Question

As you increase the temperature of a gas in a sealed, rigid container, what happens to the density of the gas? Would the results be the same if you did the same experiment in a container with a movable piston at a constant external pressure? Explain.

Short answer

In a sealed, rigid container, the density of the gas remains constant when the temperature increases as the volume is fixed. However, in a container with a movable piston at a constant external pressure, the density of the gas decreases as the temperature increases due to the increase in volume.

Step-by-step solution

Understand the Ideal Gas Law

The ideal gas law is given by the equation \(PV=nRT\), where P is the pressure, V is the volume, n is the number of moles of the gas, R is the ideal gas constant, and T is the temperature in Kelvin. In our scenario, we know that the temperature T is increasing, and we need to analyze its effect on the density of the gas.

Understand density relationship

Density (\(\rho\)) is defined as mass (m) divided by volume (V). In the context of an ideal gas, density can be expressed as: \[\rho=\frac{m}{V}=\frac{nM}{V}\], where M is the molar mass of the gas and n is the number of moles.

Analyze Scenario 1: Sealed, Rigid Container

In this scenario, the volume of the container is fixed (constant). As the temperature increases, the pressure within the container must increase to maintain the ideal gas law with a constant volume. Since volume remains constant, the density (based on the formula \(\rho=\frac{m}{V}=\frac{nM}{V}\)) also remains constant, as mass and the number of moles do not change. Therefore, the density of the gas in a sealed, rigid container remains constant while increasing the temperature.

Analyze Scenario 2: Container with Movable Piston

In this case, since the container has a movable piston at a constant external pressure, the volume of the container will change as the temperature increases. The change in volume depends on the ideal gas law \(PV=nRT\). When the temperature increases, the volume of the container will also increase to maintain the constant external pressure. As the volume increases, the density of the gas will decrease because \(\rho=\frac{m}{V}=\frac{nM}{V}\). Mass and the number of moles of the gas do not change, but the volume increases, resulting in a decrease in density. Thus, in a container with a movable piston, the density of the gas decreases as the temperature increases. To sum up, for a gas in a sealed, rigid container, the density remains constant when the temperature increases. However, for a gas in a container with a movable piston at constant external pressure, the density of the gas decreases as the temperature increases.

Key concepts

Gas Density

Gas density is a measure of how much mass of gas is present in a given volume. It can be calculated using the formula \( \rho = \frac{m}{V} \), where \( m \) is the mass and \( V \) is the volume. For gases, this relationship can also be expressed using the ideal gas law: \( \rho = \frac{nM}{V} \). Here \( n \) is the number of moles and \( M \) is the molar mass.

• Density can be affected by changes in volume, mass, and number of moles.
• In chemical reactions or physical changes, understanding gas density helps in predicting behavior under different conditions.

Changing conditions such as pressure or temperature influences gas density, making it a critical factor in many applications from industrial processes to meteorology.

Sealed Container

A sealed container is one where no gas can enter or exit. The rigidity ensures that the volume remains constant. This setup affects how changes in temperature impact the gas inside.

• When temperature increases in a sealed container, the pressure increases to maintain the ideal gas relationship \( PV = nRT \).
• Since neither the mass nor the volume changes, the density remains constant.

This property is important when studying reactions or processes that need a controlled environment, as it provides consistent variables, except for temperature changes.

Movable Piston

A movable piston allows for the change in volume of a container without altering the mass of the gas inside. When temperature increases, the piston adjusts to maintain constant pressure.

• This movable nature means that an increase in temperature leads to an expansion in volume.
• According to \( PV = nRT \), as volume increases to counteract the rise in temperature, the density \( \rho = \frac{m}{V} \) decreases.

This situation is quite common in engines and other mechanical systems where pressure control is essential for safe and efficient operation.

Pressure-Volume Relationship

The pressure-volume relationship for a gas is an important concept derived from the ideal gas law, \( PV = nRT \). It describes how, for a given amount of gas at constant temperature:

• If the volume increases, the pressure decreases, assuming no other variables change.
• Conversely, if volume decreases, pressure increases.

In practical terms, this relationship helps in understanding behavior under different conditions, such as in weather systems or pneumatic devices. It is also critical in designing systems that rely on gas compression or expansion.