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Ladies and Gentlemen, welcome to the course "Physical Chemistry 101" My name is Dr. Lauth,
and today's topic is: "How to express a gas in numbers macroscopically?"
Thermodynamics aims to describe systems. Particularly simple systems are gaseous systems. "to express
a gas macroscopically in numbers means: a phenomenological description of the state
variables without any underlying model , a mathematical description with a function connecting
the state variables A single component system can be represented
by a surface in pVT-space. At high temperatures and low densities, so in this area marked
in green - the system is homogeneous and gaseous and may be described in a particularly simple
way. If we zoom out this area we have the phase diagram of a gas - or more precisely
of an ideal gas or perfect gas. If we examine a gas experimentally, we find
very simple relationships between the state variables p, V and T (pressure, volume and
temperature) Let´s examine the compressibility kappa of a gas, like the scientists Boyle
and Mariotte did several hundred years ago. Examining, how volume depends on pressure,
we find that at constant temperature, a doubling of pressure causes the volume to be reduced
to half. We can say that pressure and volume are inversely
proportional. The product (p • V) is a constant. p (1) * V (1) is equal to p = (2) * V (2)
p is inversely proportional to V. This is Boyle's law, with its help you can
easily calculate the compressibility of an ideal gas to 1 over p.
If we represent Boyle's law graphically, the PV diagram shows the following figure: (1) is
the initial state - large volume V1 small pressure p1 - (2) is the final state - smaller
volume V2, larger pressure p2 - - The two points lie on a hyperbola, mathematically
described as p • V. (p * V = constant) times 12.4 liters times 2 bars is make exactly 1
bar times 24.8 liters. The same representation in the pVT phase diagram
gives the red line a sectional view of the surface at a constant
temperature. At higher temperature, the isotherm would run in this form (yellow).
If we examine the thermal expansion coefficient alpha of a gas, we come to the laws of Charles
and Gay-Lussac. If we keep pressure constant and change temperature, the volume will be
proportional to temperature. We observe similar behavior for changing the
pressure at constant volume. Pressure is proportional to temperature (in Kelvin) for isochoric heating
This is the so-called Charles law. P over T is constant. p is proportional to T.
If we plot this in a diagram, we obtain a straight line. State (1) corresponds to a
lower pressure p1 at a low temperature T1, state (2) corresponds to a higher pressure
p2 at a higher temperature T2. The two states are on a zero-point straight
line: If we extend the line to a temperature of 0 Kelvin (negative 273.15 ° C), both the
pressure and volume of an ideal gas would disappear.
Both Charles´ isochores or Gay-Lussac's isobars can be represented in the three-dimensional
phase diagram and result in straight lines on the curved surface
If we keep both pressure and temperature constant, and only change the amount of substance of
a gas, then we´ll find a very simple relationship, namely: V is proportional to n. in fact, a
trivial relationship. One mole of a gas has half the volume as two
moles of a gas have. But the molar volume of a gas is a very special property. According
to Avogadro the number of molecules or atoms in a specific volume of ideal gas is independent
of their size. So molar volume of a gas is independent of
gas type: At IUPAC standard state one mole of any gas or gas mixture take 24.8 liters;
The molar volume of a gas at DIN normal conditions is 22.4 liters per mole
If we combine these four gas laws, we obtain the so-called ideal gas law.
This law is represented mathematically bei this curved surface in pVT space. The ideal
gas law is a function of two variables; it´s the equation of state for a gaseous system.
p is equal to R times T over Vm, we describe pressure p as a function of temperature T,
and the molar volume Vm. The proportionality constant R is the universal gas constant and
is the same for all gases. By mathematical transformation we easily can
derive Gay-Lussac's law and Boyle-Mariotte law from the combined gas law. Graphically,
this corresponds to the pT-plane projection or the pV plane projection of the pVT surface.
The ideal gas law describes the states of all gases and gas mixtures. It is often formulated
as p * V = n * R * T: p: pressure, V: volume, n: the number of moles, T: the temperature
and R: the gas constant. In SI units the gas constant is 8.314 (J / (mol
* K) or 0.082 L * atm / (mol * K)) If we do not want to use the number of moles
but the number of gas particles N, we have to use Boltzmann constant k instead of R
With the ideal gas law, we are able to calculate any State on this pVT surface. The ideal gas
law can be used to determine the molar mass of a gas. A gaseous nitrogen oxide with the
nitrogen / oxygen ratio of 1:2, either has the formula NO2 or N2O4.
An investigation of a state of this gas using the ideal gas law could clarify this. We determine
the mass of a sample of nitric oxide, do measurements of pressure, volume and temperature.
We now may determine the molecular weight using these four data.
The molar mass M is the quotient of mass m by the amount of substance n
According to the ideal gas law n equals p * V / (R * T). In summary, we get this formula.
By plugging in all variables in SI units, we calculate a molecular weight of 0.092 kg
/ mol. The gas is obviously the gas dinitrogen tetroxide N2O4
The ideal gas law also holds for gas mixtures: If we calculate the total pressure of this
mixture, we take the entire amount n (tot), multiply by R * T and divide by V.
If we virtually remove all components but one (the red one), we may also calculate a
virtual pressure -- which was called partial pressure by Dalton. The partial pressure of
the red component is the pressure that this component would exert, if it was the only
component in the volume. Each component has a partial pressure in a
gas mixture. Although the measurement of the partial pressure requires some effort - selective
pressure sensors have to be used - but the calculation can be done by the simple ideal
gas equation. If we divide partial pressure relative by
total pressure, we get n over n total, which is the mole fraction y.
The partial pressures p (i) add up to the total pressure- just like the moles n (i)
do to the total amount of substance. These are the two formulations of Dalton's Law.
An important gas mixture is air. Dry air consists of about 78 mole% nitrogen and 21 mole% oxygen.
At a total pressure of 1 bar, the partial pressure of nitrogen p (N2) is equal to 0.78
bar and the partial pressure p of oxygen (O2) is equal to 0.21 bar.
Let´s summarize todays lecture: The surface of state in a pVT diagram of a
gas is mathematically well described by the ideal gas law. p is equal to R times T over
Vm. In gas mixtures we may define partial pressures p(i) which add up to the total pressure.
Thanks for watching.