10.1 Ideal gas law and its applications

The ideal gas law is a crucial equation in thermodynamics, linking pressure, volume, temperature, and amount of gas. It's based on assumptions about gas behavior at the molecular level, making it a powerful tool for understanding and predicting gas properties.

While the ideal gas law is incredibly useful, it has limitations. Real gases deviate from ideal behavior under certain conditions. Despite this, the law finds wide application in engineering, atmospheric science, and everyday situations involving gases.

Ideal Gas Law

Ideal gas law fundamentals

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• Equation describing relationship between pressure, volume, temperature, and amount of an ideal gas
  • Ideal gas hypothetical gas perfectly following assumptions of kinetic molecular theory (no particle volume, no intermolecular forces, elastic collisions, kinetic energy proportional to temperature)
• Ideal gas law equation $PV = nRT$
  • $P$ pressure (pascals or atmospheres)
  • $V$ volume (cubic meters or liters)
  • $n$ amount of gas (moles)
  • $R$ ideal gas constant (8.314 J/mol·K or 0.08206 L·atm/mol·K)
  • $T$ absolute temperature (Kelvin)

Problem-solving with ideal gas law

• Rearrange ideal gas law to solve for any of four variables
  1. Solving for pressure $P = \frac{nRT}{V}$
  2. Solving for volume $V = \frac{nRT}{P}$
  3. Solving for amount of gas $n = \frac{PV}{RT}$
  4. Solving for temperature $T = \frac{PV}{nR}$
• Use appropriate units for each variable when solving problems
• Convert temperatures from Celsius to Kelvin by adding 273.15 ($T_K = T_C + 273.15$)

Assumptions and limitations of ideal gas law

• Assumptions of ideal gas law
  • Gas particles have negligible volume compared to container
  • Gas particles do not interact with each other (no attractive or repulsive forces)
  • Collisions between gas particles and container walls are perfectly elastic
  • Average kinetic energy of gas particles directly proportional to absolute temperature
• Limitations of ideal gas law
  • Real gases deviate from ideal behavior at high pressures and low temperatures
  • Ideal gas law does not account for intermolecular forces or finite volume of gas particles
  • Accuracy of ideal gas law decreases as gas approaches condensation point (phase change from gas to liquid)

Real-world applications of ideal gas law

• Calculating density of a gas under specific conditions
  • Density $\rho = \frac{m}{V} = \frac{nM}{V}$, where $M$ is molar mass of gas
• Determining molar mass of an unknown gas
  • Molar mass $M = \frac{m}{n} = \frac{PV}{nRT}$
• Analyzing behavior of gases in combustion engines (internal combustion engines) and heating/cooling systems (HVAC)
  • Calculating pressure change in cylinder during compression stroke of engine
• Estimating altitude based on atmospheric pressure changes
  • Pressure decreases with increasing altitude due to decreasing weight of air column above (atmospheric pressure gradient)