Thermal Physics Interactions and Implications

Northeastern Illinois University

Greg Anderson Department of Physics & Astronomy Northeastern Illinois University

Spring 2020

c 2004-2020 G. Anderson Thermal Physics – slide 1 / 47 Northeastern Illinois Overview University

Temperature Paramagnetism Thermal Equilibrium & Temperature Pressure The Thermodynamic Identity Diﬀusive Equilibrium

c 2004-2020 G. Anderson Thermal Physics – slide 2 / 47 Northeastern Illinois Statistical Mechanics University

Fundamental assumption of statistical mechanics: All accessible microstates of a system are equally likely. Thus, the probability of ﬁnding the system in a macrostate q:

P ∝ Ω(q)

Any large state in thermal equilibrium will be found in the macrostate q of greatest multiplicity Ω(q). Given Boltzmann’s deﬁnition of entropy: S ≡ k lnΩ (entropy) The entropy of an isolated system always increases: Thermal Equilibrium: The spontaneous ﬂow of energy stops when a system is at or near its most likely macrostate.

c 2004-2020 G. Anderson Thermal Physics – slide 3 / 47 Northeastern Illinois University

Statistical Mechanics

Temperature Entropy ES Entropy Plot (NA = 150, NB = 150, q = 12) Multiplicities of Einstein Solids Temperature from Entropy Temperature Entropy and Heat

Paramagnetism Thermal Equilibrium & Temperature

Pressure The Thermodynamic Identity Diﬀusive Equilibrium

c 2004-2020 G. Anderson Thermal Physics – slide 4 / 47 Northeastern Illinois Entropy Plot (NA = 200, NB = 100, q = 12) University

S = k lnΩ. In equilib.: ∂S = ∂S = 0. ∂SA = ∂SB = 1 . ∂qA ∂UA ∂UA ∂UB T 60 S 50 k

30 B T = A

20 T SA SB Entropy10 in Units of TA TB 0 0 1 2 3 4 5 6 7 8 9101112 qA

c 2004-2020 G. Anderson Thermal Physics – slide 5 / 47 Northeastern Illinois Entropy Plot (NA = 150, NB = 150, q = 12) University

S = k lnΩ. In equilib.: ∂S = ∂S = 0. ∂SA = ∂SB = 1 . ∂qA ∂UA ∂UA ∂UB T 60 S 50 k B

40 T = A

30 T

20 SA SB Entropy in Units of 10 TA TB

0 0 1 2 3 4 5 6 7 8 9101112 qA

c 2004-2020 G. Anderson Thermal Physics – slide 6 / 47 Northeastern Illinois Multiplicities of Einstein Solids University

qA ΩA SA/k qB ΩB SB/k Ω S/k 0 1 0 100 2.8 × 1081 187.5 2.8 × 1081 187.5 1 300 5.7 99 9.3 × 1080 186.4 2.8 × 1083 192.1 2 45150 10.7 98 3.1 × 1080 185.3 1.4 × 1085 196.0 ...... 11 5.3 × 1019 45.4 89 1.1 × 1076 175.1 5.9 × 1095 220.5 12 1.4 × 1021 48.7 88 3.4 × 1075 173.9 4.7 × 1096 222.6 13 3.3 × 1022 51.9 87 1.0 × 1075 172.7 3.5 × 1097 224.6 ...... 59 2.2 × 1068 157.4 41 3.1 × 1046 107.0 6.7 × 10114 264.4 60 1.3 × 1069 159.1 40 5.3 × 1045 105.3 6.9 × 10114 264.4 61 7.7 × 1069 160.9 39 8.8 × 1044 103.5 6.8 × 10114 264.4 ...... 100 1.7 × 1096 221.6 0 1 0 1.7 × 1096 221.6

Table 1: Macrostates, multiplicities, and entropies of a system of two Einstein solids, one with 300 oscillators and the other with 200, sharing a total of 100 units of energy.

c 2004-2020 G. Anderson Thermal Physics – slide 7 / 47 Northeastern Illinois Temperature from Entropy University

A theoretical deﬁnition of temperature:

∂S −1 1 ∂S T ≡ or ≡ ∂U T ∂U N,V Example: Einstein solid at large temperature U = ǫq:

