Thermodynamics

This applet presents a simulation of four simple transformations in a contained ideal monoatomic or diatomic gas. The user chooses the type of transformation and, depending on the type of transformation, adds or removes heat, or adjusts the gas volume manually. The applet displays the values of the three variables of state P, V, and T, as well as a P-V or P-T graph in real time.

A few notes:

The rest of this page presents some results of thermodynamics and sketches of theorem proofs.

Symbols and definitions

Definitions of terms
  • P: Absolute pressure, in units of Pascals (Pa).
  • V: Volume, in units of cubic meters (m3).
  • T: Absolute temperature, in units of Kelvins (K).
  • n: The number of moles of gas.
  • N: The number of molecules of gas.
  • NA: Avogadro’s number, the number of items in a mole of items, approximately equal to 6.022 x 1023.
  • R: The gas constant, approximately equal to 8.314 J/mole.K.
  • kB: Boltzmann constant, approximately equal to 1.38 x 10-23 J/K
  • KE average : The average translational kinetic energy of the gas molecules, in units of Joules (J).
  • U: The internal energy of the contained gas, in units of Joules (J).
  • dq: a differential of heat added to the gas, in units of Joules (J).
  • dw: a differential of work done on the gas, in units of Joules (J).
  • Cv: The heat capacity for transformations at constant volume, in units of J/K
  • cv: The specific heat for transformations at constant volume, in units of J/mole.K
  • Cp: The heat capacity for transformations at constant pressure, in units of J/K
  • cp: The specific heat for transformations at constant pressure, in units of J/mole.K
  • gamma:The ratio of Cp to Cv, equal to the ratio of cp to cv

Some of the fundamental principles of thermodynamics

Some of the laws of thermodynamics
  • The Ideal Gas Law; a combined statement of Charles's, Gay-Lussac's, and Boyle’s laws:
    PV = nRT
    or
    PV = NkBT
  • The first law of thermodynamics:
    first law of thermodynamics
  • The principle of equipartition of energy:
    The internal energy of the contained gas is equal to the sum of the translational and rotational kinetic energies of the gas molecules along or around the three directions in space. Furthermore, the contributions of these individual kinetic energies to the internal energy are equal.

A few theorem proofs

Kinetic theory

The kinetic theory is an analysis of the concept of pressure applied by a contained gas on the walls of the container based on the mechanical aspect of the repeated collisions and "bouncing off" of the gas molecules against the walls of the container.

Kinetic theory
The result of the Kinetic Theory:
Kinetic theory
Proof:
The change in momentum in the x-direction for each individual molecule and each collision is

In a time interval Δt the number of collisions for a given molecule is
,
where L is the box length in the x-direction. Thus, the momentum change for an individual molecule over a time interval Δt is
.
Generalizing to N molecules,
.
Using spatial symmetry:
.
The force applied in the x-direction is:
.
Using the definition of pressure:
,
or

Two additional results that follow (more or less directly):
The temperature is a direct measure of the average translational kinetic energy of molecules:

Each degree of freedom (i.e. each axis of rotation or translation that can contribute to the total kinetic energy of the molecule) contributes the following amount to the total kinetic energy:

Work done on a contained fluid

The following expresses the work done on a contained fluid in terms of it state variables.

The mechanical expression of work done on fluids
The mechanical expression of work done on a fluid:
Proof
Work is defined as W=F x displacement. Here F = P.A, so W = P.A ΔL, or W = P ΔV. Finally, the sign of the work done on the gas is positive when its volume decreases, so:

Internal energy of a contained gas

The following expresses the internal energy of a contained ideal gas in terms of its state variables.

The internal energy of a contained ideal gas.

U as a function of the state variables for gases
for monoatomic gases
for diatomic gases.

Proof
From the kinetic theory and the equipartition of energy, each degree of freedom contributes 1/2 kBT to the total enery of a system. A monoatomic gas has 3 degrees of freedom (3 translations, 0 rotation) and a diatomic gas has 5 degrees of freedom (3 translations, 2 rotations)

A derivation of the formulas used in the four transformations used in this applet

Derivation of results for four types of transformations

Isochoric transformations

Isochoric means constant volume. P and T are the only changing variables.
We have the following results:

  • From the mechanical definition of work:
  • From the result of kinetic theory:
    (Monoatomic)
    (Diatomic)
  • First principle of thermodynamics:
By simple substitution
(Monoatomic)
(Diatomic)
By integration, we get:
(monoatomic)
(diatomic)

We can then find P using the gas Law.

This result can also be written

with cv= 3/2 R for monoatomic gases and cv = 5/2 R for diatomic gases.

Isobaric transformations

Isobaric means constant pressure. V and T are the only changing variables.
We have the following results:

  • From the mechanical definition of work:
  • By differentiating the Gas Law at constant pressure:
  • From the result of kinetic theory:
    (Monoatomic)
    (Diatomic)
  • From first principle:
By simple substitution:
(monoatomic)
(diatomic)
or
(monoatomic)
(diatomic)
By integration, we find:
(monoatomic)
(diatomic)
We can then find V using the gas Law.
This result can also be written

with cp= 5/2 R for monoatomic gases and cp = 7/2 R for diatomic gases.
Two additional results:
cp-cv = R for both mono- and di-atomic gases.
= 5/3 for monoatomic gases and = 7/5 for diatomic gases.

Isothermal transformations

Isothermal means constant temperature. P and V are the only changing variables.
We have the following results:

  • From the Gas Law:
  • From the result of kinetic theory
    dU = 0
  • From first principle:
  • From the expression of work:
By simple substitution:

By integration,

Adiabatic transformations

Adiabatic means no heat exchange.
We have the following results:

  • By differentiating the Gas Law:
  • From mechanical aspect of work:
  • From the result of kinetic theory:
  • Because the transformation is adiabatic:
  • From the first principle of thermodynamics:
By simple substitution:

separating the variables:

By integration: