Thermodynamics: Microscopic vs. Macroscopic

(Chapters 16, 18.1-5 )

• Matter and Thermal Physics

• Thermodynamic quantities:

• Volume V and amount of substance

• Pressure P

• Temperature T:

• Zeroth Law of Thermodynamics

• Temperature scales

• Phase diagrams

• Ideal gas

• Macroscopic description:

• Thermodynamic processes

• PV diagrams

• Micro/Macro description: the kinetic theory of gases

• Molecular speeds

• Equipartition Theorem

Added energy (heating)

Matter – Phases

• Physics deals with the states, processes and interactions involving matter in various

phases (solid, liquid, gas, plasma). The study of the various phases and scales of

matter splits Physics into various branches with specific methods

• Phase transitions – driven by energy transfer – determine a restructuring of the net

microscopic energy content of the substance, called:

• E includes the atomic vibrational and rotational kinetic energy, and

the electric or nuclear potential

energy of atoms, molecules and

lattice

Melting

Freezing

Boiling

• E includes the atomic

translational, vibrational and

rotational kinetic energy, and the

potential energy of atoms and

molecules

• E includes the atomic

translational kinetic energy (plus

vibrational and rotational for a

molecular gas), and the potential

energy of atoms

Def: The internal energy E of an object is the net energy necessary to

build it from its elementary constituents: the kinetic energy K of its

micro-parts and the potential energy U associated with the various bonds E K U

Solid Gas Liquid

Sublimation

Condensation

What is Thermal Physics?

• Thermal Physics is the study of system interactions involving transfer of energy as

heat or/and work leading to the variation of the thermodynamics state of the matter

Ex: Say that the moving box discussed on one of the previous slides is metallic: the heat

released by friction is absorbed by the material increasing the temperature, which will

consequently determine an increase in volume and surface melting: these processes are

thermodynamic, only in complementary relationship with the mechanical motion of the box

• Thermal Physics describes the various phases based on the interdependence of the

macroscopic and microscopic scales of matter

• For instance, it uses statistical mechanics based on probability theory applied to large

ensembles of particles to model macroscopic properties inferred from averaged

microscopic mechanical, electromagnetic and quantum mechanical characteristics of

the atomic and molecular constituents of the system

• Besides internal energy, the macroscopic thermodynamic state of a simple system is

given by a relationship between several thermodynamic parameters:

1. Volume, V: describes the space filled by a certain amount of substance

2. Pressure, P: describes the force per unit area exerted in a material

3. Temperature, T: is a conventional parameter describing a thermal state

• In a thermodynamic process resulting in a change in the thermodynamic state, all or

only two of these parameters may change in a correlated fashion

Thermodynamic parameters – Volume and Quantity of substance

• Solids and liquids occupy fixed volumes of space, while gases take the volume of

the containing vessel

• The mass density describes how a substance of mass m is distributed within the

respective shape of the object m V

m A

volumetric mass density:

superficial mass density:

linear mass density: m L

• Thus, ff the mass is distributed uniformly

within a volume V, or plane of area A, or

along a line of length L, we define

• In the case of gases, it is customary to express the amount of gas in a given volume

in terms of the number of moles, n

• One mole of any substance is characterized by:

1. the same number of particles, called Avogadro number,

NA = 6.02×1023 particles/mol

2. a molar mass, M: for the monoatomic gases M is given numerically by the

atomic mass of the element expresses in gram/mol

3. a molar volume, VM: if the gas density is ρ, we have VM = M/ρ

A M

m N Vn

M N V

• Consequently, the number of moles in a mass m of gas

containing N particles within a volume V is given by

Exercise 1: molar calculations

A container of volume V = 10–4 m3 is filled with a mass m = 20 grams of argon gas.

a) How many moles of argon are in the container?

b) How many argon atoms are in the container?

c) What is the molar volume of argon?

Quiz:

1. Which contains more molecules, a mole of hydrogen gas (H2) or a mole of water vapors

H2O?

a) The hydrogen.

b) The water.

c) They each contain the same number of molecules.

