Question

Given a fixed quantity of a gas at constant temperature, calculate the new volume the gas would occupy if the pressure were changed as shown in the following table. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Initial } \\ \text { Volume } \end{array} & \begin{array}{c} \text { Initial } \\ \text { Pressure } \end{array} & \begin{array}{c} \text { Final } \\ \text { Pressure } \end{array} & \begin{array}{c} \text { Final } \\ \text { Volume } \end{array} \\ \hline 6.00 \mathrm{~L} & 3.00 \mathrm{~atm} & 5.00 \mathrm{~atm} & ? \\ \hline 40.0 \mathrm{~mL} & 60.0 \text { torr } & 90.0 \text { torr } & ? \\ \hline 2.50 \mathrm{~mL} & 40.0 \text { torr } & 255 \text { torr } & ? \\ \hline \end{array} $$

Short answer

The final volumes for the gas are 3.60 L, 60.0 mL, 0.392 mL respectively.

Step-by-step solution

Identify the initial volume, initial pressure and final pressure

For each row in the given table, the initial volume \(V_1\), initial pressure \(P_1\), and the final pressure \(P_2\) are given, while the final volume \(V_2\) is what needs to be calculated.

Use the formula of Boyle’s Law

The final volume \(V_2\) can be calculated by rearranging Boyle’s law as \(V_2 = \frac{P_1V_1}{P_2}\). Substitute the values of \(V_1\), \(P_1\), and \(P_2\) from the chart into the equation and simplify.

Repeat for each row

Repeat steps 1 and 2 for each row in the chart to obtain the final volume \(V_2\) for each case.

Key concepts

Gas Laws

Gas laws play a crucial role in understanding the behavior of gases under different conditions. They describe the relationship between pressure, volume, and temperature in gases. One of the most fundamental gas laws is Boyle’s Law, which states that the pressure of a gas is inversely proportional to its volume when the temperature and the amount of gas remain constant. This means that if you increase the pressure on a gas, its volume decreases, and vice versa.

Other important gas laws include Charles's Law, which relates volume and temperature, and Avogadro's Law, which connects volume and the amount of gas. Together, these laws combine to form the Ideal Gas Law, a more general equation that describes the state of an ideal gas under various conditions. This law is expressed in the equation \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

Understanding these laws helps to predict how a gas will respond to changes in pressure, volume, or temperature, which is essential in many scientific and industrial applications.

Pressure-Volume Relationship

The relationship between pressure and volume in a gas is essential to understanding how gases behave. Boyle's Law specifically addresses this relationship, stating that the product of pressure and volume remains constant when temperature is constant. This can be expressed as \( P_1V_1 = P_2V_2 \), where \( P_1 \) and \( P_2 \) are the initial and final pressures, and \( V_1 \) and \( V_2 \) are the initial and final volumes.

This relationship can be observed in everyday experiences as well. For example:

• When you compress air in a bicycle pump, the pressure increases as the volume of air decreases.
• Diving deeper underwater increases pressure, causing the volume of air in your lungs or a balloon to decrease.

Understanding the pressure-volume relationship provides valuable insights into how gases operate under pressure changes. It's a direct application of Boyle's Law, emphasizing the crucial inverse relationship: more pressure equals less volume, and less pressure equals more volume.

Ideal Gas Behavior

Ideal gas behavior is a theoretical concept that simplifies the study of gases. An ideal gas is imagined to be composed of many randomly moving point particles that interact only through elastic collisions. Though no real gases fit this description perfectly, most gases exhibit ideal behavior under normal conditions.

The concept of ideal gas behavior involves several assumptions:

• Gas molecules are in constant, random motion.
• There are no forces of attraction or repulsion between the gas molecules.
• The volume occupied by the gas molecules is negligible compared to the volume of the container.

These assumptions allow us to use the Ideal Gas Law, \( PV = nRT \), to predict the behavior of gases under varying conditions of pressure, volume, and temperature.

However, when gases are at very high pressures or low temperatures, they tend to deviate from ideal behavior. In these situations, the interactions between molecules become significant, and modifications to the ideal gas equations, like the Van der Waals equation, may be necessary for accurate predictions. Understanding ideal gas behavior and its limitations helps in applying gas laws appropriately in both theoretical studies and practical applications.