Gasses

Gas Laws and Principles

Boyle’s Law

• Equation: $PV=k$
  • ( P ) = pressure
  • ( V ) = volume
  • ( k ) = constant at specific temperature
• Important Concept:
  • This law describes how pressure and volume are inversely proportional for a fixed amount of gas at a constant temperature.

Charles’ Law

• Equation: $V=bT$
  • This law states that the volume of a gas is directly proportional to its temperature at constant pressure.

Avogadro’s Law

• Equation: $V=cn$
  • This law states that the volume of gas is directly proportional to the number of moles at constant temperature and pressure.

Combined Gas Law

• Equation: $PV=nRT$
  • Combines Boyle’s, Charles’, and Avogadro’s laws.
  • ( R ) = ideal gas constant

Dalton’s Law of Partial Pressures

• Equation: $P_{\text{total}}=P_1+P_2+P_3+\dots$
  • The total pressure of a gas mixture is the sum of the partial pressures of each component.
• Mole Fraction (( x )):
  • $x_1=\frac{n_1}{n_{\text{total}}}$
  • Relates the part of each gas to the total moles in the mixture.

Real Gases

• Concept:
  • Real gases deviate from ideal behavior.
  • Volume and Attraction Adjustments:
    • $P^{\prime }=\frac{nRT}{(V-nb)}$ where ( b ) is a constant accounting for molecule size.
  • Corrected Equation:
    • Van der Waals Equation: $\left[P_{\text{obs}}+a{\left(\frac{n}{V}\right)}^2\right]=nRT$

Kinetic Molecular Theory (KMT) of Gases

• Main Points:
  • Gases consist of small particles in constant, random motion.
  • Volume of gas particles is negligible.
  • Collisions are perfectly elastic and there are no intermolecular forces.

Molecular Velocities

• Formulas:
  • Root-mean-square velocity: $u_{\text{rms}}=\sqrt{\frac{3RT}{M}}$
  • Most probable velocity: $u_{\text{mp}}=\sqrt{\frac{2RT}{M}}$
  • Average velocity: $u_{\text{avg}}=\sqrt{\frac{8RT}{\pi M}}$

Effusion

• Definition: The process by which gas molecules escape through a small hole into a vacuum.
• Graham’s Law of Effusion:
  • $\text{Rate of effusion}\propto \sqrt{\frac{1}{M}}$
  • Compares effusion rates of two gases:
  • $\frac{{\text{Rate}}_1}{{\text{Rate}}_2}=\sqrt{\frac{M_2}{M_1}}$

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Formula Sheet & When to Use Them

1. **Boyle’s Law: $PV=k$

   • Use when pressure and volume change at constant temperature.

2. **Charles’ Law: $V\propto T$

   • Use when volume and temperature change at constant pressure.

3. **Avogadro’s Law: $V=cn$

   • Use when volume and moles change at constant temperature and pressure.

4. **Ideal Gas Law: $PV=nRT$

   • Use for calculations involving any four variables: ( P, V, n, ) and ( T ).

5. Dalton’s Law of Partial Pressures:

   • Use for determining the total pressure from partial pressures.

6. Van der Waals Equation:

   • Use for real gases, especially at high pressures and low temperatures.

7. Molecular Velocities:

   • Use these to predict kinetic properties like speed in gases at different conditions.

8. Graham’s Law of Effusion:

   • Use to compare the rates of effusion for different gases.