2. Consider two diatomic gases, hydrogen H2 and oxygen O2, containing equal number of

molecules. What is the relationship between the masses of the two gases?

a) mH = mO/8

b) mH = 8mO

c) mH = mO/16

d) mH = mO

• The pressure is due to the collisions of molecules with the

walls of its container: as each molecule strikes the wall, its

momentum changes due to the force Fw→m acted by the

wall paired with a force Fm→w acted by the molecule on the

wall:

• The pressure in a thermally stable gas is the same in

every point throughout the gas

Def: The pressure P associated with a force F exerted uniformly across a surface

A is FP

A 2N m Pascal Pa

• Hence, the pressure determines how the effects of applying a force will be different

depending on how it is distributed on the surface

Thermodynamic parameters – Pressure

Ex: Pressure in gases

• gases contain a large number of molecules moving randomly.

When in a vessel, the gas fills the entire available volume

m wF

afterpbeforep

after before

m w w m

p ppF F

t t

w mF Newton’s 3rd Newton’s 2nd

• Dalton Law: The net pressure in a gas mixture is equal to the sum of the partial pressures

exerted by each gas independently occupying the entire volume of the container

Exercise 2: Pressure in a stationary liquid

In order to operate with the concept, let’s see how we can calculate the pressure at a depth h

under the surface of a liquid of uniform density ρ. Consider for instance, a calm volume of sea

water of density 1030 kg/m3. For start, consider an arbitrary surface of are A at depth h.

a) What force exerts a pressure onto the surface A? How can this force be expressed in terms

of given quantities?

b) Hence, what is the pressure at depth h in terms of given quantities?

Temperature – Zeroth Law of Thermodynamics

• Qualitatively, temperature T is a measure of how cold or warm is a certain

substance, associated with a subjective perception of its content of thermal energy

• In order to conceptualize temperature we must introduce the definitions:

1. Two objects are called in thermal contact if they can exchange heat

2. If two objects in thermal contact do not exchange energy, they are called in

thermal equilibrium

• Then, temperature can be defined using

• Hence, the temperature is the property that determines whether or not an object is in

thermal equilibrium with other objects, giving consistency to the zeroth law: two

systems are in thermal equilibrium if they have the same temperature

• Quantitatively, the measurement of temperature makes necessary an instrument

called a thermometer calibrated using a conventional scale of temperature

The Zeroth Law of Thermodynamics: If objects A and B are separately in

thermal equilibrium with a third object, C, then A and B are in thermal equilibrium

with each other

Temperature – Thermometers

• Thermometers are used to measure the temperature of an object or a system, making

use of physical properties that change with temperature

• Here are some temperature dependent physical properties that can be used: volume

of a liquid, length of a solid, pressure of a gas held at constant volume, volume of a

gas held at constant pressure, electric resistance of a conductor, color of a very hot

object, etc.

Ex: The mercury thermometer is an example of a common

thermometer that uses the variation of volume with temperature:

• The level of the mercury rises due to thermal expansion.

• Temperature can be defined by the height of the mercury column

• In order to quantify temperature, the thermometers are calibrated

using different temperature degree scales – such as Celsius °C,

Kelvin K or Fahrenheit °F – all of which are purely conventional

Ex: A thermometer using Celsius scale can be calibrated following the procedure:

1) Dip it into a mixture of ice and water at atmospheric pressure: the reading is assigned a

value of 0 °C. 2) Then, bring it to the temperature of boiling water: the reading is assigned a

value of 100 °C. 3) Impart the space between 0 and 100 into 100 segments, each representing a

change of temperature of one °C.

