Intro to Thermal Physics

Heat, Temperature and Thermal Energy

The Atomic Hypothesis

“All things are made of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed together.”

– R. Feynman

Microphysics and Macrophysics• All matter is made of a myriad of moving

particles, (atoms).Pressure arises from the collision of the atoms with the walls of the container.The energy associated with the random internal motions of the atoms is called Thermal Energy, Eth.The temperature, T, of an object is a measure of its thermal energy. Heat is the exchange of energy by the collision of atoms with each other.

Goals of Thermal Physics

• To use our knowledge of microphysics and the statistics of large numbers to understand and predict the macroscopic properties of materials.

• To develop a “macrophysics” of the properties of objects that will allow us to predict the behavior of substances based on macroscopic quantities.

State Variables

• Volume• Pressure• Density (mass and number)• Thermal energy• Temperature• Heat Capacity• etc.

Moles

• N = number of molecules• NA = Avogadro’s Number =6.02 x 1023

• n = number of moles

ANNn =

Temperature

Heat and Thermal Energy

• Heat is energy that flows from a high-temperature object to a low-temperature object due to the difference in temperature.

• Heat is a transfer of energy: objects do not have “heat” –they have thermal energy.

Heat and Temperature

• When two objects are placed in thermal contact they will exchange heat until they reach the same temperature.

• This is why your coffee gets cold.

Q

ThotTcold

Twarm

Thermal Equilibrium

• When two objects are at the same temperature they no longer exchange energy (heat).

• Temperature differences determine whether or not there is heat flow.

• Objects that at the same temperature are said to be in thermal equilibrium.

The 0th Law of Thermodynamics

• If objects A and B are separately in thermal equilibrium with C, then objects A and B are in thermal equilibrium with each other.

• Temperature is a good measure of thermal equilibrium for all objects.

Intro to Thermal Physics

Measuring Temperature

The Kelvin Scale

• The SI unit of temperature is the Kelvin (K).

T = TC + 273.15

Absolute Zero

• When a gas is heated in a constant volume its pressure increases.

Absolute Zero

• If the pressure is plotted as a function of temperature for a variety of different gases we find the graph below:

Absolute Zero

• This implies that there is a lower limit to the temperature of an object. The lowest possible temperature is called Absolute Zero (0 K).

The Ideal Gas

A macroscopic model

The Ideal Gas

• We know that the pressure of a gas increases at constant volume when the temperature is increased.

The Ideal Gas

• For an ideal gas there is a relationship between the state variables of P, V, n, and T:

PV = nRT

• R = Gas Constant = 8.315 J/mol⋅K• This is the equation of state for an Ideal

Gas.

Ideal Gas

• Another way of writing the ideal gas law is in terms of the number of atoms or molecules rather than in moles. In this case

PV= kBNT

kB = Boltzmann’s constant = 1.38 x 10-23 J/K

The pV diagram

Ideal-gas processes

The pV diagram

Processes on a pV diagram

Processes on a pV diagram

Reversible and Irreversible Processes

Isochoric (Constant Volume)

Isobaric (Constant Pressure)

Isothermal (Constant Temperature)

Thermodynamics

Energy, Work and Heat

The Mechanical Equivalent of Heat

• James Joule showed that heat was a form of energy transfer.

• He demonstrated that when a given amount of mechanical work was done a given amount of water was heated, as indicated by the increase in its temperature.

• 1 cal = 4.186 J

Heat

• Heat is energy transfer so the SI unit of heat is Joules (J).• Other common units are

• Calorie (Cal) = 1 kilocalorie • calorie (cal)

Internal Energy

intEUKEsys ++=

K+++= nuclearchemicalthermal EEEEint

First Law of Thermodynamics

QWEEE thmechsys +=∆+∆=∆

• Q = heat• W = work

Thermal Energy and Temperature

• Thermal energy is a function of temperature.• Heat Capacity:

To raise the temperature of a system by a small amount ∆T, an amount of energy ∆E = C ∆T must be added to the system.C is the Heat Capacity of the system.

dTdEC th=

Work in the Ideal Gas

Work

∫

∫−=

=

2

1

2

1

V

V

s

s

pdVW

FdsW

Work and pV diagram

• W = negative of area under the pV curve

Expanding gas Compressed gas

Isochoric

• W = 0

Isobaric

• W = -p∆V

Isothermal

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

i

f

VV

nRTW ln

Kinetic Theory

Microscopic to Macroscopic

An Atomic Model of a Gas• The number of molecules is large. (N ~ NA)• The distant between them is large compared to the size of

the molecules.• Their motions are random.• They make elastic collisions between the walls and each

other.• The force between them is negligible except during

collision.• All the molecules are identical.

