the container they are in, meaning they don't have a fixed volume.

• Chlorine gas has a molar mass of \( 70.906 \text{ g/mol} \).
• It participates in various chemical reactions, and being a gas, it diffuses quickly into the environment.
• Safety is essential when handling chlorine gas due to its reactive nature and potential health hazard upon inhalation.

Understanding these properties of chlorine gas helps us apply gas laws effectively when dealing with real-world problems of gas expansion and reaction.

Mass-Volume Relationship

The mass-volume relationship of a gas is crucial in linking the amount of substance with the space it occupies. When dealing with gases, the mass of a sample affects its volume, provided the gas is under constant temperature and pressure conditions. According to Avogadro's Law, under the same temperature and pressure, equal volumes of gases contain the same number of molecules.

In the provided exercise, we begin with \( 1.04 \mathrm{~g} \) of chlorine gas occupying a volume of \( 872 \mathrm{~mL} \). When the mass increases to \( 2.08 \mathrm{~g} \) under the same conditions, we need to determine the new volume according to the same mass-volume ratio:

• Doubling the mass of chlorine gas (from \( 1.04 \mathrm{~g} \) to \( 2.08 \mathrm{~g} \)) means doubling its volume, given the pressure and temperature remain constant.
• The direct relationship between mass and volume is evident as the calculation results in a proportional increase in volume from \( 872 \mathrm{~mL} \) to \( 1744 \mathrm{~mL} \).

This close relationship highlights the predictable nature of gaseous phase behavior, facilitating calculations related to gas reactions and expansions.

Proportionality Relationship

The proportionality relationship in gases is an essential concept for predicting how changes in mass or other conditions will affect a gas's behavior. In this context, the relationship between the volume and the mass of a gas, like chlorine, is directly proportional provided the temperature and pressure are constant.

From the exercise, we can see how this relationship is applied to find a new volume after changing the mass. By using the formula \( V_2 = V_1 \times \frac{m_2}{m_1} \), we calculate the new volume:

• \( V_1 = 872 \mathrm{~mL} \) - The initial volume.
• \( m_1 = 1.04 \mathrm{~g} \) - The initial mass of the gas.
• \( m_2 = 2.08 \mathrm{~g} \) - The new mass of the gas.
• \( V_2 = 1744 \mathrm{~mL} \) - The resulting volume after the mass changes.

By doubling the mass, the volume also doubles, which perfectly illustrates the concept of direct proportionality in this context. This principle is not only limited to chlorine gas but applies broadly to all ideal gases following similar conditions.