ice and

water

boiling

Temperature – The Kelvin Scale

• The Kelvin scale is the most common in science: it is defined in a fundamental way

rather than based on the properties of a certain substance

Ex: One way to define the Kelvin scale is by using a constant-volume

gas thermometer: a flask of some gas kept at constant volume in

thermal contact with the bath to be measured

• The bath temperature is varied and recorded (say in °C), and the

pressure in the gas is monitored using the height h of mercury column,

P = P0 + ρgh: the pressure will vary linearly with the temperature

• Irrespective of the starting pressure or the gas, if the lines are

extrapolated to zero pressure, they will intersect in the same point

which correspond to absolute zero temperature, –273.15 °C

• The Kelvin scale takes this temperature as zero, while one Kelvin is

taken to be equal to one degree Celsius, such that the two scales are

related by 273.15CT T

• In SI, the unit Kelvin is defined in terms of absolute zero

and the triple point of water – the point at 0.01 ºC where

water can exist as solid, liquid, and gas. Therefore

Def: One Kelvin is 1/273.16 of the temperature of the triple point of water

Temperature – Comparison between scales

• The most common scale used by public in the US is Fahrenheit scale related to the

Celsius scale by 9 55 932 32F C C FT T T T

Ex: Thus, we see that the temperatures of freezing water, of boiling water, and absolute zero

are given respectively by

95 0 C 32 32 F 95

100 C 32 212 F 95 273.15 C 32 459.67 F

• Notice that, while one Kelvin and one

Celsius degree are the same (since Celsius

is just an offset of Kelvin scale), a change

of temperature of one Fahrenheit degree is

9/5 times smaller than a change of one

degree Celsius, that is the variations of

temperature in the two scales are related by

95

1.8F C CT T T

Ex: If a quantity of water is warmed by 5 ºC,

the change in temperature expressed in ºF is

95 5 F 9 FFT

a)

b) (where ai are volume and gas dependent constants)

c)

4. An object is warmed up. Which of the following is true about the relationship between the

change in temperature expressed in Celsius, Fahrenheit and Kelvin scales?

a)

b)

c)

31 2 273dPdP dP

dT dT dT

273i iP aT

273i iP a T

Quiz:

3. Three gases are cooled down at constant volume, while the pressure in the gas is monitored.

The pressure winds out varying with temperature as on the figure. Which of the following is

true about the temperature dependency of the pressures Pi =1,2,3 in the three gases?

95C F K

T T T

C F KT T T

59C K F

T T T

Temperature – Phase transitions

• Pressure and temperature allow a more nuanced description of phase transitions

since each transition will occur at critical temperatures depending on the pressure

• Hence, a substance can be characterized using a phase diagram: a P vs. T map

showing regions of uniform phase bordered by boundaries of critical temperatures at

various pressures

Ex: Phase diagram of water

• Notice that the familiar critical temperatures of water (0 °C

for melt/freeze and 100 °C for boil/condense) are true only at a

normal atmospheric pressure P0 = 1 atm

• At lower pressures (such as when climbing at higher altitudes)

the interval between freeze and boil shrinks

• For lower pressures (P < 0.006 atm), the water cannot exist in

liquid form at any temperature: ice sublimates, that is, changes

into vapors directly

• Notice the point (0.006 atm, 0.01 °C) called the triple point of

water where all three phases coexist

Quiz:

5. What happens if, at a certain temperature, the pressure of liquid water is suddenly dropped

from above 1 atm?

a) The water definitely freezes b) The water definitely boils

c) Depending on temperature, the water may either freeze or boil

Ideal Gas – Characterization

• The ideal gas is a thermodynamic theoretical model used to emulate the behavior of

most gaseous systems, and as a startup and reference for more complex models

Ex: Most low density gases well above their condensation points behave like an ideal gas

• Macroscopic characteristics:

1. If not in a vessel, an ideal gas does not have a fixed volume or pressure

2. In a vessel, the gas expands to fill the container independent on the presence of

another gas in the same container

• Microscopic characteristics:

1. Collection of atoms or molecules that move randomly

2. Each atom or molecule is considered point-like

3. The particles exert no long-range force on one another

• If the particles of an ideal gas contain only one atom, the gas is called monoatomic.

