Thermodynamics Exam 2

Intensive variables

do not depend on the size of the system

ex: pressure, temperature

Extensive variables

directly proportional to the size of the system

ex: volume, energy, mass

How do you make an intensive property from an extensive property?

extensive/n = intensive property

When is the ideal gas law applicable?

1. low pressure

• gas is diluted enough that its molecules are far apart from each other on average that wen can ignore there intermediates (IMFs)

2. High Temperature

• at low temperature, gas becomes a liquid because the gases can fee the IMFs (the molecules slow down to interact

Ideal gas equation of state

PV = RT

V= v bar, which equals V/n

van der Waals equation (corrections/what it accounts for)

accounts for deviations from the ideal gas law

1. low temperature

2. high temperature

corrects:

1. attractive forces

2. molecular volume

Figure 16.3

Shows that the ideal gas equation is not valid at high pressure

Compressibility factor

Z > 1 : very high pressure

Vbar real > Vbar ideal

*If repulsive interactions are dominating

Z < 1 : intermediate pressure

Vbar real < Vbar ideal

*When attractive interactions are dominating

Figure 16.4

Shows that the effect of molecular attraction becomes less important at higher temperatures

Coexistence curve

Any point within this curve corresponds to liquid and gas coexisting in equilibrium with each other

• At any point on or outside the curve, only one phase is present

Law of Corresponding States

the properties of all gases are the same if we compare them under the same conditions relative to their critical point

• predicted by Pc*V(bar)a/R*Tc which equals 3/8, indicating that this should be the same for all substances

van der Waals equation (a and b)

a: corrects for attractive forces

• Preal < Pideal

• frequency and force of collisions with the walls are both reduced by attractive forces

• Units: dm^6 * atm * mol^-2

b: corrects for the size of the molecules

• Units: dm³ * mole^-1

• Preal > Pideal

van der Waals equation in terms of Z

(don’t need to memorize equation)

High pressure: first term dominates

• volume of the molecules is not negligible

• Zreal > Zideal

Low pressure: second term dominates

• Zreal < Zideal

Isotherms

Holding temperature constant

Figure 16.7

dashed line is a coexistence curve

Isotherm: temperature is held constant

• at a higher and higher temperature, you get closer to the ideal gas law

• dot = critical point

• at a temperature greater than the critical point (T>Tc), the gas can’t be condensed

• Experiment!!

Figure 16.8

Pressure-volume isothenns of carbon dioxide around its critical temperature, as calculated from the van der Waals equation

• Theory!!

Second virial coefficient

B2v(T)

• the second virial coefficient is the most important because it reflects the first deviation from ideality as the pressure of gas increases, or the volume decreases

• can be determined experimentally from the slope of a Z against P plot ( Z-1 = B_2P(T)*P)

Boyle Temperature

B2v(T) = 0

• the repulsive and attractive parts of the intermolecular interactions cancel each other, and the gas appears t behave ideally

At the critical point…

Critical point

max temperature before the gas becomes a liquid

• any T scale we use to describe a gas is necessarily arbitrary. A gas does “know” its critical temperature and is therefore aware of its temperature relative to its critical temperature (its reduced T, TR)

Figure 16.10

Z = (Vbar_R)/(Vbar_R - 1/3) - 9/(8 * Vbar_R * T_R)

Virial Coefficients Depend on..

the forces of interaction between the molecules of the gas

• if no forces, the second and third virial coefficient = 0

B_2v(T) = Vbar - Vbar_ideal measures…

IMFs

Consider 2 interacting molecules, their interactions depend on…

1. orientation

2. distance

Perturbation theory with large values of r…

• models intermolecular forces

Repulsive ad attractive part of perturbation theory…

u(r) = + C₁₂/r₁₂ - C₆/r₆

first term is repulsive part and the second term is the attractive part

• usually written as the Lennard-Jones Potential

Lennard Jones Parameters

ε = well depth

• measure of how strongly molecules attract each other

σ = separation at which u(r) = 0

• measure of the size of the molecules

Order of magnitude of ε…

it will be way less than bond strength (100-1000 KJ/mol) because it correlates to breaking intermolecular forces rather than intramolecular forces

ε ~ 0.24 Kcal/mol

ε_imf « bond strength

4 ways molecules can store energy

1. Translational

2. Rotational

3. Vibrational

4. Electronic

Translational is approximately a…

continuum: the levels are so close and almost on top of each other

Spectroscopy

interactions of matter with light

E_photon = hv = hc/lamda

Boltzmann factor

If a system has states with energies E1, E2, E3, etc, the probability P_j, that the system will be in the state E_j depends exponentially on the energy of that state

Type of Energy + Model

Translational → particle in a box

Rotational → rigid rotator

Vibrational → harmonic oscillator

beta =

1/K_BT

The system partition function, Q, tells us…

the number of available states

We postulate that the average ensemble energy is equal to…

the observed energy of a system

The partition function of a system of independent, distinhuishable molecules is…

the product of molecular partition functions

Fundamental Postulate of Physical Chemistry

the average of any quantity, as calculated using the probability function, is the same as the experimentally observed value of that quantity

• experimentally observed energy of the system = u (which equals the average of E/n or E bar), or the internal energy