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UCI / Physics / PHYS 60 / What is the equipartition theorem?

What is the equipartition theorem?

What is the equipartition theorem?


School: University of California - Irvine
Department: Physics
Course: Thermal Physics
Professor: J. feng
Term: Fall 2015
Tags: thermal physics uci
Cost: 25
Name: Week 1 Book Notes - Thermal Physics
Description: Week 1 of book notes! If you don't feel like reading the Thermal Physics textbook, just don't have time during the first week, want to hit the more important parts of each chapter, or just want to review, then download my notes and study away! :)
Uploaded: 10/01/2015
4 Pages 48 Views 1 Unlocks

Physics 60  

What is the equipartition theorem?


Week 1  

Ch. 1.3 Equipartition of Energy  

o Equipartition theorem: At temperature T, the average energy of any quadratic degree of  freedom is (1/2)kT  

- The equipartition theorem concerns all forms of energy that has formula as a quadratic  function of a coordinate or velocity  

- Degrees of freedom f refer to the forms of energy that constrain how energy moves (such as  translational motion, rotational motion, vibrational motion, and elastic potential energy)  - Total thermal energy: Uthermal = N*f*(1/2)kT  

- Safest to apply equipartition theorem to changes in energy due to changes in temperature  that do not break bonds between particles  

- Best to learn degrees of freedom f through example  

Thermodynamics focuses on what?

▪ Monatomic particles, such as helium or argon, have f = 3 (only translational motion  counts)  

▪ Diatomic particles, such as oxygen (O2) or nitrogen (N2), can also rotate, but rotation  does not count because it can reach an equilibrium value (higher temperatures mean  faster-moving molecules)  

∙ Rotational energy balances out with translational energy  

∙ Diatomic particles can also vibrate, which counts for two degrees of freedom (one  for vibrational kinetic energy and one for potential)  

∙ More complicated molecules can stretch, flex, twist, etc., each of which counts for  two degrees of freedom  

∙ Air molecules (O2 or N2) have five degrees of freedom in room temperature  - Vibrational modes count as degrees of freedom at higher temperatures, but at  rooŵ teŵpeƌatuƌe, theLJ aƌe ͞fƌozeŶ out͟ ďeĐause theLJ doŶ’t ĐoŶtƌiďute  

What refers to any spontaneous flow of energy from one object to another due to difference in temperature of the objects?

Don't forget about the age old question of Who is mao zedong?

▪ Each atom in a solid has six degrees of freedom (three kinetic and three potential  energy) because each atom can vibrate in three perpendicular directions  

Ch. 1.4 Heat and Work  

o Thermodynamics focuses on closely-related concepts of: temperature, energy, and heat  - Temperature - fuŶdaŵeŶtallLJ a ŵeasuƌe of aŶ oďjeĐt’s teŶdeŶĐLJ to spoŶtaŶeouslLJ ƌelease  energy  We also discuss several other topics like Which process passes food from one organ to the other?

- Energy – complex fundamental dynamical concept of various types (kinetic, electrostatic,  gravitational, chemical, nuclear)  

▪ Conservation of energy: total amount of energy in the universe cannot change despite  conversion of energy  

∙ Energy can be put or taken out of a system by the thermodynamic mechanisms of  heat and work

▪ Heat: any spontaneous flow of energy from one object to another due to difference in  temperature of the objects  

▪ Work: any other transfer of energy into or out of a system  

∙ UŶlike ǁith heat, theƌe is usuallLJ aŶ ideŶtifiaďle ͞ageŶt͟ that is ͞aĐtiǀelLJ͟ puttiŶg  energy into the system  

∙ Both heat and work discuss energy in leaving or entering a system  

o U: total energy in a system; Q: heat that enters a system; W: work that enters a system  - Negative if leave a system  

- Fiƌst laǁ of theƌŵodLJŶaŵiĐs: ΔU = Q+W

- Joules: SI unit of energy; heat is traditionally measured in calories (1 cal = 4.186 J)  o Process of heat transferring in three categories  

- conduction: transfer of heat by molecular contact  

- convection: bulk motion of a gas or liquid  

- radiation: emission of electromagnetic waves  If you want to learn more check out What does microeconomics focus on?

Ch. 1.5 Compression Work  

o Most important work done on a system is compressing it  

o Recall work is W = �⃗∙�⃗r  

✔ Note: in thermodynamics, �⃗r refers to the point of contact  

- Recall the piston arrangement in Section 1.2  

▪ In the piston sLJsteŵ, foƌĐe is paƌallel to displaĐeŵeŶt, so W = FΔdž Don't forget about the age old question of How do rocks melt?

∙ For F to be replaced by PA, we must assume that contained gas is always in internal  eƋuiliďƌiuŵ, ǁhiĐh ŵeaŶs the pistoŶ’s ŵotioŶ ŵust ďe sloǁ ;ƋuasistatiĐ: sloǁ  

volume change)  

∙ It ďeĐoŵes W = PAΔdž, which then becomes W = -PΔV ǁheŶ ƋuasistatiĐ

- Can also be W = - ʃViVf P(V)dV for quasistatic  

❖ Compression of an Ideal Gas  

o When you compress a container of gas, you do work on it, which means adding energy  - Generally means temperature increases, but when compressed slowly, heat has time to  escape, which prevents heat from rising in the system  

o Two idealized way to compress an ideal gas: isothermal compression and adiabatic compression  - Isotheƌŵal ĐoŵpƌessioŶ: ĐoŵpƌessioŶ so sloǁ that teŵpeƌatuƌe of gas doesŶ’t rise at all  ▪ Formula P = NkT/V for constant T is a concave hyperbola (a.k.a. an isotherm)  ∙ Work done: W = = - ʃViVf PdV = -NkT= - ʃViVf (1/V)dV = -NkT(lnVf – lnVi) = (NkT)ln(Vi/Vf)  ✔ Note Q is negative because heat leaves the gas in compression  

- Adiabatic compression: compression so fast that no heat escapes from the gas during the  process of compression  

▪ Like isothermal compression, assume compression is quasistatic  

▪ Woƌk doŶe ǁithout heat esĐapiŶg, iŶteƌŶal eŶeƌgLJ ΔU = Q + W = W We also discuss several other topics like What are the types of chromosomes?
Don't forget about the age old question of Why did america declare independence?

