Chapter 15 - Ideal gases

• One mole is defined as the amount of substance that contains as many elementary entities as there are atoms in 12 grams of carbon-12
• One mole is defined as the amount of substance that contains as many elementary entities as there are atoms in 12 grams of carbon-12

  $N =$$n \times N_A$
• Molar mass is the mass of one mole of a substance


  m = n $\times$ M
• The ideal gas assumptions:

  • The gas contains a very large number of atoms or molecules moving in random directions with random speeds
  • The atoms or molecules of the gas occupy a negligible volume compared with the volume of the gas
  • The collisions of atoms or molecules with each other and the container walls are perfectly elastic
  • The time of collisions between the atoms or molecules is negligible compared to the time between the collisions
  • Electrostatic forces between atoms or molecules are negligible except during collisions
• Atoms in a gas are always moving and when the collide with the walls of a container the container exerts a force on them changing their momentum so that the total change in momentum is -2$mu$
• Boyle's law states that for a gas with a constant temperature and mass, the pressure of an ideal gas is inversely proportional to its volume 


  $p \propto \frac{1}{V}$ or $pV =$$constant$
• Charles's law states that for a gas with a constant mass and volume, the pressure of an ideal gas is directly proportional to its absolute temperature 


  $p \propto T$ or $\frac{p}{T} =$$constant$
• Estimating absolute zero

  Using the equipment below the temperature of the water bath can be increases, and the resulting increase of pressure inside the sealed container can be recorded. At absolute zero, pressure must also be zero. A pressure temperature graph is plotted and then through extrapolation to where pressure is zero, an estimate for absolute zero can be gained.

  
• For an ideal gas


  $\frac{p_1 V_1}{T_1} =$$\frac{p_2 V_2}{T_2}$
• The ideal gas law

  pV = nRT
• For a graph of pV against T for a fixed amount of gas, the gradient of the line is nR where R is the molar gas constant
• The root mean square speed is the square root of the mean square speed of all the particles in a gas
• Root mean square speed provides a way to describe the motion of particles in a gas as it is an average. However the random motion of particles means that the particles have a wide range of speeds. This can be shown with the Maxwell-Boltzmann distribution
• Increasing the temperature of a sample, will increase the average speeds and spread out the distribution more.

  
• The Botzmann constant relates the mean kinetic energy of the atoms or molecules in a gas to the gas temperature

  $k=$$\frac{R}{N_A}$
• At a given temperature the atoms or molecules in different gases have the same average kinetic energy however they have different masses and therefore different rms speeds
• The internal energy of a gas is the sum of the kinetic and potential energies of the particles inside the gas. In an ideal gas it is assumed that electrostatic forces between particles in the gas are negligible except during collisions. Therefore all the internal energy in a gas is kinetic meaning that temperature, kinetic energy and internal energy are all proportional