Question

Show how Boyle's gas law can be derived from the ideal gas law.

Short answer

To derive Boyle's gas law from the ideal gas law, begin with the equation \(PV = nRT\). Since both temperature and the amount of gas are constant in Boyle's law, replace \(nR\) with a constant \(C\), resulting in \(PV = CT\). Next, isolate the pressure and volume relationship by dividing both sides by \(V\) and \(C\), yielding \(P = \frac{CT}{V}\). Combine the constants \(T\) and \(C\) into a single constant \(k\), giving the equation \(P = \frac{k}{V}\), which demonstrates that pressure is inversely proportional to volume when temperature and the amount of gas remain constant, as stated in Boyle's law.

Step-by-step solution

State the ideal gas law

Write down the ideal gas law equation: \(PV = nRT\).

Set up constants

According to Boyle's law, the temperature and the amount of gas should remain constant. Thus, we can say that \(nR\) is a constant. We can also say that the product of pressure and volume, \(PV\), is constant. Let's replace \(nR\) with the constant \(C\). So, the equation now becomes: \(PV = CT\).

Derive Boyle's law

As per Boyle's law, we want to show that the pressure of a gas is inversely proportional to its volume at a constant temperature and a constant amount of gas. To demonstrate this, we will isolate the pressure and volume relationship in the equation. Divide both sides of the equation by \(V\) and \(C\). By doing so, we get: \(P = \frac{CT}{V}\) Now, since \(T\) and \(C\) are both constants, we can combine them into a single constant, say \(k\): \(P = \frac{k}{V}\) This equation justifies Boyle's law: here, the pressure of a gas is inversely proportional to its volume, as long as temperature and the amount of gas remain constant.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in the field of chemistry and physics that relates the pressure, volume, temperature, and number of moles of an ideal gas. Represented as the equation \( PV = nRT \), where \( P \) stands for pressure, \( V \) represents volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) denotes the temperature in Kelvin.

This equation is foundational because it allows us to predict the behavior of an ideal gas under different conditions, assuming that the gas particles do not interact and occupy no volume -- which is, of course, an approximation since real gases do not perfectly follow these assumptions. Nevertheless, the ideal gas law provides a close approximation for the behavior of real gases under a range of conditions.

Gas Laws

Gas laws are the scientific laws that describe the behavior of gases, including changes in pressure, volume, and temperature. They are essential for understanding how gases interact with their environment and with each other. Some of the critical individual laws within this group are Boyle's Law, which describes the pressure-volume relationship; Charles's Law, detailing the volume-temperature relationship; Gay-Lussac's Law, for the pressure-temperature relationship; and Avogadro's Law, which provides a volumetric understanding based on the amount of gas present.

These gas laws can be considered special cases of the ideal gas law, with one or more variables held constant. For example, in Boyle's Law, temperature and the number of gas particles are constant, thereby allowing us to focus solely on the relationship between pressure and volume. Understanding these laws helps in various practical scenarios, from designing combustion engines to predicting weather patterns and even in the medical field with respiratory therapies.

Pressure-Volume Relationship

The pressure-volume relationship, often encapsulated by Boyle's Law, is a key concept in gas behavior, stating that for a fixed amount of gas at constant temperature, the pressure exerted by the gas is inversely proportional to its volume. In equation form, it is represented as \( P \propto \frac{1}{V} \) when \( T \) and \( n \) are constant.

What this means is that if you decrease the volume of a gas container, the gas particles have less space to move around, thus hitting the container walls more frequently, which increases the pressure. Conversely, if you increase the volume, the gas particles spread out, hitting the walls less frequently, leading to decreased pressure. This principle is crucial to many applications such as in syringes, hydraulic lifts, and even explains how we breathe: our chest cavity changes volume to alter pressure, drawing air in and out of the lungs.