The Microscopic View

An equation of state is a macroscopic description of matter. Thermodynamic state functions such as the energy are averaged properties of a large number of particles.

For example, the pressure is an average quantity determined by the forces of microscopic particles that exchange momentum with the walls of a container. The kinetic theory of gases describes the connection between velocity of individual molecules and pressure.

The momentum of a particle is mu_x (in the x-direction).

For an elastic collision with the wall of the container the momentum of the particle is reversed and is now –mu_x. The change in momentum is D(mu_x) = mu_x – (-mu_x) = 2mu_x. If the distance between the two walls perpendicular to the x-direction is a, then the time elapsed between collisions is Dt = 2a/u_x because the molecules travels a distance of 2a to arrive back at the right-hand wall. The rate of change of momentum is equal to the force so Since the area of wall is bc the pressure exerted on the wall is

since V = abc.

Pressure is a force per unit area

One molecule cannot really exert a pressure, but the extension of the above reasoning to a container with N molecules shows that

The choice of the x-direction is arbitrary and the results for the y- or z-direction must be equivalent

Since the mean square speed of any molecule is

we have

for the mean square speed in one direction.

The fundamental equation of the kinetic theory of gases is

This equation relates a macroscopic property, the PV product, on the left hand side with a microscopic (or molecular) property m<u²> on the left-hand side. The kinetic theory of gases shows that

And therefore the equation also is equivalent to

The ideal gas law is an equation of state. It relates the macroscopic variables of n, V, and T to the pressure.

The ideal gas law was derived assuming that the particles do not interact. There is no force acting between them and hence no potential energy. Recall from physics that F = -dU/dx (the force is the gradient of the potential).

Real gases have finite size and forces acting between them. The microscopic description of these forces has a consequence for the macroscopic equation of state here too.
The equation of state is modified by molecular dimension and intermolecular potential energy.

The attractive force is a mutual electrostatic interaction.

The repulsive force is due to the finite dimension of molecules.

The potential energy surface may be a Lennard-Jones potential.

The Molecular Potential

The r^-12 term accounts for repulsion due to molecular radii. The r^-6 term accounts for attraction at larger distances.

Contributions to the r^-6 term include:

Dipole-dipole interactions are attractive on average.

where m is the ground state dipole moment. Note that the factor of kT in the denominator signifies that thermal fluctuations tend to disrupt interactions between the dipole moments and to reduce the magnitude of this term.

Dipole-induced dipole interactions are attractive.

where a is the molecular polarizability.

Induced-dipole-induced-dipole interactions are dominant if the dipole moment is zero.

where I is the ionization potential. This term is also known as the dispersion term (or the London dispersion attraction). Surprisingly, this is the dominant term in the attractive part of the potential energy surface. See McQuarrie and Simon’s calculation of these terms for HCl in section 16-6 for an example.

Properties of Gases

If a gas is sufficiently dilute it obeys the ideal gas law

The ideal gas law can also be written

A molar quantity is indicated by the bar across the top. The ideal gas law is an equation of state. An equation of state relates the pressure, volume, and temperature of the gas given a quantity of n moles of gas. The properties of a gas are of two type.

Extensive variables are proportional to the size of the system.

Intensive variables do not depend on the size of the system.

Extensive variables: volume, mass, energy

Intensive variables: pressure, temperature, density

If we divide an extensive quantity by the number of moles (or number of particles), we obtain an intensive quantity. For example, volume V (L) is an extensive quantity, but molar volume V/n (L/mole) is an intensive quantity.

Pressure in the ideal gas law has units of N/m² which is corresponds to force per unit area. When thinking about the unit of atmospheres which corresponds to the pressure at sea level, the force is that due to the weight of the atmosphere above the surface of the earth.

Problem: what is the mass of the atmosphere above 1 m² of the earth surface?

Solution: The pressure is a force per unit area. The force in this case is the weight of the atmosphere (F = mg). If we assume that g is a constant for the entire column of the atmosphere above 1 m² we have

m = PA/g = (1.0325 x 10⁵ N/m²)(1 m²)/(9.8 m/s²)

m = 1.05 x 10⁴ kg.

SI units of pressure refer to the N/m² as the Pascal (Pa). There are 1.0325 x 10⁵ Pa per atm. A new standard unit of pressure is the bar, where 1 bar = 10⁵ Pa.

If m is the mass of the liquid and g is the gravitational

acceleration, the force is F = mg. The pressure is

where r is the density, r = m/V. Note that the area of the object cancels exactly as in our example above for the mass of the atmosphere. The application of P = rhg is most common in liquids.

Problem: what is the weight of water above 1m² of a swimming pool that is 10 m deep? (You can solve it).

Temperature is perhaps the most difficult quantity to conceptualize. We shall see that temperature depends on microscopic motions of molecules. However, the temperature can be defined macroscopically based on the ideal gas law. Since the PV product cannot be less than zero, T cannot be less than zero. This implies an absolute zero of temperature.

We can consider a definition of a thermometer based on the following theorem. We consider object A that is in thermal equilibrium with object B. Further we can consider object B in equilibrium with C. The zeroth law of thermodynamics states that A is in equilibrium with C. This law implies that object C can act as a thermometer for other objects. The definition of a temperature scale based on the properties of water is known as the Kelvin scale. The triple point of water is defined to be at 273.16 K. K represents the degrees in the Kelvin temperature scale. The triple point is the unique point in the phase diagram where solid, liquid, and vapor coexist. The boiling point of water at 1bar of pressure is defined as the being 100 degrees higher (373.16 K).