Real Gases and Deviations from Ideality

Real Gases

• Ideal gases are a fundamental assumption in gas laws and theories.
• Real gases have particles with non-negligible volumes and measurable interactions.
• The ideal gas law approximates real gas behavior, but deviations occur, especially at high pressure, low volume, or low temperature.

Deviations Due to Pressure

• Increased pressure pushes gas particles closer.
• As condensation pressure is approached, intermolecular attractions increase until the gas condenses into a liquid.
• At moderately high pressures (a few 100 atm), a gas's volume is less than predicted by the ideal gas law due to intermolecular attraction.
• At extremely high pressures, particle size becomes significant, causing the gas to occupy a larger volume than predicted by the ideal gas law.
• The ideal gas law assumes gases can be compressed to zero volume, which isn't physically possible.

Deviations Due to Temperature

• Decreased temperature reduces the average speed of gas molecules, increasing the significance of intermolecular forces.
• As the condensation temperature is approached, intermolecular attractions cause the gas to condense into a liquid.
• Near the boiling point, intermolecular attraction causes gases to have a smaller volume than predicted by the ideal gas law.
• Gases closer to their boiling point behave less ideally.
• At extremely low temperatures, gases occupy more space than predicted because particles cannot be compressed to zero volume.

Van der Waals Equation of State

• Corrects deviations from ideality.
• Equation: $(P + \frac{n^2a}{V^2})(V - nb) = nRT$
• $a$ and $b$ are physical constants determined for each gas experimentally.
• The $a$ term corrects for attractive forces between molecules:
  • Smaller for small, less polarizable gases (e.g., Helium).
  • Larger for larger, more polarizable gases (e.g., Xenon, $N_2$).
  • Largest for polar molecules (e.g., $HCl$, $NH_3$).
• The $b$ term corrects for the volume of the molecules themselves; larger molecules have larger $b$ values.
• Numerical values for $a$ are generally much larger than those for $b$.

Example: Ammonia

• Problem: By what percentage does the real pressure of one mole of ammonia in a one liter flask at 227 degrees C deviate from its ideal pressure? ($R = 0.0821 \frac{L mes atm}{mol mes K}$, for $NH_3$, $a = 4.2$, $b = 0.037$)

Solution:

1. Ideal Gas Law:
   $P = \frac{nRT}{V} = \frac{(1 mol)(0.0821 \frac{L mes atm}{mol mes K})(500 K)}{1 L} = \frac{0.0821 mes 1000}{2} = \frac{82.1}{2} = 41.5 atm$
2. Van der Waals Equation of State:
   $P = \frac{nRT}{V-nb} - \frac{n^2a}{V^2} = \frac{(1 mol)(0.0821 \frac{L mes atm}{mol mes K})(500 K)}{1 L - (1 mol)(0.037)} - \frac{(1 mol)^2 (4.2)}{(1 L)^2}$
   P = \frac{41.5}{0.963} - 4.2 \approx 41.5 + 4\% \tmes 41.5 \approx 43 - 4 \approx 39 atm
3. Actual Value: 38.8 atm
4. Pressure Difference: $41.5 - 38.8 = 2.7 atm$
5. Percentage Error: \frac{2.7 atm}{41.5 atm} \tmes 100\% = \frac{3}{40} \tmes 100\% = 7.5\%

Conclusion

• The ideal gas law ($PV=nRT$) shows the mathematical relationship among pressure, volume, temperature, and the number of moles of a gas.
• Special cases of the ideal gas law exist where pressure or volume is held constant.
• Henry's law explains the dissolution of gases in liquids and gas exchange in biological systems.
• Dalton's law relates the partial pressures of a gas to its mole fraction and the sum of the partial pressures to the total pressure.
• The kinetic molecular theory explains the behaviors of ideal gases.
• Real gases deviate from ideal behaviors; the van der Waals equation corrects for deviations caused by molecular interactions and volumes.
• Gases are important in everyday life and biological systems (e.g., oxygen and carbon dioxide exchange).