∼ N S/k = lnΩ ∼ (q + N)ln q +(q + N) q − N ln N − q ln q ∼ N ln q/N + N + O(N 2/q) ∼ qe ∼ N ln N ∼ Ue e ∼ N ln Nǫ = N ln U + N ln Nǫ Active Learning: Show that U = NkT

c 2004-2020 G. Anderson Thermal Physics – slide 8 / 47 Northeastern Illinois Entropy and Heat University

At constant V , N: dU d¯Q dT dS = = = C T T V T The increase in entropy:

Tf Tf dT Tf ∆S = dS = C ∼ C ln V T V T ZTi ZTi i

c 2004-2020 G. Anderson Thermal Physics – slide 9 / 47 Northeastern Illinois University

Statistical Mechanics

Temperature

Paramagnetism Paramagnetism Two State Paramagnet Two State Paramagnet (s = 1/2) Entropy: Two Paramagnetism State Paramagnet Temperature of a Paramagnet Plot: S(U) Negative Temperature Plot: S(T ) Plot: U(T ) Plot: CB (T ) Plot: M(T ) Plot: M(T ) II Plot: S(U) Negative Temperature Plot: S(T ) Plot: U(T ) Plot: CB (T ) Curie’s c 2004-2020 Law G. Anderson Thermal Physics – slide 10 / 47 Northeastern Illinois Paramagnetism University

Paramagnet: a material in which constituent particles behave like tiny compass needles which line up parallel to an externally applied magnetic ﬁeld.

c 2004-2020 G. Anderson Thermal Physics – slide 11 / 47 Northeastern Illinois Two State Paramagnet University

Consider N, identical spin-1/2 dipoles, e.g. electrons:

Electron Spin: ↓ ↑ ↑

N↑ = spin up, N↓ = spin down ↑ ↑ ↓ N electrons: ↑ ↓ ↑

N = N↑ + N↓

The total energy of the system is a function of N↑. Multiplicity of states:

N N! N! Ω= = = N N !(N − N )! N !N ! ↑ ↑ ↑ ↑ ↓

c 2004-2020 G. Anderson Thermal Physics – slide 12 / 47 Northeastern Illinois Two State Paramagnet (s =1/2) University +µB B ↑ ↓ ↓ ↑ ↑ ↑ ↓ ↓ ↑··· U −µB N = N↑ + N↓ Total Energy| {z }

U = µB (N↓ − N↑)= µB (N − 2N↑)= −M · B

Magnetization, M: U M = µ (N − N )= − ↑ ↓ B Multiplicity

N N! N! Ω= = = N N !(N − N )! N !N ! ↑ ↑ ↑ ↑ ↓

c 2004-2020 G. Anderson Thermal Physics – slide 13 / 47 Northeastern Illinois Entropy: Two State Paramagnet University

Multiplicity

N N! N! Ω(N )= = = ↑ N N !(N − N )! N !N ! ↑ ↑ ↑ ↑ ↓ Using Stirling’s approximation and S = k lnΩ:

S ≈ N ln N − N↑ ln N↑ − (N − N↑)ln(N − N↑) − N + N↑ + N↓

≈ N ln N − N↑ ln N↑ − (N − N↑)ln(N − N↑)

c 2004-2020 G. Anderson Thermal Physics – slide 14 / 47 Northeastern Illinois Entropy: Two State Paramagnet University

Using Stirling’s approximation and S = k lnΩ:

S ≈ N ln N − N↑ ln N↑ − (N − N↑)ln(N − N↑) − N + N↑ + N↓

≈ N ln N − N↑ ln N↑ − (N − N↑)ln(N − N↑)

The energy of a two state paramagnet is:

U = µB (N↓ − N↑)= µB (N − 2N↑)

From which we can write: 1 U 1 U N = N − , N = N + ↑ 2 µB ↓ 2 µB Entropy in terms of U:

2N U N − (U/µB) S/k ≈ N ln + ln 2 − 2 2µB N +(U/µB) N (U/µB) ! c 2004-2020 G. Anderson Thermal Physics – slide 14 / 47 p Northeastern Illinois Entropy: Two State Paramagnet University

Entropy in terms of U:

2N U N − (U/µB) S/k ≈ N ln + ln 2 − 2 2µB N +(U/µB) N (U/µB) ! From the deﬁnitionp of temperature: ∂S k N − U/µB T −1 = = ln ∂U 2µB N + U/µB V,N

c 2004-2020 G. Anderson Thermal Physics – slide 14 / 47 Northeastern Illinois Entropy: Two State Paramagnet University

From the deﬁnition of temperature:

∂S k N − U/µB T −1 = = ln ∂U 2µB N + U/µB V,N Starting with the previous expression

1 1 N − U/µB = ln kT 2µB N + U/µB Multiply by 2µB and exponentiate to ﬁnd

N − U/µB e2µB/kT = N + U/µB Solving for U

1 − e2µB/kT µB U = NµB = −NµB tanh 1+ e2µB/kT kT c 2004-2020 G. Anderson Thermal Physics – slide 14 / 47 Northeastern Illinois Temperature of a Paramagnet University

Entropy

S ≈ N ln N − N↑ ln N↑ − (N − N↑)ln(N − N↑)

Temperature

1 ∂S ∂N ∂S 1 ∂S ≈ = ↑ = − T ∂U ∂U ∂N 2µB ∂N N,B ↑ ↑ In terms of U 1 k N − U/µB = ln T 2µB N + U/µB Solving for U µB U = −NµB tanh kT

c 2004-2020 G. Anderson Thermal Physics – slide 15 / 47 Northeastern Illinois Entropy of a Paramagnet S(U) University

T −1 = ∂S/∂U S/k

∂U/∂S

−N 0 N U/µB

2N U N − (U/µB) S/k ∼ N ln + ln 2 − 2 2µB N +(U/µB) N (U/µB) !

c 2004-2020 G. Anderson p Thermal Physics – slide 16 / 47 Northeastern Illinois Negative Temperature University

Negative temperature requires: • Thermal equilibrium • Requires a ﬁnite upper bound on energy spectrum. e.g., nuclear paramagnets. • System must be energetically isolated from pos. temp. states. Understanding negative temperature

• From Inverted population e.g. N↓ >N↑. • Negative temps. correspond to higher energies. They are “hotter” than positive temperatures. Entropy is more fundamental than temperature.

c 2004-2020 G. Anderson Thermal Physics – slide 17 / 47 Northeastern Illinois Entropy of a Paramagnet S(T ) University

S Nk

0 0 1 2 3 4

x = µB/kT

S/(Nk) = ln2(cosh x) − x tanh x

c 2004-2020 G. Anderson Thermal Physics – slide 18 / 47 Northeastern Illinois Energy of a Paramagnet U(T ) University

µB U = −NµB tanh kT

0 U NµB

-1 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

τ = kT/µB

c 2004-2020 G. Anderson Thermal Physics – slide 19 / 47 Northeastern Illinois Heat capacity of a spin-1/2 paramagnet University

Heat capacity at constant magnetic ﬁeld:

∂U ∂NµB tanh(µB/kT ) (µB/kT )2 CB = = − = Nk ∂T ∂T cosh2(µB/kT ) B B

0.4 Schottky anomaly 0.3 CB 0.2 0.1 0 01234567 τ = kT/µB

c 2004-2020 G. Anderson Thermal Physics – slide 20 / 47 Northeastern µB Illinois Magnetization M = Nµ tanh University kT

1 ) M/Nµ

Curie’s Law (T ≫ µB/k) ∼ µB M ∼ Nµ kT Magnetization (

-1 -3 -2 -1 0 1 2 3

x = µB/kT

c 2004-2020 G. Anderson Thermal Physics – slide 21 / 47 Northeastern Illinois Magnetization of a Paramagnet University

1 M Nµ kT µB

0 tanh Nµ = M

-1 M -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 − Nµ

τ = kT/µB

c 2004-2020 G. Anderson Thermal Physics – slide 22 / 47 Northeastern Illinois Entropy of a Paramagnet S(U) University

T −1 = ∂S/∂U S/k

∂U/∂S

−N 0 N U/µB

2N U N − (U/µB) S/k ∼ N ln + ln 2 − 2 2µB N +(U/µB) N (U/µB) !