If there are more bounded atoms per particle (molecules), the gas is called polyatomic

Ex: Noble gases, such as He, Ne, Ar, etc. are monatomic; H2 or O2 gases are diatomic

• The ideal gas model can be extended to describe non-chemical systems such as the

free electrons in metals, but the model fails to describe gases at very low

temperatures or high pressure, or heavy gases such as water vapors

Ideal Gas – Ideal gas processes

• Imagine an experiment on n moles of ideal gas, allowing the

independent modification of volume V, pressure P and temperature T

• Experimental observations:

1. isothermal (T = const.) PV = const. (Boyle’s law)

2. isobaric (P = const.) V/T = const. (Charles’s law)

3. isochoric (V = const.) P/T = const. (Gay-Lussac’s law)

• These results can be integrated into the equation of thermodynamic state

for the ideal gas given by

n, V, P, T

Ideal Gas Law: If n moles of ideal gas are confined in

a volume V under a pressure P at a temperature T, then

where R is the universal gas constant R = 8.31 J/mol.K,

and T must be in Kelvins

PV TnR

• Alternative forms:

A

Bm N

PV RT R VT NkM N

P T where kB is Boltzmann constant

kB = R/NA = 1.38×10–23 J/K

T1 T2 T3

T4

Ideal Gas – PV diagrams

• It is customary to represent ideal gas processes on P

vs. V graphs called PV diagrams

• Processes of n moles of gas are paths between states

characterized by sets of parameters (P, V, T)

• The processes are considered slow or quasi-static,

such that each point on a path is in thermal equilibrium

• Temperatures can be shown using isotherms: sets of

hyperbolas characterized by constant temperature

• There are infinitely many paths between any two

states, and they are reversible: that is, they can be

traveled in both directions

V

P

Increasing

temperature

Isotherms P1,V1,T1

V2 V1

P1

P2

Ex: PV diagrams for different types of processes

P

V

Isobaric:

P = const.

P2,V2,T2

compression

expansion

Isochoric:

V = const. P

V

Vi=Vf

fi

i f

VV

T T fi

i f

PP

T T

Isothermal:

T = const.

Path

P

V Pi=Pf

Ti=Tf

i i f fPV P V

Generic:

P

V

f fi i

i f

P VPV

T T

Adiabatic: the graph is steeper

than an isotherm P

V

i i f fPV P V

Quiz:

6. A cylinder of gas has a frictionless but tightly sealed piston of mass M. A

small flame heats the cylinder, causing the piston to slowly move upward. For

the gas inside the cylinder, what kind of process is this?

a) Isobaric.

b) Isochoric.

c) Isothermal.

7. A cylinder of gas floats in a large tank of water. It has a frictionless but

tightly sealed piston of mass M. Small masses are slowly placed onto the top

of the piston, causing it to slowly move downward. For the gas inside the

cylinder, what kind of process is this?

a) Isobaric.

b) Isochoric.

c) Isothermal.

Exercise 3: PV diagrams

The blue curves on the adjacent diagram are isothermals of a

monoatomic ideal gas, with equal temperature differences

between them. Identify each of the thermodynamic paths

represented on the figure as isobaric, isothermal, isochoric or

adiabatic.

P A

V

B C

D

Problems:

1. Molar volume: What is the volume occupied by one mole of any ideal gas in the so called

normal conditions, that is at atmospheric pressure P0 ≈1.01×105 Pa and temperature 0°C?

3. Thermal and mechanical equilibrium: A vertical cylinder of cross-sectional area A is

fitted with a tight-fitting, frictionless piston of mass m. There are n mols of an ideal gas in the

cylinder at temperature T0.

a) What is the height h0 at which the piston is in mechanical equilibrium?

b) Say that the piston is slowly pushed in by a constant force to a gas height H. What is the

final temperature T of the gas?

c) Say that the cylinder is dipped in a water tank at temperature T0. Then the piston is pushed

in just a bit and then released. Are the ensuing oscillations simple harmonic?

