Question

Consider two different containers, each filled with 2 moles of Ne(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, when the temperature increases, the pressure will increase, and the density will remain the same. In the flexible container, when the temperature increases, the pressure will remain the same, and the density will decrease.

Step-by-step solution

Recall the ideal gas law

The ideal gas law is given by: \( PV = nRT \) Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in kelvin.

Analyze the rigid container

Since the volume of the rigid container is constant, we can use the ideal gas law to find the effect of increasing the temperature on the pressure and density of the gas. The density of the gas is given by: \( \rho = \frac{m}{V} = \frac{nM}{V} \) Where M is the molar mass of the gas. Since the volume and moles remain constant and the temperature increases: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \) As the temperature increases, the pressure will also increase, which can be seen from the equation above. Since the volume is constant, the density of the gas will remain the same in the rigid container.

Analyze the flexible container

In the flexible container, the external pressure (P) and internal pressure are equal to each other and remain constant. Again, we can use the ideal gas law to investigate the effect of increasing the temperature on the pressure and density of the gas. Since the pressure is constant and the temperature increases: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \) As the temperature increases, the volume of the gas in the flexible container will also increase, which can be seen from the equation above. Now let's analyze the density of the gas in the flexible container. Since the molar mass, moles, and temperature are constant, we can rewrite the equation for the density change as: \( \rho_1 = \frac{nM}{V_1} \) and \( \rho_2 = \frac{nM}{V_2} \) As the volume increases (due to the temperature increase), the density of the gas in the flexible container will decrease.

Conclusion

In the rigid container, when the temperature increases, the pressure will increase, and the density will remain the same. In the flexible container, when the temperature increases, the pressure will remain the same, and the density will decrease.

Key concepts

Rigid Container

A rigid container is a container that does not change its volume, which means that the volume remains constant even if other conditions like temperature or pressure change. When discussing gases, this means the gas particles are restricted to a fixed space regardless of changes in external conditions. The behavior of gases in a rigid container is elegantly explained by the Ideal Gas Law, which is represented as \( PV = nRT \). Here, the volume (V) of the container does not change, so if the temperature (T) increases, the pressure (P) must also increase. Hence, in a rigid container:

• When the temperature increases, the pressure inside the container increases.
• The density remains constant because the mass and volume don’t change even as pressure increases.

This constant volume property makes rigid containers particularly useful in scenarios where maintaining a constant condition is crucial, despite external environmental changes.

Flexible Container

Unlike a rigid container, a flexible container can change its volume. This means when the temperature of the gas inside it changes, the container will adjust its volume to keep the internal pressure equal to the external pressure. For a flexible container, the Ideal Gas Law still applies, but with pressure (P) staying constant and volume (V) adjusting with temperature (T). Hence, for a flexible container:

• The volume increases if the temperature increases.
• The pressure remains constant since the container expands or contracts to match external conditions.

This characteristic of flexible containers makes them suitable for situations where pressure balance with the environment is necessary, like in balloons. The adaptability of their volume is key when using flexible containers in dynamic settings.

Gas Density

Gas density is a measure of how much mass of the gas exists in a given volume. It's given by the formula \( \rho = \frac{nM}{V} \), where \( n \) is the number of moles, \( M \) is the molar mass, and \( V \) is the volume. For a rigid container, because the volume does not change when the temperature increases, the gas density remains constant. However, in a flexible container, since the volume increases as temperature rises, the density decreases. As a flexible container expands, there's more space for the same number of gas molecules, thus reducing the density.

• Rigid Container: Density remains unchanged with increasing temperature.
• Flexible Container: Density decreases as volume increases with temperature.

Understanding gas density helps predict how gases behave under different conditions, enabling control over their applications in various industries, such as in balloons or airships.

Temperature Effects

Temperature has a significant impact on gas behavior, as seen in both the rigid and flexible containers. When temperature increases, molecules move faster, increasing the kinetic energy of the gas, which affects pressure and volume.

• In a rigid container, increased temperature leads to increased pressure because the gas molecules hit the container walls more frequently and with greater force, as volume is fixed.
• In a flexible container, the temperature increase allows the container to expand, increasing volume and keeping pressure constant. This expansion provides a relief mechanism whereby the pressure doesn’t build up, allowing safety and stability.

These effects highlight the importance of temperature control in practical applications, such as heating systems or environmental management, where it’s crucial to anticipate how materials will respond to thermal fluctuations.

Pressure Change

Pressure in a gas is the force exerted by the gas molecules as they collide with the surfaces of their container. It is a crucial aspect of gas behavior governed by the Ideal Gas Law. In a rigid container, pressure changes with temperature due to volume constancy. As the temperature increases, so does the pressure because the molecules move faster, exerting greater force on the container walls.

• In rigid containers, pressure increases as the temperature increases while the volume remains unchanged.
• In flexible containers, the pressure remains constant because the volume adjustment offsets any potential increase in pressure.

Understanding how pressure changes under different conditions helps us design efficient systems in industries ranging from chemical processing to HVAC systems. These insights are also crucial in situations involving large pressure differences, providing safety and performance optimization.