Question

A mathematical expression that summarizes Boyle's law is _________ .

Short answer

A mathematical expression that summarizes Boyle's law is \( PV = k \).

Step-by-step solution

Understand Boyle's Law

Boyle's law states that the pressure of a given quantity of an ideal gas is inversely proportional to its volume when the temperature is held constant. Mathematically, this relationship can be written as P ∝ 1/V.

Formulate the equation for Boyle's law

Since the pressure is inversely proportional to the volume (P ∝ 1/V), we can express this relationship as an equation by introducing a constant of proportionality, k: P = k(1/V) To make this expression more familiar, we can rewrite it by multiplying both sides of the equation by V: PV = k This mathematical expression summarizes Boyle's law: \[ PV = k \]

Key concepts

Gas Laws

The behavior of gases under various conditions is described by what physicists and chemists call the gas laws. These laws are empirical observations that, over time, have been cast into precise mathematical relationships. They describe how gases tend to expand when heated, how they exert pressure on their surroundings, and how different factors like pressure, volume, and temperature are interrelated. In study or practice, understanding these laws is crucial for explaining how gases behave in natural processes and industrial applications, such as in engines or the Earth's atmosphere.

The core gas laws include Boyle's Law, which focuses on the pressure-volume relationship, Charles's Law addressing the volume-temperature relationship, and Gay-Lussac's Law about pressure-temperature relationship. In addition to these, the Combined Gas Law integrates these three laws into one comprehensive equation by considering two of the variables constant.

Pressure-Volume Relationship

At the heart of Boyle's Law lies the pressure-volume relationship. This relationship is of central importance in understanding how gases will respond when they are compressed or allowed to expand. The law asserts that for a fixed amount of an ideal gas kept at a constant temperature, the pressure and volume are inversely proportional. This means that if you increase the volume of the gas, there will be more space for gas particles to move, and they will exert less pressure on their container. Conversely, if you decrease the volume by compressing the gas, the particles will be in closer proximity and collide more often with the container walls, thus increasing the pressure.

This concept is critical not only in theoretical physics but also in practical applications such as syringes, hydraulic lifts, and even in how we breathe with our lungs. To visualize this concept, think of a balloon: when you press on it to decrease its volume, you feel more resistance, which is indicative of increased pressure inside the balloon.

Ideal Gas Equation

While Boyle's Law is specific about the pressure-volume relationship at a constant temperature, the ideal gas equation generalizes the behavior of gases into a single expression that considers pressure (P), volume (V), temperature (T), and the amount of gas in moles (n). The equation is written as \( PV = nRT \) where R is the ideal gas constant. This equation is derived from the combined gas laws and it assumes that gases consist of a large number of small particles which are in random motion and do not interact except when they collide elastically.

The ideal gas equation provides a good approximation of the behavior of real gases under many conditions, although it becomes less accurate at high pressures and low temperatures, where the assumptions of the model are no longer valid. Nonetheless, it serves as a fundamental cornerstone in understanding gas behavior and is used extensively in chemistry, physics, engineering, and other disciplines that deal with gases.