Ideal gas equation

$$\begin{array}{c} \begin{aligned} pV &= nRT \end{aligned} \\ \hspace{-9em}\textbf{where} \\ \begin{array}{rl} p & \text{pressure in $\text{Pa} = \text{Nm}^{-2} $ or $\text{atm} = 1.01 \times 10^5 \text{Pa}$} \\ V & \text{volume in $\text{m}^3$} \\ n & \text{amount in mol} \\ R & \text{molar gas constant $= 8.31\text{JK}^{-1}\text{ mol}^{-1}$} \\ T & \text{absolute temperature in K} \end{array} \end{array}$$

Assumptions

• Gas particles occupy negligible volume as compared to the volume of the container
• The forces of attraction and repulsion between gas particles are negligible
  • i.e. there are negligible intermolecular forces
• Gas particles collide elastically
  • i.e. there is no loss in kinetic energy after collision
• Gas particles are in constant random motion
• The average kinetic energy of the gas particles is directly proportional to the absolute Kelvin temperature
  • At a particular temperature, all particles possess the same average kinetic energy

Deviations from ideal gas behaviour

• Real gas particles occupy significant volume

• Real gas particles experience significant intermolecular forces of attraction between them

• Pressure

  • Low Pressure
    • Few gaseous particles exist in a unit volume; the volume of particles is negligible compared to the container volume
    • Molecules are far apart, making intermolecular forces insignificant
  • High Pressure
    • Many gaseous particles exist in a unit volume; the volume of particles cannot be considered negligible
    • Molecules are close together, making intermolecular forces significant

• Temperature

  • High Temperature
    • Gaseous particles have great speed and average kinetic energy
    • Particles can easily overcome intermolecular forces, which are thus insignificant
  • Low Temperature
    • Gaseous particles have less speed and average kinetic energy
    • Particles are less able to overcome intermolecular forces, which are thus significant

• Conditions for Ideal and Non-Ideal Behaviour

  • Real gases behave most ideally at low pressures and high temperatures
  • Real gases deviate most from ideal behaviour at high pressures and/or low temperatures

• Gases that Deviate Significantly from Ideal Behaviour

  • Large molecules (e.g., $\ce{CO2}$): Have a large electron cloud and thus stronger intermolecular forces between molecules
  • Polar molecules (e.g., $\ce{HCl}$ and $\ce{NH3}$): Have stronger intermolecular forces of attraction such as permanent dipole-permanent dipole interactions or hydrogen bonding

• Deviation from ideal gas is compared with graph of $\frac{Pv}{RT}$ against P

  • The graphs begins at 1 and dips before rising above the line of ideal gases
  • dip
    • cause by intermolecular forces of attraction which pull molecules together, causing the gas to compress more easily than an ideal gas
    • The actual volume of the gas is less than the expected ideal volume, making $pV$ smaller and causing the graph to dip below 1
  • rise
    • At high pressures, molecules are packed tightly and resist further compression
    • The space occupied by the molecules becomes a significant fraction of the total container volume
    • The actual volume is greater than the expected ideal volume, making $pV$ larger and causing the graph to rise above 1

• For smaller molecules like $\ce{H2}$, they do not dip at all as the dispersion forces between them are too weak

  

• When changing temperature, deviation is greatest at low temperatures

  

Van der Waals equation

$[P+a\left( \frac{n}{v} \right)^2](V-nb)=nRT$

• a (intermolecular forces correction)
  • unit is $atm\cdot L^2\cdot mol^{-2}$
• b (molecular volume correction)
  • unit is $L\cdot mol^{-2}$
• $1L = 1dm^3$
• $1\text{bar} = 100000Pa$
• since $1L\cdot \text{bar} = 100J$, molar gas constant can also be in $0.0831\ L\ \text{bar}\ K^{-1}\ \text{mol}^{-1}$
• will provide during exam