Pressure and Atoms

Pressure and Atoms

• Pressure is caused by collisions between the molecules and the walls.

A Box of Atoms

• Consider one of the atoms inside a box of volume V with sides of length d.

• The atom has mass m and velocity v.

Change in Momentum

The atom colliding with the wall experiences a change in momentum:∆px = -2mvxIt feels an average force:F1 = -2mvx/ ∆t

Time between collisions

• The time between collisions with the same wall is ∆t = 2d/vx

• The force becomesF1 = -2mvx/ ∆t

= -mvx2/d

Force on the wall

• The force on the wall is equal and opposite the force on the atom by Newton’s 3rd Law:F1wall = mvx

2/d

• The force due to all the atoms in the gas is F1wall = (m/d)(vx1

2 + vx22 + vx3

2 + …)

Average Force

• The average of the square of the velocity in the x-direction for N atoms is

• The total average force is N

vvvv Nxxxx

222

212 +++

=L

2xv

dNmF =

Average Velocity

• For a given atom:

• The average value of velocity squared for all the atoms in the box is

2222zyx vvvv ++=

2222zyx vvvv ++=

Average Square Velocity

• The motion is completely random so the atoms are just as likely to move in the y or z direction as in the x direction. Therefore

22 3 xvv =

Force on the wall

• The total average force

• becomes

2xv

dNmF =

2

3v

dNmF =

Pressure

• The pressure is F/A so

232 3

1 vmdN

dF

AFP ⎟

⎠⎞

⎜⎝⎛===

)(32 2

21 vm

VNP ⎟⎠⎞

⎜⎝⎛=

Pressure

• The pressure is proportional toThe number of molecules NThe average translational kinetic energy

)(32 2

21 vm

VNP ⎟⎠⎞

⎜⎝⎛=

Temperature

• From the ideal gas law PV = NkBT

)(32 2

21 vmNTNkPV B ==

221

23 vmTkB =

Temperature and Energy

• Temperature is a direct measure of average kinetic energy of the molecules.

• The internal (thermal) energy of the gas is

221

23 vmTkB =

nRTTNkvmNE B 23

232

21

int ===

The Equipartition of Energy

Spinning and Vibrating

• In addition to translating a diatomic molecule can rotate and vibrate.

• Each of these motions can share energy with each other.

Degrees of Freedom• Each motion has a number of “degrees of

freedom”Translation has 3: x, y, z motionRotation has 2: around x-axis, around z-axis.Vibration has 2: kinetic and potential energy

Equipartition Theorem• Each degrees of freedom contributes (1/2)kBT to

the internal energy.• Eint = (3 + 2 + 2) (1/2)kBT = (7/2)kBT

Equipartition Theorem• At “room temperature”:

Eint = (5/2)kBT • Vibrational motions don’t contribute as ∆Evib is

too large.

The Speeds of Molecules

The Maxwell – BoltzmannDistribution

Speeds of Molecules in a Gas

• Root-Mean-Square Speed

• Average Speed

• Most Probable Speed

mTkvv Brms /32 ==

mTkv B π/8=

mTkv Bmp /2=

Maxwell-Boltzmann Distribution

Evaporation

• Molecules with high velocities may leave a liquid becoming vapor.

The 2nd Law of Thermodynamics

Statistical Equilibrium

• On average balls will move from box with higher number of balls to box with lower number of balls.

• When they have the same number they will be in statistical equilibrium.

Statistical Equilibrium

• On average collisions will transfer energy from the high temperature side to the low temperature side..

• When they reach the same temperature, (when the atoms on each side have the same average kinetic energy), they will be in statistical equilibrium.

Reversible Processes

Microstates and Macrostates

• For any given macrostate (T,P,V) there are many microstates (x1, x2, …,v1,v2, …)

Statistical Equilibrium

• The equilibrium state is the most probable macrostate.

• Irreversible macroscopic behavior arises from reversible microscopic events because some macroscopic states are vastly more probable than others.

• The equilibrium state is the state with the most possible microstates.

• Entropy is a measure of the probability of a macrostate

Entropy and Order

The 2nd Law of Thermodynamics

• For an isolated system the entropy never decreases.

• Heat always moves spontaneously from high to low temperature.

• For an isolated system order turns into disorder.