Question

Given a fixed quantity of a gas at constant temperature, calculate the new pressure the gas would exert if the volume were changed as shown in the following table. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Initial } \\ \text { Pressure } \end{array} & \begin{array}{c} \text { Initial } \\ \text { Volume } \end{array} & \begin{array}{c} \text { Final } \\ \text { Volume } \end{array} & \begin{array}{c} \text { Final } \\ \text { Pressure } \end{array} \\ \hline 602 \text { torr } & 405 \mathrm{~mL} & 1512 \mathrm{~mL} & ? \\ \hline 0.00100 \text { torr } & 1.50 \mathrm{~L} & 15.0 \mathrm{~mL} & ? \\ \hline 0.832 \mathrm{~atm} & 805 \mathrm{~L} & 37.5 \mathrm{~L} & ? \\ \hline \end{array} $$

Short answer

The final pressures are 160.85 torr for the first row, 0.1 torr for the second row, and 17.82 atm for the third row.

Step-by-step solution

Understanding Boyle's Law

Boyle's Law states that when temperature is kept constant, pressure and volume have an inversely proportional relationship. This can be represented by the equation: P1*V1=P2*V2. Here, P1 and V1 are initial pressure and volume, and P2 and V2 are final pressure and volume.

Applying Boyle's Law to the first row

For the initial condition, P1 equals 602 torr, V1 equals 405 mL and V2 equals 1512 mL. We need to find P2. So, rearranging the equation, we get P2=P1*V1/V2 = 602 torr * 405 mL / 1512 mL.

Calculate the final pressure for the first row

Calculate the equation to find out P2. In this case, P2 = 602 torr * 405 mL / 1512 mL = 160.85 torr. Therefore, the final pressure for the first row is 160.85 torr.

Repeat the process for the remaining rows

Repeat the process for next rows. For the second row: P2 = 0.00100 torr * 1.50 L / 0.015 L = 0.1 torr. And for the third row: P2 = 0.832 atm * 805 L / 37.5 L = 17.82 atm.

Key concepts

Gas Laws

Understanding gas laws is fundamental in studying the behavior of gases under various conditions. These laws describe the relationships between pressure, volume, temperature, and the amount of gas.

Boyle's Law is one such gas law specifically addressing the pressure-volume relationship for a constant temperature and mass of gas. Other gas laws include Charles's Law, which focuses on volume and temperature, and Avogadro's Law that correlates volume with the amount of gas, among others.

By mastering these concepts, students can predict how changes in one variable may affect another when working with gases. This knowledge is widely applied in fields like meteorology, engineering, and respiratory medicine.

Pressure-Volume Relationship

The crux of Boyle's Law lies in the pressure-volume relationship it describes. When the temperature of a gas is held constant, a decrease in volume leads to an increase in pressure, and vice versa.

This inverse relationship is essential in many practical applications such as syringes, where pulling back on the plunger increases the volume inside the syringe, thereby reducing the pressure and drawing fluid into it. Similarly, compressing the plunger decreases the volume which increases the pressure, forcing the fluid out.

This principle is also seen in the functioning of lungs, automobiles' brake systems, and even in scuba diving equipment.

Chemistry Education

Chemistry education plays a pivotal role in developing critical thinking and problem-solving skills through the study of matter and its interactions. Educators focus on breaking down complex concepts, such as the behavior of gases, into manageable segments.

Using real-world examples, such as the exercise provided, helps students grasp abstract principles by relating them to tangible scenarios. Through practice and application of these laws, students not only learn to solve problems theoretically but also appreciate the relevance of chemistry in everyday life.

Effective teaching strategies might include demonstrations, simulations, and problem-solving sessions to solidify students' understanding of concepts like Boyle's Law.

Inverse Proportionality

Inverse proportionality is a foundational concept in mathematics and science. It describes a relationship between two variables where one increases as the other decreases.

In the context of Boyle's Law, pressure and volume are inversely proportional, meaning that as one increases (e.g., volume), the other (pressure) decreases, and this occurs in such a way that their product remains constant for a given mass of gas at a constant temperature.

A clear understanding of inverse proportionality enables students to tackle exercises that involve variable changes, like those in the provided table. It's a concept that underscores not only gas behaviors but also various phenomena across physics and beyond.