Assumptions of the Ideal Gas Model and Its Applications
Assumptions of the Ideal Gas Model
- The ideal gas model is based on five key assumptions.
- These assumptions simplify the complex behavior of real gases, allowing us to predict their behavior using mathematical relationships.
1. Gas Particles Are in Constant, Random Motion
- Gas particles are never at rest; they move in straight lines until they collide with another particle or the walls of their container.
- This constant, random motion explains why gases fill any container they occupy, regardless of its shape.
Analogy
When you spray perfume in a room, the gas molecules disperse evenly, filling the available space.
Tip
Visualize gas particles as tiny billiard balls moving in random directions and bouncing off walls without losing energy.
2. Collisions Between Gas Particles Are Perfectly Elastic
- When gas particles collide with each other or with the walls of their container, no energy is lost as heat or sound.
- These are perfectly elastic collisions, meaning the total kinetic energy of the system remains constant.
- This explains why the pressure exerted by a gas on its container walls doesn’t decrease over time, as long as temperature and volume remain constant.
Common Mistake
Students often assume that gas particles lose energy during collisions, but in the ideal gas model, energy is always conserved.
3. Gas Particles Have Negligible Volume Compared to the Space They Occupy
- Although gas particles have mass and volume, their size is so small compared to the distance between them that we treat their volume as negligible.
- This assumption explains why gases are compressible and why their behavior can be described using simple equations.
Example
- Vaporized water occupies about 1600 times the volume of liquid water at standard temperature and pressure (STP).
- This dramatic expansion illustrates how much empty space exists between gas particles.
4. No Intermolecular Forces Act Between Gas Particles
- In an ideal gas, particles neither attract nor repel each other.
- This assumption allows gas particles to move independently of one another.
- As a result, an ideal gas cannot condense into a liquid, no matter how much the temperature is lowered.
Note
In real gases, intermolecular forces like van der Waals forces become significant at low temperatures or high pressures, causing deviations from ideal behavior.
5. The Kinetic Energy of Gas Particles Is Proportional to Temperature (in Kelvin)
- The average kinetic energy of gas particles is directly proportional to the gas's absolute temperature.
- This means that as temperature increases, particles move faster, increasing the pressure they exert on their container walls (if volume is constant).
- This relationship is why temperature is always expressed in Kelvin when dealing with gases.
Analogy
Think of temperature as a speedometer for gas particles: the higher the temperature, the faster the particles move.
Applications of the Ideal Gas Model
- The ideal gas model provides a foundation for understanding and predicting the behavior of gases under a wide range of conditions.
- Here are some key applications:
1. Explaining Gas Behavior Under Standard Conditions
- Under standard conditions (STP: 0°C and 100 kPa), most gases behave similarly, regardless of their chemical identity.
- The molar volume of an ideal gas at STP is 22.7 dm³ mol⁻¹, meaning one mole of any ideal gas occupies this volume.
- This uniformity simplifies calculations in chemistry, such as determining the amount of gas produced in a reaction.
Example
- Consider a reaction that produces 2.00 mol of carbon dioxide gas at STP.
- Using the molar volume, you can calculate the volume of CO₂ as:
$$V = n \times V_m = 2.00 \, \text{mol} \times 22.7 \, \text{dm}^3 \, \text{mol}^{-1} = 45.4 \, \text{dm}^3$$
2. Deriving the Ideal Gas Equation
- The ideal gas model forms the basis of the ideal gas equation, which relates pressure ($p$), volume ($V$), temperature ($T$), and the amount of gas ($n$): $$pV = nRT$$
- Here, $R$ is the universal gas constant ($8.31 \, \text{J K}^{-1} \text{mol}^{-1}$).
- This equation enables us to predict how gases respond to changes in conditions, such as pressure changes during weather balloon ascents.
Self review
Can you rearrange the ideal gas equation to solve for $n$, the amount of gas in moles?
3. Predicting Deviations from Ideal Behavior
- The ideal gas model also provides a benchmark for identifying when real gases deviate from ideal behavior.
- At low temperatures or high pressures, intermolecular forces and the finite volume of gas particles become significant.
- Under these conditions, real gases condense into liquids or exhibit non-ideal pressure-volume relationships.
Example
At 100 K and $1 \times 10^6$ Pa nitrogen gas deviates from ideal behavior due to intermolecular attractions, which reduce the pressure exerted by the gas.
Tip
Always ensure temperature is converted to Kelvin and pressure is in consistent units when using the ideal gas equation.
Self review
- A gas sample occupies 3.00 dm³ at 25°C and 100 kPa. Calculate the number of moles of gas in the sample.
- Explain why real gases deviate from ideal behavior at high pressures.
- Compare the behavior of helium (He) and water vapor (H₂O) under identical conditions. Which is more likely to behave as an ideal gas, and why?
