(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 \(\mathrm{L}-\mathrm{bar} / \mathrm{mol}-\mathrm{K}\).

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.

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

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)

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.