Heat and Waves Summary: Key Concepts and Formulas for PHYS 232 at McGill University, Summaries of Thermodynamics

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McGill University

PHYS 232 Heat and Waves

Heat Summary

-A Summary of Key Concepts and Formulas-

by Clara Costello and Gabriel Gonzalez Ascencio

261086278 and 261179992

April 15, 2024

Key Terms and Summary

Equipartition : the equal distribution of basic states with equal energy. All basic states have equal probability.

Basic State (Microstate) is described using a complete specification of all the constitutent particles, for example, the position of each atom.

Macrostate is described using only quantities that can be determined by measurements made on scales much larger than molecular sizes.

Thermal Equilibrium : the condition in which all parts of a system are at the same temperature. In this condition, the thermal energy is equipartitioned over the system and the entropy of the system is at its maximum.

The Zeroth Law of Thermodynamics : if two objects are in thermal equilibrium with a third, then they are in thermal equilibrium with each other. This is a direct consequence of equipartition.

The First Law of Thermodynamics : total energy is conserved in a closed system.

The Second Law of Thermodynamics : a closed system always evolves to maximize the number of basic states. As the number of basic states increases, so does the entropy. At equilibrium, the number of basic states reaches its maximum value and the entropy of the system no longer changes (∆S = 0).

The Third Law of Thermodynamics : zero temperature cannot be reached. If it could be reached, a perfect crystal at zero temperature (absolute zero) would have zero entropy.

Heat Capacity : the amount of heat applied to an object to produce a unit change in its temperature. In other words, the number of heat units needed to raise the temperature of a body by one degree. Units of J/K. (Oxford Languages)

Specific Heat Capacity : the number of heat units needed to raise the temperature of one gram of a body by one degree. Units of J/kgK.

Boson : every particle in the universe fits into two broad categories, bosons and fermions. Bosons (such as photons and phonons) carry energy and forces throughout the universe.

Fermion : every particle in the universe fits into two broad categories, bosons and fermions. Fermions (such as protons, neutrons, and electrons) can be thought of as the building blocks of matter.

Constants and Definitions

List of key constants and definitions for symbols that appear frequently in the formulas found in Table 4, Table 5, Table 6, Table 7, Table 8, and Table 9. Altough some symbols are obvious in their meaning, it is important to clarify their definitions in order to establish coherency.

Table 1: Symbols used throughout the formulas.

Name Symbol Unit

Pressure P Pa

Length L m

Area A m^2

Volume V m^3

Number of particles N unitless

Degrees of freedom of particle D unitless

Heat Capacity at constant volume CV JK−^1

Heat capacity at constant pressure CP JK−^1

Specific heat capacity c JK−^1 kg−^1

Thermal conductivity k Wm−^1 K−^1

Average velocity of particle ¯v ms−^1

frequency ω Hz

Number of basic states for a system Ω unitless

Rate of heat generation inside the element (power) e˙gen Js−^1

Emissivity ε unitless

Einstein frequency ωE Hz

Debye frequency ωD Hz

Temperature of: matter, hot reservoir, cold reservoir T, Th, Tc K

Table 2: Definitions and equivalences used throughout the formulas.

s

Description Relationship Units

Volume density ρ = dmdV kgm−^3

Einstein temperature TE ≡ ℏ kωBE K

Debye temperature TD ≡ ℏ kωBD K

Substitution I used in Debye equation x ≡ (^) kℏBω T unitless

Substitution II used in Debye equation xD ≡ (^) kℏωBD T unitless

Heat capacity const.P and heat capacity const.V ratio γ ≡ C CPV unitless

Table 3: Important constants used throughout the formulas and for this section in general.

Name Value

Reduced Plank’s constant ℏ = 2 hπ = 1. 05 × 10 −^34 Js

Boltzmann’s constant kB = 1. 38 × 10 −^23 JK−^1

Stefan-Boltzmann constant σB = 5. 67 × 10 −^8 Wm−^2 K−^4

Avogadro’s number NA = 6. 022 × 1023

Speed of light c = 3 × 108 ms−^1

Standard atmospheric pressure 1 atm = 1. 01 × 105 Pa

Standard temperature 0 C◦^ = 273.5 K

Molar volume at STP 1 mol of air = 22.4 l

Water sound velocity v = 1500 ms−^1

Mass of a nucleus m = 1. 66 × 10 −^27 kg

Regarding the Einstein and Debye heat capacity equations, there are interesting results when looking the high and low temperature limits, T << TD/E and T >> TD/E respectively. Einstein’s model is a bad fit a the low temperature limit, so only Debye’s model will be considered. Both models approximate to the same value at the high temperature limit.

Table 5: Heat capacity models at high and low temperature limits.

Limit Approximation at Limit

Low temperature limit T << TD CV D^ ≃ 12 π 4 5 N kB

T TD

High temperature limit T >> TD/E CV ≃ 3 N kB

Thermal Conductivity and Heat Equation

Table 6: Key formulas for thermal conductivity and heat equation.

Name Formula

Heat Flux Q˙ = −k AL ∆T = −kA dTdx

Effective thermal conductivity (kef f )−^1 = (^) Ltotal^1

Pn i=

Li ki

Thermal conductivity of a gas kgas = 13 C VV L¯v = D 2 PT

Thermal loss in a vacuum (Stefan-Boltzmann expression) Q˙ = AεσB (T 4 − T 04 )

Heat equation* ∂ (^2) T ∂x^2 +^

e˙gen k =^

ρc k

∂T ∂t

Diffusive Equilibrium and Chemical Potential

Table 7: Key formulas for diffusive equilibrium and chemical potential.

Name Formula

Chemical potential μ ≡ −kB T

∂S

∂N

U,V

Generalized thermodynamic identity dU = kB T dS − P dV + μdN

The following is a quick summary of the three types interactions considered in this section, and the associated variables and partial-derivative relations.

Table 8: Three types of interactions.

Type of interaction Exchanged quantity Governing variable Formula

Thermal Energy Temperature (^) kB^1 T =

∂S

∂U

V,N

Mechanical Volume Pressure (^) kBP T =

∂S

∂V

U,N

Diffusive Particles Chemical potential (^) kBμ T = −

∂S

∂N

U,V

Engines and Refrigerators

Table 9: Key formulas for engines and refrigerators.

Name Formula

Efficiency of a Heat engine* e = benef itcost = (^) QWh ≤ 1 − T Tch

Efficiency of Carnot cycle* e = 1 − T cT h^ ln(ln(VV^23 /V/V^14 ))

Adiabatic expansion relation for ideal gases P V γ^ = const.

Coefficient of operation COP ≤ (^) ThT−cTc