Question

Of the values for \(P, V, n\), and \(T\), which ones remain constant during the following changes? (a) Air in a sealed glass bulb is heated from \(25^{\circ} \mathrm{C}\) to \(32^{\circ} \mathrm{C}\). (b) An He-filled balloon is cooled from \(23^{\circ} \mathrm{C}\) to \(10^{\circ} \mathrm{C} .\) (c) The amount of \(\mathrm{CO}_{2}\) in a sealed steel cylinder is increased by adding \(50.0 \mathrm{~g}\) of \(\mathrm{CO}_{2}\).

Short answer

(a) V and n remain constant. (b) P and n remain constant. (c) Only V remains constant.

Step-by-step solution

Scenario A: Sealed Glass Bulb

In this scenario, the glass bulb is sealed, implying that the volume (V) and the number of moles (n) stay constant as no air is allowed to enter or leave the bulb. The temperature (T) changes as the air is heated, and the pressure (P) will change as a consequence of the change in temperature. So, V and n remain constant.

Scenario B: He-filled Balloon

In this case, the balloon is filled with He, and the temperature (T) is changing. Since the balloon can expand and contract in response to temperature changes, the pressure (P) can be assumed to remain constant. Furthermore, the number of moles (n) won't change as we don't add or remove any gas from the balloon. So, P and n remain constant.

Scenario C: Sealed Steel Cylinder

In this scenario, the steel cylinder is sealed, which means that no gas can escape or enter, keeping the Volume (V) constant. The amount of \(\mathrm{CO}_{2}\) is increased by adding \(50.0 \mathrm{~g}\) of \(\mathrm{CO}_{2}\), meaning that the number of moles (n) is changing. As a result of the increase in the amount of \(\mathrm{CO}_{2}\), the pressure (P) will increase, and we have no information about the temperature (T) changes. So, only V remains constant.

Key concepts

Ideal Gas Equation

The ideal gas equation, represented by the formula \( PV = nRT \), is a cornerstone in understanding the behavior of gases under various conditions. In this equation, \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles of gas, and \( T \) for temperature. The variable \( R \) is the ideal gas constant, a value that relates all these variables together and is the same for all ideal gases.

When working with gas problems, it's important to note which variables are kept constant and which are allowed to change. For instance, in a sealed system where the volume is constant (as in the steel cylinder with added \( CO_2 \)), adding more gas particles (increasing \( n \)) will directly increase the pressure \( P \), provided the temperature \( T \) remains unchanged. This relationship, demonstrated in the exercise, is neatly captured by the ideal gas equation.

Chemistry of Gases

Understanding the chemistry of gases involves looking into the microscopic interactions and behaviors of gas particles. Gases are composed of tiny particles in constant, random motion, and they exert pressure on the walls of their container due to collisions. The basic assumptions of an ideal gas—which include negligible particle volume and no intermolecular forces—allow us to use the ideal gas equation as a model for predicting and explaining these behaviors.

While the ideal gas law is a helpful tool, real gases can deviate from ideal behavior at high pressures and low temperatures, where intermolecular forces and particle volume become significant. In chemistry, gas laws are crucial for understanding and predicting reactions that involve gases, as well as collecting products of a reaction or measuring reactant quantities.

Pressure-Volume-Temperature Relationships

The pressure-volume-temperature relationship in gases is described by various gas laws, which can be combined into the general ideal gas equation. These relationships help us predict how a gas will react to changes in one or more of its physical properties. For example, Charles's Law explains that the volume of a gas is directly proportional to its temperature at constant pressure, which is relevant to the unsealed He-filled balloon whose volume can change.

Boyle's Law, on the other hand, states that pressure and volume are inversely proportional to each other at a constant temperature. This is applicable to the sealed steel cylinder scenario, where an increase in the number of moles of gas leads to an increase in pressure at a constant volume. It is essential to grasp these fundamental relationships to solve complex problems in chemistry and to understand the behavior of gases in practical applications like air conditioning or the internal combustion engine.