Question

Indicate what happens to the pressure of a gas with the following changes: (a) The volume increases. (b) The temperature increases. (c) The moles of gas increase.

Short answer

a) Pressure decreases; b) Pressure increases; c) Pressure increases.

Step-by-step solution

Understanding the Relationships

According to the ideal gas law, which is expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin, we can explore how changes in these variables affect pressure.

Change in Volume (Part a)

If the volume \( V \) increases while keeping the number of moles \( n \), temperature \( T \), and gas constant \( R \) constant, pressure \( P \) decreases. This is because pressure and volume have an inverse relationship according to Boyle's Law: \( P \propto \frac{1}{V} \).

Change in Temperature (Part b)

If the temperature \( T \) increases while keeping volume \( V \), the number of moles \( n \), and the gas constant \( R \) constant, pressure \( P \) increases. This follows from Gay-Lussac's Law, which states \( P \propto T \), meaning pressure is directly proportional to temperature.

Change in Moles of Gas (Part c)

When the number of moles \( n \) increases while keeping volume \( V \), temperature \( T \), and the gas constant \( R \) constant, pressure \( P \) increases. Pressure is directly proportional to the number of moles, as shown in \( PV = nRT \), indicating that more moles result in greater pressure for a given volume and temperature.

Key concepts

Boyle's Law

Understanding how gases behave under different conditions often begins with Boyle's Law. This principle deals directly with the relationship between pressure and volume. It states that the pressure of a gas is inversely proportional to its volume, as long as the number of moles of the gas and the temperature are held constant. In simple terms, if you increase the volume of a gas by using a bigger container and don't change anything else, the pressure will go down. This effect can be mathematically expressed as \[ P \propto \frac{1}{V} \] In practical terms:

• If you compress a gas into a smaller space, the pressure goes up.
• Conversely, if you allow it more room by increasing volume, the pressure drops.

Think of a syringe: moving the plunger in reduces the volume and increases pressure, pushing it out does the opposite.

Gay-Lussac's Law

Gay-Lussac's Law describes the relationship between pressure and temperature when volume remains constant. This law tells us that if the temperature of a gas increases, so does its pressure, assuming the volume and moles of gas are unchanged. Simply put: \[ P \propto T \] This means that pressure is directly proportional to temperature. Here's why:

• As a gas heats up, its molecules move faster.
• Faster molecules hit the walls of their container with more force, thus increasing pressure.

An everyday example includes heating an aerosol can; the rise in temperature leads to increased pressure inside the can, sometimes to dangerous levels if not controlled.

Pressure and Volume Relationship

The intricate dance between pressure and volume is beautifully captured in real-world applications. When thinking about pressure and volume within the context of gases, it's crucial to remember that they are inversely related. This basically means:

• If volume is reduced by half, the pressure doubles, provided temperature and moles remain constant.
• If volume is doubled, the pressure is halved, under the same conditions.

Such a relationship is key when studying gases in closed systems where temperature doesn't change, such as in diving tanks or home HVAC systems, where mismanagement can lead to pressure mishaps. Engineers and chemists constantly consider these relationships to optimize safety and efficiency.

Pressure and Temperature Relationship

The connection between pressure and temperature unveils itself vividly in Gay-Lussac's Law. Whenever the temperature of a gas held at a constant volume is increased, its pressure climbs as well. This occurs because:

• Higher temperatures impart more energy to gas molecules, increasing their movement.
• This surge in molecular activity means more frequent and forceful collisions with the container's sides, elevating pressure.

Understanding this relationship is pivotal in various fields, such as atmospheric science and automotive engineering. Consider how car tires can become over-inflated on hot days because the air inside warms up and increases in pressure.

Pressure and Moles Relationship

In the realm of gases, the number of gas molecules, or moles, plays a crucial role in determining pressure when considering a specific volume and temperature. This relationship is encapsulated in the ideal gas equation and shows: \[ P \propto n \] Here, pressure (P) is directly proportional to the number of moles (n) of a gas. Simply put:

• If you add more gas (increase moles), the pressure rises.
• Conversely, removing some of the gas (decreasing moles) reduces the pressure, provided volume and temperature remain constant.

This principle is evident in practices like inflation of balloons — more gas molecules mean a fuller, tighter balloon, increasing the internal pressure.