c 2004-2020 G. Anderson p Thermal Physics – slide 23 / 47 Northeastern Illinois Negative Temperature University

• From Inverted population e.g. N↓ >N↑. • Negative temps. correspond to higher energies. They are “hotter” than positive temperatures. Entropy is more fundamental than temperature.

c 2004-2020 G. Anderson Thermal Physics – slide 24 / 47 Northeastern Illinois Entropy of a Paramagnet S(T ) University

S Nk

0 0 1 2 3 4

x = µB/kT

S/(Nk) = ln2(cosh x) − x tanh x

c 2004-2020 G. Anderson Thermal Physics – slide 25 / 47 Northeastern Illinois Energy of a Paramagnet U(T ) University

µB U = −NµB tanh kT

0 U NµB

-1 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

τ = kT/µB

c 2004-2020 G. Anderson Thermal Physics – slide 26 / 47 Northeastern Illinois Heat capacity of a spin-1/2 paramagnet University

Heat capacity at constant magnetic ﬁeld:

∂U ∂NµB tanh(µB/kT ) (µB/kT )2 CB = = − = Nk ∂T ∂T cosh2(µB/kT ) B B

0.4 Schottky anomaly 0.3 CB 0.2 0.1 0 01234567 τ = kT/µB

c 2004-2020 G. Anderson Thermal Physics – slide 27 / 47 Northeastern Illinois Magnetization & Curie’s Law University

Active learning: Starting from the general form of the magnetization of a two-state paramagnet:

µB M = Nµ tanh kT Show that the high temperature limit, µB ≪ kT , satisﬁes Curie’s law: Nµ2B M ∼ kT Hint: ex − e−x tanh(x)= ex + e−x

c 2004-2020 G. Anderson Thermal Physics – slide 28 / 47 Northeastern Illinois Magnetization & Curie’s Law University

Active learning: Starting from the general form of the magnetization of a two-state paramagnet:

µB M = Nµ tanh kT Show that the high temperature limit, µB ≪ kT , satisﬁes Curie’s law: Nµ2B M ∼ kT Hint: ex − e−x (1 + x + ··· ) − (1 − x + ··· ) tanh(x)= = ex + e−x (1 + x + ··· )+(1 − x + ··· )

c 2004-2020 G. Anderson Thermal Physics – slide 28 / 47 Northeastern Illinois Magnetization & Curie’s Law University

Active learning: Starting from the general form of the magnetization of a two-state paramagnet:

µB M = Nµ tanh kT Show that the high temperature limit, µB ≪ kT , satisﬁes Curie’s law: Nµ2B M ∼ kT Hint: 2x tanh(x) ∼ 2

c 2004-2020 G. Anderson Thermal Physics – slide 28 / 47 Northeastern Illinois University

Statistical Mechanics

Temperature

Paramagnetism Thermal Equilibrium & Temperature Ideal Gas: Entropy Thermal Equilibrium & Two Ideal Gasses: S Entropy ES Entropy Plot Temperature (NA = 150, NB = 150, q = 12)

Pressure The Thermodynamic Identity Diﬀusive Equilibrium

c 2004-2020 G. Anderson Thermal Physics – slide 29 / 47 Northeastern Illinois Entropy of an Ideal Gas University

Multiplicity of a monatomic ideal gas:

1 2πm 3N/2 Ω= f(N)V N U 3N/2, f(N) ∼ N!(3N/2)! h2 Sakur-Tetrode Eqn. for Entropy:

4πmU 3/2 5 S = Nk ln V − ln N 5/2 + 3h2 2 ( " # ) Isolating the dependence on U and V : 3 S = Nk ln V + Nk ln U + k ln f(N) 2

c 2004-2020 G. Anderson Thermal Physics – slide 30 / 47 Northeastern Illinois Entropy of two Ideal Gasses University

Entropy of two a monatomic ideal gasses in thermal contact at constant volume, U = UA + UB: 3 3 S = S + S = N k ln U + N k ln U + const. A B 2 A A 2 B B S = SA + SB b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b T b b b b b b b b B b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b S b b b B b b b b b b b b b S b A b b b TA TA < TB TA > TB Temperature: 012345678910UA 1 ∂S 3Nk 3 ≡ = ⇒ U = NkT T ∂U 2U 2 N,V

c 2004-2020 G. Anderson Thermal Physics – slide 31 / 47 Northeastern Illinois Entropy Plot (NA = 200, NB = 100, q = 12) University

