Question

(a) What is an ideal gas? (b) Show how Boyle's law, Charles's law, and Avogadro's law can be combined to give the ideal-gas equation. (c) Write the ideal-gas equation, and give the units used for each term when \(R=0.08206 \mathrm{~L}-\mathrm{atm} / \mathrm{mol}-\mathrm{K}\). (d) If you measure pressure in bars instead of atmospheres, calculate the corresponding value of \(R\) in \(\mathrm{L}-\mathrm{bar} / \mathrm{mol}-\mathrm{K}\).

Short answer

An ideal gas is a hypothetical gas with no intermolecular forces and perfectly elastic collisions between particles. The ideal gas equation is derived by combining the Boyle's law, Charles's law, and Avogadro's law: PV = nRT. When R = 0.08206 L atm/mol K, having P in pressure, V in volume, n in the amount of gas, and T in temperature. If the pressure is measured in bars, R's value is approximately 0.0807 L bar/mol K.

Step-by-step solution

(a) Definition of Ideal Gas)

An ideal gas is a hypothetical gaseous substance whose behavior can be described using a few simple relationships between pressure, volume, temperature, and the amount of gas particles. Ideal gases are assumed to have no intermolecular forces and collisions between particles are perfectly elastic. In reality, there is no gas that perfectly follows the ideal gas behavior; however, many gases behave like ideal gases under a range of temperature and pressure conditions.

(b) Deriving the Ideal Gas Equation from Boyle's, Charles's, and Avogadro's Laws)

To derive the ideal gas equation, we'll first briefly explain each law: 1. Boyle's Law: It states that the pressure of a given amount of gas held at a constant temperature is inversely proportional to its volume (P ∝ 1/V when n, T are constant). 2. Charles's Law: It states that the volume of a given amount of gas is directly proportional to its temperature when the pressure is held constant (V ∝ T when n, P are constant). 3. Avogadro's Law: It states that the volume of a given gas is directly proportional to the amount of gas when the pressure and temperature are held constant (V ∝ n when P, T are constant). Let’s combine these laws to derive the ideal gas equation: Since P ∝ 1/V, V ∝ 1/P Since V ∝ T and V ∝ n, we can write V ∝ nT Now, combining these two relationships, we get: V ∝ nT / P To convert the proportionality to equality, we need a proportionality constant: V = R × (nT / P) Rearranging the equation to have P on one side, we get the ideal gas equation: P × V = n × R × T

(c) Writing the Ideal-Gas Equation and Stating the Units)

The ideal gas equation can be written as: PV = nRT Where: 1. P: pressure in atmospheres (atm) 2. V: volume in liters (L) 3. n: amount of gas in moles (mol) 4. R: ideal gas constant, equal to 0.08206 L atm/mol K 5. T: temperature in Kelvin (K)

(d) Finding the Value of R in L bar/mol K)

To find the value of R when pressure is measured in bars, we first need to know the relationship between atm and bar. The conversion between atm and bar is: 1 atm = 1.01325 bar Now, we can calculate the value of R in L bar/mol K: \(R_{atm} = 0.08206 \frac{L \cdot atm}{mol \cdot K}\) We can convert it to L bar/mol K by multiplying by the conversion factor for atm to bar: \(R_{bar} = R_{atm} * \frac{1}{1.01325}\) \(R_{bar}\)= \(\frac{0.08206 \frac{L \cdot atm}{mol \cdot K}}{1.01325}\) \(R_{bar}\)≈ \(0.0807 \frac{L \cdot bar}{mol \cdot K}\) So when the pressure is measured in bars, the corresponding value of R is approximately 0.0807 L bar/mol K.

Key concepts

Boyle's Law

Boyle's law is one of the fundamental principles governing the behavior of ideal gases. In simple terms, it describes how pressure and volume are related for a fixed amount of gas under constant temperature conditions. According to Boyle's law, when the temperature remains unchanged, the pressure of a gas is inversely proportional to its volume.
What this means is that when you increase the volume of a gas, its pressure decreases, and vice versa. Mathematically, it is expressed as \(P \propto \frac{1}{V}\), or \(PV = k\), where \(k\) is a constant.
This principle is crucial for understanding how gases compress and expand in response to changes in pressure. Boyle's law is practical in various real-world applications, such as breathing, where the lungs expand and contract to manage the pressure of air inside them.

Charles's Law

Charles's law focuses on the relationship between the volume and temperature of a gas, provided the pressure remains constant. It states that the volume of a gas is directly proportional to its absolute temperature (measured in Kelvin). Having a direct relationship implies that as the temperature increases, the volume increases.
The formula can be written as \(V \propto T\), or \(\frac{V}{T} = k\), where \(k\) is a constant for a given amount of gas. It's crucial to remember that temperature must be in Kelvin to avoid negative values.
This law helps explain how gases behave under temperature changes. For instance, a balloon will expand in a hot environment due to increased kinetic energy, leading to greater volume as per Charles's law.

Avogadro's Law

Avogadro's law provides important insights into the relationship between the quantity of gas (in moles) and its volume. It asserts that the volume of a gas is directly proportional to the number of moles when temperature and pressure are constant.
In equation form, this is expressed as \(V \propto n\), or \(\frac{V}{n} = k\), where \(k\) is constant. This principle highlights that equal volumes of gases, at the same temperature and pressure, contain an equal number of moles.
Avogadro's law is instrumental in deducing the molar volume of gases, especially under standard conditions. It provides a foundational understanding for working with gaseous reactions in chemistry, such as those involving stoichiometric calculations.

Ideal Gas Equation

The ideal gas equation is a key mathematical relationship used to describe the behavior of ideal gases. It combines Boyle's, Charles's, and Avogadro's laws into a single equation: \(PV = nRT\).
Each term in this equation represents a critical property of the gas:

• \(P\) is the pressure of the gas
• \(V\) is the volume
• \(n\) is the number of moles
• \(R\) is the ideal gas constant
• \(T\) is the temperature in Kelvin

This powerful equation allows scientists and engineers to predict how a gas will respond to changes in pressure, volume, and temperature. Though real gases may deviate under certain conditions, the ideal gas equation remains a valuable tool in understanding gas behavior over a broad range of conditions.

Ideal Gas Constant

The ideal gas constant, denoted as \(R\), is a fundamental constant in the ideal gas equation \(PV = nRT\). It provides a link between the macroscopic properties of gases.
The value of \(R\) depends on the units of measurement for pressure and volume. Commonly, \(R = 0.08206 \frac{L \cdot atm}{mol \cdot K}\), when pressure is in atmospheres and volume in liters. If pressure is measured in bars, for instance, the constant becomes approximately \(0.0807 \frac{L \cdot bar}{mol \cdot K}\), ensuring consistent results across different measurement systems.
Knowing \(R\) allows for accurate calculations and predictions about how ideal gases behave under various conditions, with its universality offering ease across diverse scientific and engineering applications.