2. An isochoric process: A spray can of volume V0 =125 cm3 containing a propellant gas at

twice the atmospheric pressure, is initially at T0 = 22°C. The can is tossed in an open fire.

Assuming no change in volume, when the temperature in the gas reaches T = 195°C, what is

the pressure P inside the can?

4. Chain of thermodynamic processes: A quantity of n = 2 mol of monoatomic ideal gas

first expands isothermally from V0 = 0.003 m3 to V1 = 0.010 m

3 at a temperature of T0 = 0°C,

and then it is returned to the original volume V0 by means of an isobaric process. Then the

system returns to the original state via the shortest path on the PV-diagram.

a) Sketch the PV diagrams of all processes

b) Calculate the unknown parameters for each of the visited states.

Kinetic Theory – Assumptions

• The macroscopic properties of the ideal gas can be obtained using a model for the

behavior of the gas at microscopic level, as predicated by the kinetic theory

• Assumptions made by the kinetic theory about the molecules in a gas:

1.Their number is large and the average separation between them is large

compared to their dimensions

2. They obey Newton’s laws of motion, and move randomly

3. They interact only by short-range forces during elastic collisions

4. They interact with the walls by elastic collisions

5. The gas under consideration is a pure substance: all the molecules are identical

• So, individual molecules travel in zigzag: the average

distance traveled between successive collisions is called

mean free path λ, and depends on the concentration of

particles (N/V) and the size of each molecule such as the

radius r if the molecules are considered spherical:

21

4 2 N V r

Kinetic Theory – Molecular velocity

• The molecule speeds in an ideal gas are

distributed as described by Maxwell-Boltzmann

distribution: a bell-shaped curve that peaks at the

most probable speed vmp

• The peak flattens, broadens and shifts to higher

vmp as the gas temperature increases

• Notice that the area under the curve is actually the

total number of molecules N in the sample so it

doesn’t change with temperature

• The motion of the molecules can be

also described using the average speed

vavg of the molecules or their root-mean

square speed vrms given by the square of

the average of the speed squares:

• So, the average speed is not the most

probable speed:

Ex: Speed distribution in a sample

of nitrogen gas (N ≈ 105 atoms)

Most probable

speed

300 K

900 K

Equal

areas (N)

vmp vrms

2

rmsv v

mp avg rmsv v v

vavg

Quiz:

8. A gas experiences an isobaric increase in temperature. What happens with

the mean free path of the molecules?

a) Increases.

b) Decreases.

c) Doesn’t change.

Exercise 4: Molecular speeds

A six-molecule gas is confined in a container. A snapshot of the

gas reports the velocity vectors shown in the figure. Calculate

the average velocity, average speed and root-mean square speed.

21

4 2 N V r

Kinetic Theory – Deriving macro parameters from micro analysis

• Using the simplifying assumptions about the motion of large ensembles of molecules

in the gas, one can derive a relationship between the macro-characteristics (pressure P

and volume V) and micro-characteristics of the gas (such as the average translational

kinetic energy of a molecule):

2122

23 2

312

rms

r

NV N

avgV

am gs v

P v

mP

m

v

Physical situation: consider a volume V of ideal gas with N “spherical” molecules, each of

mass m. As we learned earlier, the pressure in the gas is due to collisions.

• Consider Nc such molecules within a distance Δx from a wall

of area A. Each time Δt, the average force on the wall is

• In the time Δt between two successive collisions, each

molecule travels distance 2Δx. Hence, Nc/Δt comes from

_x before xp mv

_x after xp mv

F

A

x

z

_ _2

x after x before

c c xc

p ppF

tm

t

NN N v

t

12x

x v t

y

12

2

c c cx

x

N N N NAv

N

A x Av t V Vt

• Therefore, since the molecules move identically in all directions, the pressure on the wall is

Average translational

kinetic energy per

molecule

2

xvAmV

NF

2

2 2 2 2 21 13 3

x

x y z rms

F NP mv

A V

v v v v v

Kinetic Theory – Molecular interpretation of temperature

• Comparing the kinetic expression for the pressure with the equation of state for the

ideal gas, we see that:

1. Temperature is proportional to the average translational kinetic energy per

molecule

2. The net kinetic energy of N molecules is proportional to the absolute temperature

3 3 32 2 2net avg B A B

A R

NK N Nk T N k T nR

NT

3 32 2 B

E nRT Nk T

23 22 2 2 1

3 3 3 2B B

avg

avg B avg rmsk k

B

PV NN Nk T mv

PV Nk TT

• Therefore, in a monoatomic ideal gas – where molecules cannot rotate or vibrate, and

there are no potential energies – the net translational kinetic energy provides all the

internal energy E of the ideal gas:

• Hence, we see that the internal energy – or thermal energy – of an certain amount of

ideal gas depends only on temperature, that is, if the temperature is constant, the

internal energy is conserved, and vice-versa

• However, be cautious: these particular formulas work only for ideal gases

Kinetic Theory

Ideal Gas Law

Thermal Energy – In Gases and Solids

• Notice that the thermal energy per particle of monoatomic gas contains the same

amount of energy attributed to each direction available for translation, x, y or z:

• The available distinct and independent motions or configurations of an object are

called degrees of freedom. So we see that a monoatomic gas gets ½kBT energy per

each available translational degree of freedom. This predication can be generalized by:

The Equipartition Theorem: At thermal equilibrium, the net internal energy stored

in a system is evenly shared by all its available modes, or degrees of freedom

21 12 23avg rms Bmv k T

Ex: Thermal energy equipartitioned in gases and solids

Diatomic gas:

Deg. of freedom: 3 transl. + 2 rot

For N molecules:

5 52 2

avg BE N Nk T nRT 62

3 3avg B BE N Nk T Nk T nRT

z

y y

x x

z

Simple Solid:

Deg. of freedom: 3 vibr.

Bonds: 3

Total: 6 modes

For N atoms:

Kinetic Theory – Molecular speeds and temperature

• By the fact that the internal energy of the ideal gas is given by the net kinetic energy,

we can estimate the temperature dependency of the root-mean-square speed vrms:

• Based on this result, we conclude that, at a given temperature, lighter molecules

move faster, on average, than heavier ones

2 2312 23 B

rms B rms

k TE N mv Nk T v

m

3 3Brms

k T RTv

m M

Quiz:

9. A rigid container holds both hydrogen gas (H2) and nitrogen gas (N2) at 100 °C. Which

statement describes the average translational kinetic energies of the molecules?

a) εavg of H2 < εavg of N2.

b) εavg of H2 = εavg of N2.

c) εavg of H2 > εavg of N2.

10. How about the rms speeds of the molecules in the gases above?

a) vrms of H2 < vrms of N2.

b) vrms of H2 = vrms of N2.

c) vrms of H2 > vrms of N2.

Problems:

5. Kinetic theory and internal energy: A vessel of volume V0 = 2.5

liter contains n = 0.20 mol of helium gas at temperature T0 = 10°C.

The container is warmed up to temperature T = 60°C.

a) How come even the molecules that do not make contact with the

container wall increase their speed in average?

b) Give examples of mechanical quantities that increase such that the

pressure on the walls increases?

c) Calculate the increase in internal energy of the gas

d) Calculate the rms speed at T0

6. Conservation of internal energy: A thermally isolated container of

volume V and temperature T is separated by a piston into two equal

compartments containing the same type of monoatomic ideal gas.

However, the left half has n moles, and the right half has 2n moles.

The piston is removed.

a) What happens with the temperature of the gas and why?

b) Write out the final internal energy of the gas in terms of n, R and T

c) Find the pressure in the expanded gas in terms of n, R, V and T

T0 T

T

n, P1, V/2 2n, P1, V/2

T

P, V