When looking at the macroscopic world and gases, we need to have macroscopic properties to measure the gases with. This can be:
- Volume, V.
- Mass, m.
- Density, p (rho).
- Temperature, T.
- Pressure, p.
Pressure vs VolumeTo perform an experiment of pressure vs volume, we need to make sure the mass and temperature of the gas stays constant. By changing the pressure, we can see how the volume changes. How are they proportional?
From this table and graph, it is clear it follows a typical y = mx graph where the y is 1/V and x is pressure. This is Boyle's law of pressure vs volume. If pressure is inversely proportional to volume, them the equation for pressure vs volume is:
p = k/V where k is a constant.There is no '+ c' in the equation because the graph passes through the origins. This also means that pressure x volume will find the constant. This brings us onto Boyle's law:
For a fixed mass of has at constant temperature, pressure is inversely portion all to volume.
The next equation we will look at for gases is the pressure/temperature law which is also known as Charles' Law.
The Gas LawsFrom producing a graph of pressure x volume = constant, while having temperature and mass constant, it will produce a straight line graph passing through the origin.
From producing a graph of volume / temperature will also produce a straight line graph passing through the origin. At this time, both pressure and mass are constants.
From producing a graph of pressure / temperature will produce a straight line graph passes through the origin. Volume and mass are constants.
From these three graphs, we can gather that pV/T will equal a constant. We can then add the number of moles of the gas to the equation to produce pV/nT = R (constant). From this, the following equation is produced:
pV = nRT
- P - pressure in Pa or N/m²
- n - number of moles of gas.
- V - volume in meters cubed.
- T - temperature in kelvin.
- R - molar gas constant = 8.3
The Molar Gas Constant, R
- pV = nRT.
- If n = 1 mole, pV = 1 x RT = RT.
- R = PV /T = 8.31 J/K
- R / Na (number of particles in a mole) = 8.31 / 6.023x10^23 = 1.38x10^-23 J/K per particle.
The Kinetic Theory Of Gases
pV = nRTThe consequence of the above relationship is that there is a lower limit to when T = absolute 0. This implies that the volume will be 0 and pressure will be 0. Gases are made of tiny particles. The particles have kinetic energy which falls as temperature moves towards 0 kelvin. When the particles collide, this involves the transfer of momentum.
Newton's third law describes equal but opposite forces on particles and the walls of a container. The collisions are elastic which creates the pressure.
From observable measurements and chemistry:
- pV = nRT from gas laws where R = 8.31 J/mol/K
- K = R /Na which is the Boltzmann constant which is 1.38x10^-23 J/K
pV = 1/3 N m vrms
- m = mass of the gas particle.
- N = number of particles in sample.
- vrms = average of the square of the speeds of the particles.
p = 1/3 ρ (rho) vrms as density = ρ = Nm / VTherefore, vrms = 3p / ρ (rho) = (3 x pressure) / density. Root mean square or 'vrms' is a microscopic quantity and pressure (p) and density (ρ) are macroscopic properties. vrms is appropriate as it is taken from a range of speeds and if we averaged only v, as it is a vector, it would equal zero.
The Boltzmann Constant and Average Kinetic Energy
pV = nRT and k = R / Na
N = nNa
- nRT = 1/3 N m vrms (N = number of particles, m = mass of one particle and vrms = average speed squared).
- But, K = R / Na and
- Therefore, we can change nRT = 1/3 N m vrms to n K Na T = 1/3 N m vrms .
- If we divide through by 'n', we get K Na T = 1/3 (N/n) m vrms. But, N/n = Na.
- Therefore, K
NaT = 1/3 Nam vrms. However, both 'Na's cancel each other out.
- Now, we will try to put the equation for kinetic energy into the equation (1/2mv²).
- 1/2 KT = 1/3 x 1/2 m vrms.
3/2 KT = 1/2 m vrms = average kinetic energy of one particleWhere K = Boltzmann constant at 1.38x10^-23, T = temperature in kelvin (+273 to any temperature in degrees Celsius), m = mass of the particle and vrms = average speed of the particle squared.
For gases, the average KE is proportional to Temperatre for one molecule or particle.
Average Thermal Energy
- In Joules per particle.
- In electron-volts where 1eV - 1.6x10^-19 J (W = QV = 1.6x10^-19 x 1V).
What Do We Really Mean By Average Energy, KT?
KT = 1/2 m vrms = Average Kinetic Energy.For a mixture of gases, at temperature T, the value of KT gives us the average KE per particle but for less massive molecules, their average speed will be greater. For example, a hydrogen atom will have a lighter mass and therefore a faster velocity. A oxygen molecule is heavier than hydrogen. Therefore, it will have a slower velocity to maintain the same KT as the hydrogen.
But, what do we really mean by average? What is the distribution of speeds that make up the average? From an experiment, we found the average energy per particle of warm water to be 5.15x10^-20 J. The energy needed to vaporize one molecule of water (ε) is 6.76x10^-20 J. We can make a ratio:
ε / kt = 6.76x10^-20 / 5.15x10^-21 ≈ 13
The Magic Ratio and the Boltzmann Factor
- Boyle's Law states pressure is inversely proportional to volume.
- Charles' law states volume is proportional to temperature.
- pV = nRT with R being the molar gas constant of 8.31 J/K.
- The Boltzmann Constant can be worked out through R / Na which equals to 1.38x10^-23 J/K per particle.
- pV = 1/3 N m vrms.
- vrms = 3 p / ρ (rho).
- N = nNa
- For gases only, 3/2 KT = 1/2 m vrms= average thermal energy per particle.
- The average energy per particle is KT.
- We can express average energy of a particle in electron-volts. 1eV = 1.6x10^-19 J.
- ε / KT is the 'Magic Ratio' which is usually 15-30 times larger than KT. If it is not (roughly) 15-30 times larger than KT, it is likely there are not any particles that have been 'lucky' enough to collide with other particles to gain extra momentum and energy to vaporize.
- The Boltzmann factor is e^-ε/KT.
- It is important to remember when putting temperature into any of the above formulas, it needs to be in kelvin. Physics and most other subjects only measure temperature in kelvin. To change your temperature from degrees Celsius to kelvin, add 273 to your current degrees Celsius temperature (0 degrees Celsius = 273 kelvin).