▪ Curve on a PV diagram of U proportional to T in an ideal gas in an adiabatic compression  connect low-temperature isotherm to higher-temperature isotherm  

∙ Start with equipartition theorem: Uthermal = Nf(1/2)kT

∙ Next, take change in energy over infinitesimal segment: dU = (1/2)fNkdT  

∙ Because work done in quasistatic compression is –PdV, (1/2)fNkdT = -PdV  

- Change equation to (f/2)(dT/T) = -(dV/V) so that you can integrate from initial Vi - and Ti to Vf and Tf  

- It becomes (f/2)ln(Tf/Ti) = -ln(Vf/Vi), and then VfTff/2 = ViTif/2  

- Simplifying further turns it into VTf/2 = constant  

- Using ideal gas law to eliminate T, it becomes VɣP = constant (ɣ, a.k.a. the  adiabatic exponent, is the abbreviation for (f+2)/f  

Ch. 1.6 Heat Capacities  

o Heat capacity: amount of heat needed to raise its temperature per degree temperature  increases C  

- C = Q/ΔT

- Heat ĐapaĐitLJ is aŵďiguous, ŵeaŶiŶg that aŵouŶt of heat to ƌaise aŶ oďjeĐt’s teŵpeƌatuƌe by a degree depends on the circumstance (i.e. if work is being done on the object)  ▪ Use first law of thermodynaŵiĐs ;ΔU = Q + WͿ to get C = Q/ΔT = ;ΔU – WͿ/ΔT

- More fundamentally specific heat capacity heat capacity per unit mass (c = C/m)  o Two likely cases that are most likely to occur:  

- 1) W = 0, where there is no work being done on the system, which usually means that  volume is constant  

▪ Heat capacity at constant volume CV = ;ΔU/ΔTͿv = (�ܷ/�ܶ)v (subscript V indicates that  volumes are understood to be held fixed)  

∙ More accurately ͞eŶeƌgLJ ĐapaĐitLJ͟ ďeĐause it is eŶeƌgLJ Ŷeeded to ƌaise oďjeĐt’s  temperature by a degree whether or not energy actually enters as heat  

∙ For example, a gram of water has CV = 1 cal/°C = about 4.2 J/°C  

- 2) objects often expand when heated, so they do work on their surroundings and W is  negative  

▪ C > CV  

▪ If pressure is constant in surroundings, then it is heat capacity at constant pressure:  CP = ;;ΔU – (- PΔVͿͿ/ ΔTͿp = (�ܷ/�ܶ)P + P(�ܷ/�ܶ)P  

∙ For solids and liquids, (�ܷ/�ܶ) can often be neglected because of how small it is  o Supposing that the system stores thermal energy in only quadratic degrees of freedom, then  using equipartition theorem ( U = (1/2)NfkT), we get CV = (�ܷ/�ܶ) = �/�ܶ(NfkT/2) = Nfk/2 (if we  assume f is independent of temperature)  

- Example: for helium (a monatomic gas), f = 3, so CV = (3/2)Nk = (3/2)nR, and heat capacity  per mole is 3R/2 = 12.5 J/K  

✔ Note R is the gas ĐoŶstaŶt ;R = 8.3ϭ J/ŵol∙KͿ  

- Rule of Dulong and Petit states that, because solids have 6 degrees of freedom per atom,  heat capacity per mole for solids is 3R  

o For constant pressure, using ideal gas law, we get (�ܸ/�ܶ)P = (�/�ܶ) (NkT/P) = Nk/P  - CP = CV+Nk = CV+nR

❖ Latent Heat  

o You can put heat into a system without increasing temperature in some situations  - Normally during phase transformation, when heat capacity then is technically infinite, so  C = Q/ΔT = Q/Ϭ = ∞

- You can still find amount of heat required to melt or boil the substance completely by  finding the latent heat L (L = Q/m), where you assume pressure is constant  

❖ Enthalpy  

o Enthalpy H is the total energy needed to create the system out of nothing to put into the  environment (i.e. energy of a system is not just U by also work (PV) done my atmosphere to fill  the space of the system were the system to be annihilated)  

- H = U + PV  

▪ ChaŶge iŶ eŶthalpLJ duƌiŶg a ĐoŶstaŶt ĐhaŶge iŶ pƌessuƌe: ΔH = ΔU + PΔV  

▪ Using first law of thermodynamics Uthermal = Q+W, ǁe ĐaŶ get ΔU = Q+;-PΔV)+Wother,  ǁhiĐh, usiŶg the eƋuatioŶ foƌ eŶthalpLJ, ďeĐoŵes ΔH = Q+Wother in constant pressure  ∙ Foƌ edžaŵple ΔH foƌ a ŵole of ǁateƌ pƌoduĐed is –286 jK  

- This is the enthalpy of formation of water because it is the enthalpy required to  form out of elemental constituents in their most stable states  

▪ Change in enthalpy per degree at constant pressure is similar to heat capacity at  constant pressure CP (CP = (��/�ܶ)P)

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