Question

Starting from the ideal gas law, prove that the volume of a mole of gas is inversely proportional to the pressure at constant temperature (Boyle's law).

Short answer

The volume of a mole of gas is inversely proportional to the pressure, according to Boyle's Law.

Step-by-step solution

Write the Ideal Gas Law

Start with the ideal gas law equation, which is given by pV = nRT. Here, p = pressure, V = volume, n = number of moles, R = ideal gas constant, T = temperature.

Substitute for One Mole of Gas

Since we are dealing with one mole of gas, set n = 1. The equation simplifies to\(pV = RT\)

Isolate Volume (V)

To express the volume (V) in terms of pressure (p) and temperature (T), rearrange the equation as follows:\(V = \frac{RT}{p}\).

Analyze the Relationship

Observe the equation \(V = \frac{RT}{p}\). When the temperature T is constant, RT is a constant. Thus, the equation simplifies to \(V \propto \frac{1}{p}\),which illustrates that volume (V) is inversely proportional to pressure (p), as stated in Boyle's law.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics, serving as a cornerstone for understanding the behavior of gases under various conditions. It is represented by the equation:\[ pV = nRT \]This equation relates four important properties of a gas:

• Pressure (p): the force exerted by the gas particles against the walls of their container, usually measured in units like atmospheres (atm) or pascals (Pa).
• Volume (V): the space occupied by the gas, typically measured in liters (L) or cubic meters (m³).
• Number of Moles (n): the amount of substance present, where one mole contains Avogadro's number of particles.
• Temperature (T): the measure of the kinetic energy of gas particles, measured in Kelvin (K).

The symbol \( R \) represents the ideal gas constant, a universal constant that bridges the relationship between these variables.

The Ideal Gas Law is useful for predicting how a gas will behave when subjected to changes in pressure, volume, or temperature. This law assumes that gas particles do not interact with each other (apart from elastic collisions) and occupy no volume, making it most accurate under low pressure and high temperature conditions.

Understanding this law is crucial for delving into more specific gas laws, such as Boyle's and Charles' laws, each of which explores particular relationships within this equation.

Pressure-Volume Relationship

The pressure-volume relationship, foundational to Boyle's Law, describes how the pressure and volume of a gas are interrelated at a constant temperature. Boyle's Law states that the volume of a gas is inversely proportional to its pressure when the temperature is kept constant. This principle can be mathematically expressed as:\[ V ropto \frac{1}{p} \]Here,\( V \) is the volume, and \( p \) represents the pressure. According to this relationship:

• If the pressure increases, the volume decreases.
• If the pressure decreases, the volume increases.

This means that when you compress a gas by increasing its pressure, it takes up less space, and when you decrease the pressure, the gas expands.

In practical scenarios, this is seen in activities like breathing. When we breathe in, our diaphragm expands, lowering the pressure in the lungs compared to the atmosphere, allowing air to fill the lungs. When breathing out, the diaphragm contracts and increases the pressure in the lungs, expelling air.

This relationship is essential for understanding how pressurizing or depressurizing gas affects its volume, hence it's widely applied in fields like engineering, meteorology, and medicine.

Gas Constant

The Gas Constant, denoted as \( R \), is a key component in the Ideal Gas Law, serving as the bridge linking pressure, volume, temperature, and moles of a gas. Its value is constant across all ideal gas calculations, making it a universal constant essential for solving gas-related equations.

The commonly used value of \( R \) in the Ideal Gas Law is approximately 8.314 J/(mol·K). This value is derived from empirical gas laws and provides consistency in calculations involving different units of measurement.

• When the pressure is in atmospheres, and volume is in liters, \( R \) can also be expressed as 0.0821 L·atm/(mol·K).
• Different unit systems might slightly alter the numerical value of \( R \), but its role and function in the equation remain unchanged.

An understanding of \( R \) is crucial since it factors into how temperature and pressure affect a gas's volume, and vice versa, under controlled conditions.

This constant emerges in many applications beyond just theoretical calculations, including real-world applications in experimental chemistry and physics, where \( R \) helps maintain consistent results as scientists and engineers manipulate gas properties for various processes.