S = k lnΩ. In equilib.: ∂S = ∂S = 0. ∂SA = ∂SB = 1 . ∂qA ∂UA ∂UA ∂UB T 60 S 50 k

30 B T = A

20 T SA SB Entropy10 in Units of TA TB 0 0 1 2 3 4 5 6 7 8 9101112 qA

c 2004-2020 G. Anderson Thermal Physics – slide 32 / 47 Northeastern Illinois Entropy Plot (NA = 150, NB = 150, q = 12) University

S = k lnΩ. In equilib.: ∂S = ∂S = 0. ∂SA = ∂SB = 1 . ∂qA ∂UA ∂UA ∂UB T 60 S 50 k B

40 T = A

30 T

20 SA SB Entropy in Units of 10 TA TB

0 0 1 2 3 4 5 6 7 8 9101112 qA

c 2004-2020 G. Anderson Thermal Physics – slide 33 / 47 Northeastern Illinois University

Statistical Mechanics

Temperature

Paramagnetism Thermal Equilibrium & Temperature

Pressure Mechanical Equilibrium & Pressure Pressure Pressure The Thermodynamic Identity Diﬀusive Equilibrium

c 2004-2020 G. Anderson Thermal Physics – slide 34 / 47 Northeastern Illinois Mechanical Equilibrium & Pressure University In equilibrium Q UA,VA,SA UB ,VB ,SB ∂S ∂S Q = =0 ∂U ∂V S A A dVA = −dVB Mech. equil. ∂S = 0: Pressure ∂VA P ∝ ∂S ∂SA = ∂SB ∂V ∂VA ∂VB

PA = PB

Temperature −1 ∂S T = ∂U

UA VA c 2004-2020 G. Anderson Thermal Physics – slide 35 / 47 Northeastern Illinois Pressure University

dVA = −dVB Deﬁne Pressure Q UA,VA,SA UB ,VB ,SB ∂S Q P = T ∂V U,N Multiplicity of a monatomic ideal gas:

Ω= f(N)V N U 3N/2 Entropy: 3 S = Nk ln V + Nk ln U + k ln f(N) 2 Pressure of monatomic ideal gas: ∂S ∂ NkT P = T = T Nk ln V = ∂V U,N ∂V V c 2004-2020 G. Anderson Thermal Physics – slide 36 / 47 Northeastern Illinois University

Statistical Mechanics

Temperature

Paramagnetism Thermal Equilibrium & Temperature Pressure The Thermodynamic The Thermodynamic Identity The Identity Thermodynamic Identity Entropy and Heat A one dimensional, two-state polymer A real, linear polymer Homework Diﬀusive Equilibrium

c 2004-2020 G. Anderson Thermal Physics – slide 37 / 47 Northeastern Illinois The Thermodynamic Identity University

Deﬁne Temperature dVA = −dVB ∂S T −1 = ∂U Q V,N UA,VA,SA UB ,VB ,SB Q Deﬁne Pressure

∂S P = T ∂V U,N Inﬁnitesimal change in entropy

∂S ∂S dU P dS = dU + dV = + dV = dS ∂U ∂V T T V U The thermodynamic identity

dU = TdS − PdV

c 2004-2020 G. Anderson Thermal Physics – slide 38 / 47 Northeastern Illinois Entropy and Heat University

The ﬁrst law and the Thermodynamic identity

dU = d¯Q + d¯W = TdS − PdV

For a quasi-static process, d¯W = −PdV :

d¯Q = TdS (quasi-static)

In general d¯Q dS ≥ T e.g. when d¯W > −PdV , free expansion of a gas.

c 2004-2020 G. Anderson Thermal Physics – slide 39 / 47 Northeastern Illinois A one dimensional, two-state polymer University

Multiplicity of microstates for ﬁxed N, N = NR + NL: N N! N N Ω= = ∼ NR NL NR NR!NL! N N R L ℓ The length of the polymer can be written: |{z} 1 L L =(N − N )ℓ = (2N − N)ℓ ⇒ N = + N R L R R 2 ℓ Inﬁntesimal change in entropy

