Question

Consider two different containers, each filled with 2 moles of \(\mathrm{Ne}(\mathrm{g})\). One of the containers is rigid and has constant volume. The other container is flexible (like a balloon) and is capable of changing its volume to keep the external pressure and internal pressure equal to each other. If you raise the temperature in both containers, what happens to the pressure and density of the gas inside each container? Assume a constant external pressure.

Short answer

In the rigid container with constant volume, when the temperature increases, the pressure increases and the density remains constant. In the flexible container with variable volume under constant external pressure, when the temperature increases, the pressure remains constant and the density decreases.

Step-by-step solution

Analyze the rigid container with constant volume

For the first container with a constant volume, the internal volume will remain the same as we increase the temperature. Therefore, our main focus is to determine the effect on pressure and density.

Determine the effect on pressure for the rigid container

Using the Ideal Gas Law equation for the first container: PV = nRT Since the volume V and the number of moles n are constant, we can rewrite the equation as: P = nRT/V When the temperature T increases, since n, R, and V are constant, the pressure P will also increase. Thus, the pressure inside the rigid container increases as the temperature increases.

Determine the effect on density for the rigid container

Density (ρ) is given by: ρ = n/V Since both the number of moles n and the volume V are constants, the density ρ remains constant as temperature increases. Therefore, the density of the gas inside the rigid container does not change when the temperature increases.

Analyze the flexible container with variable volume

For the second container, the volume can change to keep the internal pressure equal to the external pressure, which is constant. Thus, our main focus is to determine the effect on volume and density.

Determine the effect on volume for the flexible container at constant pressure

Since the pressure inside the container is equal to the constant external pressure, as temperature increases, the volume will also increase in order to maintain the constant pressure using the Ideal Gas Law equation: PV = nRT Since P is constant, we can rewrite the equation as: V = nRT/P As temperature T increases, since n, R, and P are constant, the volume V will also increase. Thus, the volume of the container increases as the temperature increases.

Determine the effect on density for the flexible container

Density (ρ) is given by: ρ = n/V As the volume V increases due to the increase in temperature (from Step-5), the density ρ will decrease. Therefore, the density of the gas inside the flexible container decreases as the temperature increases. In conclusion: - In the rigid container with constant volume, as temperature increases, the pressure increases while the density remains constant. - In the flexible container with variable volume, as temperature increases, the pressure remains constant while the density decreases.

Key concepts

Rigid Container

A rigid container is one that maintains constant volume regardless of the conditions inside or outside. This means the space the gas occupies does not change, no matter the pressure or temperature applied. In the context of the Ideal Gas Law, \[ PV = nRT \],the volume, \( V \), remains fixed.

Thus, when we change other factors like temperature, certain effects result. If we increase the temperature, since \( P = \frac{nRT}{V} \) and both \( n \) and \( V \) are constant, the pressure must rise to compensate for a higher temperature.

In a rigid container:

• Volume (\( V \)) remains constant.
• An increase in temperature leads to an increase in pressure.
• Density remains unchanged as it depends solely on the fixed volume and a constant number of moles.

Flexible Container

A flexible container is capable of changing its volume based on external conditions. Think of it like a balloon that expands or contracts to balance pressures. In such a container, the gas inside can adjust its volume to maintain the pressure equilibrium between the inside and outside.

This ability to change volume is crucial when considering changes in temperature. According to the Ideal Gas Law equation:\[ PV = nRT \],if the outside pressure is constant, the volume \( V \) must change with temperature to maintain constant pressure. Hence, as temperature rises, the container expands, resulting in an increased \( V \):

• Volume varies to adjust for temperature changes while keeping pressure constant at external levels.
• This means that as the temperature increases, the volume increases.

Pressure and Volume Relationship

The relationship between pressure and volume is an inverse one, beautifully articulated by Boyle's Law. The law states that at constant temperature, the pressure of a gas is inversely proportional to its volume. This means that if you increase the volume, the pressure decreases, and vice versa.

Within the rigid container, since the volume is constant, pressure directly decreases or increases with any change in temperature. By contrast, in the flexible container, changes in volume occur to maintain a balance with pressure, ensuring it remains constant even as temperature changes.

Key aspects:

• Rigid containers: constant volume; pressure increases with temperature.
• Flexible containers: volume changes so that the pressure remains constant.

Temperature Effects on Gases

Temperature has a profound effect on gases, primarily influencing pressure and volume. In rigid containers, any increase in temperature leads to an increase in pressure, given the constant volume. This is explained by the Ideal Gas Law, where temperature and pressure are directly correlated: as temperature increases, so does pressure if volume stays steady.

For flexible containers, a rise in temperature will prompt an increase in volume, keeping pressure constant as described in the Ideal Gas Law equation:\[ PV = nRT \]. Temperature effects:

• Rigid container: Temperature rise leads to pressure increase, volume remains the same.
• Flexible container: Temperature rise results in volume increase, keeping pressure constant.

A comprehensive grasp of these concepts allows students to predict how gases behave under varied conditions, considering each container's nature.