Question

Consider two different containers, each filled with 2 moles of $\mathrm{Ne}(g)$ . 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

When the temperature is raised in the two containers: For the rigid container with constant volume, the pressure increases while the density remains unchanged. In the flexible container with variable volume, the pressure remains constant, and the density decreases.

Step-by-step solution

1. Rigid Container with Constant Volume

For the rigid container with a constant volume, the equation of state can be written as: $P=\frac{nRT}{V}$ As we raise the temperature (T), it's clear from this equation that the pressure (P) will increase proportionally. Since the volume stays constant, the density of the gas will remain unchanged.

2. Flexible Container with Variable Volume

For the flexible container that can change its volume, we need to consider that the internal pressure and external pressure are equal. So, we can write the equation of state as: $P_{ext}=\frac{nRT}{V}$ Since the external pressure (P_ext) remains constant, as we raise the temperature (T), the volume (V) needs to increase proportionally to keep the pressure constant. Therefore, the pressure inside the flexible container remains constant. Now let's consider the relationship between density, mass, and volume for the flexible container: $\rho =\frac{m}{V}$ Since the mass (m) of the gas inside the container remains constant and the volume (V) increases, the density of the gas (ρ) will decrease as the temperature increases. To sum up, when the temperature is raised: - In the rigid container (constant volume): pressure increases and density remains unchanged. - In the flexible container (variable volume): pressure remains constant, and density decreases.

Key concepts

Pressure and Temperature Relationship

The relationship between pressure and temperature in gases is a fundamental concept in chemistry and physics. This relationship is often described by Gay-Lussac's Law, which states that, for a fixed mass of gas at constant volume, the pressure is directly proportional to its absolute temperature. This can be expressed in the formula: $P\propto T$.

In the scenario with a rigid container, the volume doesn't change. This means that if the temperature of the gas is increased, the pressure also increases because the molecules move faster and hit the walls more often.

However, in the flexible container, maintaining constant pressure means that the temperature increase leads to an expansion in volume rather than a rise in pressure. Here, the container adjusts its size to accommodate the constant pressure dictated by the surrounding environment.

It’s fascinating to see how gas laws illustrate these changes and why pressure might rise in one context and stay stable in another.

Volume and Density Relationship

Volume and density have an inverse relationship in the context of gases, as described by the equation $\rho =\frac{m}{V}$. Here, $\rho$ represents density, $m$ is mass, and $V$ is volume. When the volume of a gas increases, for the same amount of mass, its density decreases, and vice versa.

In the flexible container scenario, as the temperature increases, the volume also increases to keep the pressure constant. The mass of the gas doesn’t change, so the result is a decrease in density. This is a practical illustration of how volume changes can directly affect density.

Understanding this relationship helps explain why gases can expand and spread in less dense forms, making them behave differently under various conditions.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates pressure, volume, temperature, and the number of moles of a gas. It is expressed as $PV=nRT$, where $P$ stands for pressure, $V$ is volume, $n$ represents the number of moles, $R$ is the ideal gas constant, and $T$ is temperature in Kelvin.

This law helps us understand how changing one of these variables affects the others. In our example, with a rigid container, the volume $V$ is constant. Thus, if the temperature $T$ increases, the pressure $P$ must also increase to maintain the equation’s balance.

In contrast, in a flexible container, because pressure $P$ is constant as the temperature $T$ rises, the volume $V$ must increase. This characteristic flexibility is why balloons expand when the gas inside them is heated. The Ideal Gas Law provides a theoretical framework for predicting how gases will respond to changes in environmental conditions.

Thermodynamics in Chemistry

Thermodynamics in chemistry involves understanding energy transformations in relation to matter, particularly at the molecular level. It encompasses concepts like heat, work, enthalpy, and internal energy. In the context of gases, thermodynamics helps us understand how energy input (like heating) affects gas behavior.

The exercise demonstrates some basic principles of thermodynamics. By increasing temperature, energy is added to the system, causing gas molecules to move more. This movement has different outcomes based on whether the gas is in a rigid or flexible container.

In a rigid container, the energy leads to increased pressure, while in a flexible container, the energy causes expansion, highlighting how energy and molecular behavior are intertwined in thermodynamics.