Question

You have a balloon that contains \(\mathrm{O}_{2}\). What could you do to the balloon in order to double the volume? Be specific in your answers; for example, you could increase the number of moles of \(\mathrm{O}_{2}\) by a factor of \(2 .\)

Short answer

Double the moles of \( \text{O}_2 \), halve the pressure, or double the temperature.

Step-by-step solution

Understanding the Gas Laws

We begin by recognizing that the volume of a gas can be changed using the ideal gas law, which states \( PV = nRT \). Here \( P \) 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.

Identify Ways to Change Volume

According to the ideal gas law, the volume can be altered by changing the pressure, number of moles, or temperature, assuming the others are constant. Our goal is to identify actions that can effectively double the volume \( V \).

Doubling Moles of Gas

One way to double the volume of the balloon is by doubling the number of moles \( n \) of \( \text{O}_2 \) inside the balloon. Since volume is directly proportional to the number of moles when temperature and pressure are constant, increasing \( n \) by a factor of 2 will double \( V \). In simple terms, add more \( \text{O}_2 \).

Decreasing Pressure by Half

Another method is to decrease the external pressure \( P \) to half of its original value. Volume is inversely proportional to pressure, so reducing \( P \) by half allows the volume \( V \) to double, provided that the number of moles and temperature remain constant.

Increasing Temperature Appropriately

Alternatively, doubling the temperature in Kelvin without changing the number of moles or pressure will also double the volume of the gas. Use the relation \( V \propto T \), and if initial temperature is \( T_1 \), then set final temperature to \( 2T_1 \) to achieve this.

Key concepts

Volume of Gas

The volume of a gas is an important property and is greatly influenced by several factors according to the ideal gas law. The ideal gas law can be expressed as \( PV = nRT \), and it helps us understand how different conditions affect the amount of space that gas occupies.

Here are the key components that influence the volume of a gas based on this equation:

• Pressure \((P)\): The force exerted by the gas molecules against the walls of its container. Changes in pressure can compress or expand gas volumes.
• Number of Moles \((n)\): A measure of the amount of substance. More moles mean more particles taking up space.
• Temperature \((T)\): Higher temperatures increase the kinetic energy of gas molecules, causing them to occupy more space.
• Ideal Gas Constant \((R)\): A constant that relates energy units to physical conditions of gas.

Understanding these components helps in effectively managing and predicting changes in the volume under different conditions.

Pressure and Volume Relationship

Pressure and volume share an intriguing relationship defined by Boyle's Law. According to Boyle's Law, if temperature and the number of moles remain constant, pressure and volume are inversely proportional. This means that if the pressure of a gas is reduced by half, its volume will double, assuming temperature and moles are consistent.

Here’s a more detailed breakdown of this relationship:

• Decreasing the pressure on a gas allows the molecules more room to move, leading to an increase in volume.
• Conversely, increasing the pressure pushes the gas molecules closer together, reducing the volume.

Remembering this relationship is crucial when considering how volume changes in practical scenarios, like making a balloon bigger by reducing the pressure around it while ensuring other factors remain unaffected.

Temperature and Volume Relationship

The relationship between temperature and volume is directly proportional, as described by Charles' Law. This means that increasing the temperature of a gas, while keeping pressure and the number of moles constant, will increase its volume.

Understanding this relationship can be visualized as follows:

• When gas is heated, its particles gain energy and move more vigorously, causing the gas to expand. Thus, doubling the temperature results in doubling the volume, provided pressure and moles are constant.
• Conversely, cooling the gas causes the particles to slow down and occupy less space, leading to a decrease in volume.

This principle helps us understand phenomena such as the expansion of air inside a hot air balloon, which becomes more buoyant as the air is heated.

Moles of Gas

The concept of moles ties directly into the overall behavior and volume of gases. In the context of the ideal gas law, the number of moles \((n)\) represents the amount of gas present, which is a fundamental factor that defines the overall characteristics of the gas. A mole is a unit that measures quantity and is equal to Avogadro's number, which is approximately \(6.022 \times 10^{23}\) particles.

Let's see why this is important:

• Doubling the number of moles of gas doubles the volume, as long as temperature and pressure remain constant. This is because more gas particles require more space.
• Changing the number of moles can be achieved by adding or removing gas, thus directly influencing the gas's volume.

This concept is crucial for applications like inflating a balloon, where increasing the amount of air inside directly increases its volume.