Standard conditions

  • standard conditions are agreed upon by IUPAC
  • standard pressure is exactly $10,000\ Pa$
  • standard temperature is exactly $273.15\ K$

Definitions

  • ideal gas is a gas composed of molecules which do not interact with one another via intermolecular forces in any way
    • ideal gas cannot be condensed
    • it doesn’t exist in the real world, some molecules do come close to it though
      • the molecules need to be very small and the intermolecular forces very week
      • $O_2$, $N_2$, $CO_2$ and $H_2$
  • ideal gas interacts with its container, poushing in the walls resulting in pressure $p$
  • the speed $v$ of the particles is directly proportional to the gas' temperature $T$
  • the size of the container with the gas determines its volume $V$
  • closed system is a system which doesn’t exchange energy nor matter with its surroundings

Preliminary laws

Boyle’s law (Loi de Mariotte)

  • proven by Robert Boyle

For a constant temperature, pressure is inversly proportional to the volume of the gas in a closed system. $$P=\cfrac{k}{V}$$ $$PV=k$$

  • at a constant speed of particles (temperature), if the volume shrinks, the number of collisions of the gas and the container must increase and vice versa

Charles' law

  • discovered by Jacque Charles

For constant pressure, volume is directly proportional to temperature. $$V=kT$$ $$\cfrac{V}{T}=k$$

  • at a constant pressure, if we want to increase volume, we must also increase the speed of the particles (temperature)

Gay-Lussac’s law

  • discovered by Joseph Louis Gay-Lussac

For constant volume, pressure is directly proportional to temperature $$P=kT$$ $$\cfrac{p}{T}=k$$

  • at a constant volume, if we want to increase pressure, we also need to increase the speed of the particles (temperature)

Avogadro’s law

  • discovered by Amadeo Avogadro

The volume of an ideal gas is directly proportional to the ammount of substance $$V=kn$$ $$\cfrac{V}{n}=k$$

Ideal gas law

  • it is a combination of all already mentioned laws $$\cfrac{pV}{nT}=k$$
  • the constant $k$ is known as the unversal gas contant $R$ ($\approx{8.3145\ J\cdot{K^{-1}}\cdot{mol^{-1}}}$)
  • since the constant is universal, a mixture of multiple different ideal gasses will behave as an ideal gas still

Dalton’s law

The overall pressure of an ideal gas is the sum of the partial pressures of all the gasses. $$P=\sum_iP_i$$

  • the partial pressure can be calculated as the product of the molar fraction of the gas and the overall pressure $$P_i=x_iP$$
  • also: $$\cfrac{p_i}{p}=\cfrac{n_i}{n}$$
  • the oressure and ammount of substance of whatever kind is virtually the same for an ideal gas

Kinetic molecular theory of gases

Different forms of the ideal gas law

  • the ammount of substance in the ideal gas law can be rewritten using the number of particles: $$PV=Nk_BT$$
  • where $k_B$ is the Boltzmann constant
    • its value is approximately $1.380648\cdot{10^{-23}J\cdot{K^{-1}}}$
    • universal gas constant is the product of multiplying it and the avogadro constant $$R=N_Ak_B$$

Relation of temperature to the speed of particles

  • the mean speed of particles follows Maxwell-Boltzmann propability distribution
    • the equation counts the propability that a particle with three properties (mass, speed and temperature) exists
    • it is usually denoted as the propability $\mathbb{P}$

$$\mathbb{P}(m,v,T)=\sqrt{\left(\cfrac{m}{2\pi{k_BT}}\right)^3}\cdot{4\pi{v^2}e^{-\frac{mv^2}{2k_BT}}}$$

  • the velocities of the particals of a gas are not the same everywhere
  • the lower the temperature, the higher propability of particles moving is similar speeds
  • the higher the temperature, the higher the mean velocity of the particles
  • the lower the mass, the higher the mean velocity of the particles
  • the mean kinetic energy of the particles can be derived from the propability equation above

$$E_k=\cfrac{mv^2}{2}=\cfrac{3k_BT}{2}$$

  • at constant temperature, if we want to echange a light element for a heavier one, the kinetic energy must stay the same and the velocity drops
  • at constant velocity, if we want to exchange a light element for a heavier one, the kinetic energy must rise as does the temperature

Diffusion

  • it is the spontaneous mixing of two substances
  • the higher the temperature or the lower the mass, the faster the diffusion

Effusion

  • it is the process of a gas leaving a container with a hole in it
  • the faster the particle, the faster the effusion

Graham’s law

  • it desribes the rates of effusion in a mixture of two gasses

$$\cfrac{r_1}{r_2}=\sqrt{\cfrac{M_2}{M_1}}$$

  • $r$ is is the rate at which the gas escapes the container
    • it is proportional to the velocity of the particles

Deviations from the ideal gas law

Compressibility

  • the ideal gas law neglects the volume of the gas and its changing properties based on it
  • at low enough pressure, it is almost absurd speaking of a volume of gas when the molecules of it are so far apart
  • an ideal gas can be compressed infinitely and it never condenses
  • at high enough pressure or temperature, the propability of collision of two molecules rises significanly and thus the volume of the gas cannot be neglected anymore
  • gasses thus have a property called compressibility $z$

$$z=\cfrac{PV}{nRT}$$

  • gasses behave almost ideally only at certain conditions, from molecule to molecule

Van der Waals equation

  • derived by Johannes Diderickk van der Waals

$$\left(P+\cfrac{an^2}{V^2}\right)\cdot{(V-nb)}=nRT$$

  • the volume of the container is reduced by the volume of the molecules $nb$
    • $b$ represents a constant, the volume of one mole of the gas molecules
  • the pressure is lowered due to attractive forces of molecules
    • $a$ is a constant derived experimenally