∂S ∂S 1 F dS = dU + dL = dU − dL ∂U ∂L T T L U Recall ∂S ∂S ∂S T −1 ≡ , P = T ⇒ F = −T ∂U ∂V ∂L L,N U,N U,N

c 2004-2020 G. Anderson Thermal Physics – slide 40 / 47 Northeastern Illinois A one dimensional, two-state polymer University

dU = TdS + FdL

with a little algebra the applied force is:

∂S kT 1+ L/Nℓ kT F = −T = ln ∼ L ∂L 2ℓ 1 − L/Nℓ Nℓ2 U

c 2004-2020 G. Anderson Thermal Physics – slide 40 / 47 Northeastern Illinois A real, linear polymer University

Statistical Mechanics

Temperature

Paramagnetism Thermal Equilibrium & Temperature

Pressure The Thermodynamic Identity The Thermodynamic Identity Entropy and Heat A one dimensional, two-state polymer A real, linear polymer Homework Diﬀusive Equilibrium

Journal of the American Chemical Society, vol. 127, iss. 45, pp. 15688-15689 (2005) c 2004-2020 G. Anderson Thermal Physics – slide 41 / 47 Northeastern Illinois Homework University

Statistical Mechanics Homework: Problem 3.34 on polymers. Temperature Note: It is not possible to deﬁne a temperature without a Paramagnetism model which provides the dependence of U on the number Thermal Equilibrium & of kinks and anti-kinks in the polymer. Temperature

Pressure The Thermodynamic Identity The Thermodynamic Identity Entropy and Heat A one dimensional, two-state polymer A real, linear polymer Homework Diﬀusive Equilibrium

c 2004-2020 G. Anderson Thermal Physics – slide 42 / 47 Northeastern Illinois University

Statistical Mechanics

Temperature

Paramagnetism Thermal Equilibrium & Temperature

Pressure The Thermodynamic Diﬀusive Equilibrium Identity Diﬀusive Equilibrium Chemical Potential Chemical Potential II Identities and Partial Derivatives Interactions & Equilibria

c 2004-2020 G. Anderson Thermal Physics – slide 43 / 47 Northeastern Illinois Chemical Potential University

dNA = −dNB In diﬀusive equilibrium, dS =0 Q, N UA,VA,SA,NA UB ,VB ,SB ,NB ∂S ∂S A = B Q, N ∂N ∂N A V,U B V,U Inﬁnitesimal change in entropy

∂S ∂S ∂S dS = dU + dV + dN ∂U ∂V ∂N V,N U,N U,V The thermodynamic identity 1 P µ dS = dU + dV − dN T T T where the chemical potential is

∂S µ ≡−T ∂N U,V c 2004-2020 G. Anderson Thermal Physics – slide 44 / 47 Northeastern Illinois Chemical Potential II University dNA = −dNB

Q, N Chemical Potential UA,VA,SA,NA UB ,VB ,SB ,NB ∂S Q, N µ = −T ∂N V,U The thermodynamic identity 1 P µ dS = dU + dV − dN T T T

Solving for dU 0 µA ptls. dU = TdS − PdV + µdN µB Particles ﬂow from higher to lower chemical potential. System with larger ∂S/∂N gains particles.

c 2004-2020 G. Anderson Thermal Physics – slide 45 / 47 Northeastern Illinois Identities and Partial Derivatives University

Thermodynamic identity

dU = TdS −PdV + µdN heat mechanical work chemical work From this single identity|{z} | we{z can} immediately|{z} derive: ∂S ∂U ∂V µ = −T = = P ∂N ∂N ∂N U,V S,V U,S ∂S ∂U ∂N P = T = − = µ ∂V ∂V ∂V U,N S,N U,S

c 2004-2020 G. Anderson Thermal Physics – slide 46 / 47 Northeastern Illinois Interactions & Equilibria University

Interaction Exchanged Governing Type quantity Variable Formula

1 ∂S thermal heat energy temperature T = ∂U V,N P ∂S mechanical volume pressure T = ∂V U,N µ − ∂S diﬀusive particles chem. pot. T = ∂N U,V

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

c 2004-2020 G. Anderson Thermal Physics – slide 47 / 47