# 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 gas is ideal
• The weight on top of the piston does not take atmospheric pressure into account: it is determined as if there were a vacuum on top of the piston.
• The numerical values used in this applet are semi-arbitrary, but were chosen according to the following general ideas:
1. The piston cross section is a disk of diameter equal to 4.67 cm
2. The diameter and height of the cylinder of gas are drawn on screen in the correct ratio.
3. The size of the cylinder as it is drawn on screen is approximately true to the chosen size (of course, actual size depends on monitor size and resolution).
4. The number of moles is 1.023 x 10-3, which corresponds to a round value of nR = 0.01 J/K
5. The values of P, V and T are constrained to stay within certain bounds. The upper limit on V is given by the physical height of the cylinder. The upper limits on P and T are 200 kPa and 200K so that the PV and PT graphs stay within their boundaries on screen. The minimum value of T is 2K and the minimum value of V is 21cc - these are somewhat arbitrary, but reflect the fact that absolute zero temperature cannot be reached.

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

## Symbols and definitions

 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 : 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). : a differential of heat added to the gas, in units of Joules (J). : 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 :The ratio of Cp to Cv, equal to the ratio of cp to cv

## Some of the fundamental principles 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: 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.

 The result of the 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 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.

 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.
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,

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: