Lecture 1: Preliminaries Schroeder Ch. 1.1 1.3 Williams Ch. 1.1 1.4 - - PowerPoint PPT Presentation

Lecture 1: Preliminaries

Schroeder Ch. 1.1 – 1.3 Williams Ch. 1.1 – 1.4

Outline

• Preliminary definitions
• Temperature and thermal equilibrium
• Temperature scales and thermometers
• Macroscopic model of the ideal gas model
• Elementary kinetic theory

What is Thermal Physics?

• Thermal physics = Thermodynamics + statistical

mechanics

• Thermodynamics provides a framework of relating the

macroscopic properties of a system to one another.

• It is concerned only with macroscopic quantities and

ignores the microscopic variables that characterize individual molecules

• Statistical Mechanics is the bridge between the

microscopic and macroscopic worlds: it links the laws

• f thermodynamics to the statistical behavior of

molecules.

Thermodynamic Systems

• A thermodynamic system is a precisely

specified macroscopic region of the universe together with the physical surroundings of that region, which determine processes that are allowed to affect the interior of the region.

• A thermodynamic system can be classified in

three ways: – Open systems can exchange both matter and energy with the environment. – Closed systems can exchange energy but not matter with the environment. – Isolated systems can exchange neither energy nor matter with the environment.

Thermodynamic State

• A

thermodynamic state is the macroscopic condition

• f

a thermodynamic system as described by a suitable set of parameters known as state variables.

– Examples of state variables are temperature, pressure, density, volume, composition, and entropy.

• The state variables of a given system span

the thermodynamic phase space of the system and they define a space of possible equilibrium states of the system.

• An

essential task

• f

classical thermodynamics is to discover a complete set of state variables for a given thermodynamics system.

Thermodynamic Processes

• A thermodynamic process is any process that takes a macroscopic system

from one equilibrium state to another.

• In this course, we will usually examine quasi-static processes, which are

sufficiently slow thermodynamic processes in which all of the state variables are well-defined along any intermediate state.

– For quasi-static processes, the path in thermodynamic phase space between two states is a continuous line.

• The evolution of a thermodynamic system can be given by a

thermodynamic diagram.

• Because there is one equation of state, all processes will occur in a two-

dimensional plane, which can be spanned by any of the three possible pairs: (p,V), (p,T), and (V,T).

Temperature and Thermal Equilibrium

• Consider two thermodynamic systems, A and B, that are

brought into contact with one another.

• Over a period of time, the net exchange of energy between

both systems ceases and we say that they are in thermal equilibrium.

• Thermal equilibrium is determined by a single variable

called the temperature.

The Zeroth Law of Thermodynamics

• Thermal

equilibrium satisfies the zeroth law

• f

thermodynamics which states

– If two thermodynamic systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each

• ther.
• The zeroth law of thermodynamics ensures that thermal

equilibrium is determined solely by temperature.

Quantifying Temperature

• Q: How can we quantify temperature?
• A: We can use thermometric properties to build thermometers by

defining the scale of temperature in such a way that for any thermometric property 𝑌, 𝑈 𝑌 = 𝑈0 + 𝛽 𝑌 − 𝑌0

• We can define the two constants, 𝛽 and 𝑌0, that define this linear

scale by choosing two reproducible phenomenon that always occur at the same temperature.

• We choose

– The boiling point of pure water at sea level 𝑌𝑐, 𝑈

𝑐

– Triple point of pure water 𝑌𝑔, 𝑈

𝑔

• Using these properties, we have that

𝑈 = 𝑈

𝑔 + 𝑈𝑐 − 𝑈 𝑔

𝑌𝑐 − 𝑌𝑔 𝑌 − 𝑌𝑔

Constructing a Thermometer

• A constant volume gas thermometer measures

the pressure of the gas contained in the flask immersed in the bath.

• The height of the mercury column tells us the

pressure of the gas, and we could then find the temperature

• f

the substance from the calibration curve.

• We choose to measure two points:

– The pressure of the gas when the flask is inserted into an ice-water bath and we define this as 𝑈 = 𝑈

𝑔.

– The pressure of the gas when the flask is inserted into water at the steam point and we define this as 𝑈 = 𝑈

𝑐.

• The line connecting two points on the pressure
• vs. temperature curve serves as a calibration

curve for measuring unknown temperatures.

Temperature Scales

• Choosing 𝑈

𝑔 = 0 and 𝑈𝑐 = 100 gives

the Celsius scale 𝑈(𝐷) = 100 𝑌𝑐 − 𝑌𝑔 𝑌 − 𝑌𝑔

• If we now plot the pressure vs.

temperature as measured by our thermometer for different gases, we

• btain a series of linear curves.
• Notice that the pressure is exactly zero

at 𝑈 = −273.16°𝐷 for all cases.

• This is often called absolute zero and

serves as the basis for a new temperature scale called the Kelvin scale. 𝑈(𝐿) = 𝑈(𝐷) + 273.16

Properties of Low Density Gases

• The figures to the right examine the

properties of low density gases while holding volume, pressure, and temperature fixed, respectively.

• Examination of low density gases give

the following observations:

– When 𝑈 is constant, then 𝑄 ∝ 1/𝑊 (Boyle’s law) – When 𝑄 is constant, then 𝑊 ∝ 𝑈 (Charles’ law) – When 𝑊 is constant, then 𝑄 ∝ 𝑈 (Guy-Lussac’s law) – When 𝑈 and 𝑄 are constant, then 𝑊 ∝ 𝑜 (Avogadro’s law)

• These

relationships can be summarized: 𝑄𝑊 ∝ 𝑜𝑈

Ideal Gas Law

𝑄𝑊 ∝ 𝑜𝑈

• These observations lead to the equation of state

for an ideal gas known as the ideal gas law, which is given by 𝑞𝑊 = 𝑜𝑆𝑈

• Here, 𝑜 is the number of moles in the gas and 𝑆

is called the universal gas constant with magnitude 𝑆 = 8.31 𝐾/𝑛𝑝𝑚 ∙ 𝐿

• The ideal gas law is the equation of state for an

ideal gas because it gives a functional relationship between state variables 𝑄, 𝑊, 𝑈

Ideal Gas Law

• The ideal gas law can be written in other forms commonly

used by physicists

• 1) In terms of the number of molecules 𝑶 in the gas, the

ideal gas law becomes 𝑄𝑊 = 𝑂𝑙𝑈

– Here, 𝑙 is the Boltzmann’s constant with magnitude 𝑙 = 𝑆/𝑂

𝐵 = 1.38 × 10−23 𝑛2 𝑙𝑕 𝑡−2 𝐿−1

– 𝑂

𝐵 is known as Avogadro’s number

• 2) In terms of the density 𝝇 of the gas, the ideal gas law

becomes 𝑄 = 𝜍𝑆∗𝑈

– Here, 𝑆∗ is called the specific gas constant with magnitude 𝑆∗ = 𝑆/𝑁 – 𝑁 is defined as the molar mass of the gas in consideration (in grams).

Dalton’s Law of Partial Pressures

• In a mixture of gases, each gas has a partial pressure

which is the hypothetical pressure of that gas if it alone

• ccupied the volume of the mixture at the same

temperature.

• For a mixture of non-reacting gases, Dalton’s law states

that the total pressure exerted is equal to the sum of the partial pressures of the individual gases.

• Mathematically, the pressure of a mixture of non-

reactive gases can be defined by 𝑄𝑈𝑃𝑈𝐵𝑀 = 𝑄𝑗

𝑜 𝑗=1

Example 1

• Q: What is the specific gas constant for dry air?
• A: If the pressure and density of dry air are 𝑄𝑒

and 𝜍𝑒, respectively, the ideal gas equation for dry air is given by 𝑄𝑒 = 𝜍𝑒𝑆𝑒

∗𝑈

• Using Dalton’s law, the dry air gas constant is

given by 𝑆𝑒

∗ = 𝑆

𝑁𝑒

• 𝑁𝑒 is known as the apparent molecular weight
• f dry air

Example 1

• To determine 𝑁𝑒, we need to know the mole

fraction of each gas constituent that comprises dry air, which is given below.

• Therefore, we have

𝑆𝑒

∗ = 𝑆

𝑁𝑒 = 8.31 𝐾/𝑛𝑝𝑚 ∙ 𝐿 28.97 𝑕 ≈ 287 𝐾 𝑙𝑕 𝐿

Example 2

• Q: What is the specific gas constant for water

vapor?

• A: If the pressure and density of dry air are 𝑄

𝑤 and

𝜍𝑤, respectively, the ideal gas equation is given by 𝑄

𝑤 = 𝜍𝑤𝑆𝑤 ∗𝑈

• Here,𝑆𝑤

∗ is the specific gas constant for water

vapor, given by 𝑆𝑤 = 𝑆 𝑁𝑥 = 8.31 𝐾/𝑛𝑝𝑚 ∙ 𝐿 18.016 𝑙𝑕/𝑙𝑛𝑝𝑚 ≈ 461 𝐾 𝑙𝑕 𝐿

The Kinetic Theory of Gases

• In the previous section, we discussed the

macroscopic properties of an ideal gas.

• Now, we consider the ideal gas model

from a microscopic point of view using kinetic theory.

• The kinetic theory of gases makes the

following assumptions

– All molecules in the gas are identical – The molecules interact only through short-range forces during elastic collisions – The molecules obey Newton’s laws of motion – The number of molecules in the gas is large – The average separation between molecules is larger compared with their dimensions

The Kinetic Theory of Gases

• Consider a one-dimensional gas in a
• ne-dimensional box of length L.
• The change in momentum after the

molecule collides with the wall is ∆𝑞𝑦= 2𝑛𝑤𝑦

• Since the molecule must travel a

distance 2L before returning to the same wall, the rate at which the molecules imparts momentum to the wall is 𝐺

𝑛𝑝𝑚 = ∆𝑞

∆𝑢 = 2𝑛𝑤𝑦 2𝑀/𝑤𝑦 = 𝑛𝑤𝑦2 𝑀

The Kinetic Theory of Gases

• If there are N molecules in the box, then

the force on the wall is 𝐺 = 𝑂𝑛 𝑤𝑦2 𝑀

• The pressure on the wall is given by

𝑄 = 𝐺 𝐵 = 𝑂𝑛 𝑤𝑦2 𝑀𝐵 𝑄𝑊 = 𝑂𝑛 𝑤𝑦2

• Since the molecules are equally probable

to move in all three directions of space, then we have 𝑄𝑊 = 1 3 𝑂𝑛 𝑤2

The Kinetic Theory of Gases

• Comparing our previous result with the ideal gas law, we see that

temperature is associated with the mean kinetic energy of the molecules 𝑙𝑈 = 1 3 𝑛 𝑤2 = 2 3 𝐿𝑛𝑝𝑚

• Thus, temperature is a direct measure of average molecular kinetic

energy.

• We can also obtain a relationship between the pressure of a gas, its

density, and the root mean square speed 𝑤𝑠𝑛𝑡 ≡ 𝑤2 . 𝑄 = 1 3 𝜍𝑤𝑠𝑛𝑡2

The Equipartition Theorem

𝐿𝑛𝑝𝑚 = 3 2 𝑙𝑈

• This shows that the average kinetic energy per molecule is 3

2 𝑙𝑈.

• Since 𝑤𝑦2 = 𝑤𝑧2 = 𝑤𝑨2 = 1

3 𝑤2 , it follows that

1 2 𝑛 𝑤𝑗2 = 1 2 𝑙𝑈

• Thus, each translational degree of freedom contributes an equal

amount of energy to the gas.

The Equipartition Theorem

• In general, it can be shown

that for an ideal gas, the total energy associated with the molecules is 𝐹𝑛𝑝𝑚 = 𝑔 2 𝑙𝑈

• Here, 𝑔 is the number of

degrees

• f

freedom per molecule which is the number of ways in which a molecule can store energy.

• This result is known as the

equipartition theorem and it is a result

• f

classical statistical mechanics.

The Equipartition Theorem

• The total energy associated

with point-like molecules is due solely to its translational kinetic energy.

• Ideal gases that follow this

description are called monatomic ideal gases.

• For diatomic and polyatomic

ideal gases, additional possibilities for energy storage are available

– The internal vibration

• f

molecules – The rotation of the molecules about its center of mass

The Hydrostatic Equation

• Consider a horizontal slab of air

whose thickness (height) is 𝑒𝑨.

• If this slab is at rest, the pressure

holding up this slab must balance both the pressure from above and the weight of the slab.

• Using Newton’s 2nd law, this implies

that 𝑄 𝑨 + 𝑒𝑨 − 𝑄 𝑨 = − 𝑁𝑕 𝐵

• Rearranging gives us the hydrostatic

equation 𝑒𝑄 𝑒𝑨 = −𝜍𝑕

𝑒𝑨 𝑁𝑕

The Barometric Equation

𝑒𝑄 𝑒𝑨 = −𝜍𝑕

• Using the ideal gas law, we can write the density of the

gas as 𝜍 = 𝑁 𝑊 = 𝑂 𝑛 𝑊 = 𝑄 𝑙𝑈 𝑛

• Here 𝑛 is the average mass of the air molecules. The

hydrostatic equation becomes 𝑒𝑄 𝑒𝑨 = − 𝑛 𝑕 𝑙𝑈 𝑄

• This is known as the barometric equation

The Law of Atmospheres

𝑒𝑄 𝑒𝑨 = − 𝑛 𝑕 𝑙𝑈 𝑄

• Assuming that the temperature of the

atmosphere is independent of height, we can solve the barometric equation to obtain 𝑄 𝑨 = 𝑄 0 exp − 𝑛 𝑕𝑨 𝑙𝑈

• Since the number density is proportional to

the pressure for an ideal gas, the number density obeys a similar equation 𝑜𝑊 𝑨 = 𝑜𝑊 0 exp − 𝑛 𝑕𝑨 𝑙𝑈

• This is known as the law of atmospheres and

it states that the number density of air molecules decreases exponentially with increasing altitude.

The Boltzmann Distribution Law

𝑜𝑊 𝑨 = 𝑜𝑊 0 exp − 𝑛 𝑕𝑨 𝑙𝑈

• The exponential function can be interpreted as a

probability distribution that gives the relative probability of finding a gas molecule at some height 𝑨.

• We can write the law of atmospheres in terms of

gravitational potential energy. 𝑜𝑊 𝑉 = 𝑜𝑊 0 exp − 𝑉 𝑙𝑈

• This suggests that gas molecules in thermal equilibrium

are distributed in space with a probability that depends

• n the gravitational potential energy.

The Boltzmann Distribution Law

• This expression can be generalized to determine the

number density of molecules have energy 𝐹 𝑜𝑊 𝐹 = 𝑜𝑊 0 exp − 𝐹 𝑙𝑈

• This equation is known as the Boltzmann distribution

law and it states that the probability of finding the molecules in a particular energy state varies exponentially with the energy.

• All the molecules would fall into the lowest energy

level if the thermal agitation at a temperature 𝑈 did not excite the molecules to higher energy levels.