Ideal Gases - 4 main assumptions

Thermal Physics models use the ideal gas concept, which is based on these four main idealizations:

1) Gas atoms/molecules are point particles;

2) Intermolecular forces are negligible except during collisions;

3) Time between collisions is large compared to the average duration of a collision;

4) Molecules move with uniform velocities between collisions.

The Gas Laws

• Two gases, e.g. chlorine and argon, become ideal gases when their physical properties are independent of their chemical composition;

• The gas laws were determined experimentally for ideal gases. The equations must be modified in order to be applied to real gases;

Boyle’s Law

p α V^-1

or: pV = constant

Note: Plotting p versus (vs.) V^-1 yields straight lines.

Charles’s Law The Pressure Law

The ideal gas equation

p V = n R T

Physically, R is the constant of proportionality that relates energy and temperature. It is called the UNIVERSAL GAS CONSTANT.

R = 8.31 J/mol K

Boltzmann constant k_B = R/N_A=1.38x10^-23 JK^-1

• The internal energy of a system (U) changes only through an exchange of energy with the surroundings in the form of work (W) and/or heat (Q);

• This law expresses the equivalence of work and heat.

Equilibrium ↔ the most random arrangement of gas molecules

The tendency of a system is toward disorder.

Entropy ∆Q is the amount of energy gained or lost in a reversible process between two equilibrium states at constant temperature T. is the change in entropy

From a statistical perspective, entropy increases with the number of possible arrangements of a system;

Entropy is a measure of a system’s progress towards equilibrium.

The 2^nd Law of Thermodynamics

The entropy of an isolated system can only increase or stay the same;

OR: Heat can only flow from hot to cold objects;

OR: Heat engines cannot be 100% efficient.

Consequencies:

• At a microscopic level, an increase in entropy leads to higher randomness (disorder);

• The entropy can decrease only for a non-isolated system, but the entropy of the surroundings will increase;

• The entropy of the universe increases continuously;

• Time moves in the direction of increasing entropy.

The 3^rd Law of Thermodynamics

• Objects can never be cooled down to absolute zero (using only thermodynamic means and a finite number of steps).

• Would Heisenberg’s uncertainty principle, ΔxΔp ≥ h/4π, be valid at absolute zero?

Kinetic Theory of Gases – Assumptions

1. The number of molecules in the gas is large and the average separation between them is large compared with their dimensions.

2. The molecules obey Newton’s laws of motion, but as a whole, they move randomly.

3. The molecules interact only through short-range forces during elastic collisions.

4. The molecules make elastic collisions with the walls of the container.

5. All molecules in the gas are identical.

2 . Molecular model for pressure in an ideal gas Consider an ideal gas in a cube with edges of length d. The gas consists of N molecules with mass m and volume V collides elastically with the inside walls of the cube.

Consider a gas molecule travelling in the x- direction Before collision: After collision: velocity: -v_x +v_x momentum: -mv_x +mv_x change in momentum : The force F exerted on the wall by the molecule in time t obeys Newton’s 2nd law: To make two collisions, a single molecule travels distance 2d in time interval ∆t = 2d / vx (distance traveled is 2d)

The average value of the square of the velocity in the x-direction for N molecules is:

The total force on the wall becomes

Since the motion of the molecules is in 3-dimensions, the motion is random therefore the average value is:

= + + Using the equation for the velocity in the x-direction:

The total force on the wall now becomes:

Finally the total pressure exerted on the walls becomes: Pressure is proportional to the number of molecules per unit volume and to the average translational kinetic energy of a molecule.

Review of Ideal Gas Equation is the number of moles, and R = 8.31 J/mol K. We now introduce a new term; Boltzman’s constant; 1.38 x 10 ^-23 J/K Where: , which gives a new version of the Ideal Gas Equation: The new equation is used for molecules of the gas, instead moles of gas.

From the equation relating pressure to kinetic energy:

We obtain:

Compare this to the equation for an ideal gas:

Re-arranging:

we get:

which relates the translational molecular kinetic energy to temperature expressed as:

For a monatomic gas like He or Kr only translational motion is possible therefore the internal energy U for a monatomic gas is:

For polyatomic gases like O₂ or CH₄ other types of motion are possible eg: rotational motion

• The root-mean-square (rms) speed of the molecules is given by:

• where M is the molar mass in kilograms per mole

Maxwell-Boltzmann velocity distribution

Heat Transfer by Conduction

• T hermal conduction of heat occurs at microscopic scales through collision between particles of an object

• Less energetic particles gain energy by collision with more energetic particles, resulting in heat being transferred from hot regions to cold regions

• Metals are good thermal conductors. Heat is transferred via oscillations of atoms and motion of free electrons

• Conduction can occur only if there is a difference in temperature between two parts of a conducting medium

• The rate of energy transfer can be expressed as:

• F or a continuous change in temperature along an object of length, one can define the temperature gradient as:

Heat Transfer by Convection

• Convection corresponds to the transfer of heat through the movement of a substance (fluid)

• The system evolves towards a homogeneous temperature via mixing of hot and cold liquids or gases

• Natural convection occurs when the motion of fluids results from differences in densities (hot fluids have a lower density than cold fluids)

• Forced convection occurs when the motion of fluids is forced by an external agent (e.g. fan heater, wind etc.)

Heat Transfer by Radiation

• All objects radiate energy continuously in the form of electromagnetic waves due to thermal vibrations of their molecules

• For a blackbody, the rate of radiation is given by Stefan’s law. (next slide)

• A blackbody is an ideal body that absorbs and/or emits all radiation incident on it, and hence is a perfect absorber/emitter.

• Therefore, for a perfect blackbody, emissivity ε = 1

Stefan-Boltzmann’s Law

• Stefan-Boltzmann’s law relates the power emitted by a blackbody through radiation to its temperature according to:

• P is the power radiated by the object

• σ is Stefan-Boltzmann’s constant

(σ = 5.67 × 10^-8 W m^-2 K^-4)

• A is the surface area of the object

• T is the absolute temperature (in K)

Emission and Absorption of radiation

• Blackbodies not only emit radiation, but also absorb radiation emitted by the surroundings (also assumed to behave as a blackbody)

• The net energy emitted each second by a blackbody at a temperature T can be expressed as:

Where P_net is the net power radiated by the object

T₀ is the absolute temperature of the surroundings

• Note: When T = T₀, the net energy transfer is zero but objects still exchange energy to each other.

The case of grey bodies (emissivity < 1)

• Bodies emitting less radiation than blackbodies at the same temperature are called grey bodies

• Grey bodies are characterised by their emissivity

• The power emitted by a grey body of surface A, temperature T and emissivity ε can be expressed as:

• When an object is in equilibrium with its surroundings, it radiates and absorbs energy at the same rate, so its temperature remains constant.

Wien’s Law