Download 29 jul 2016 classical monatomic ideal gas . L10–1 Classical

29 jul 2016
classical monatomic ideal gas . L10–1
Classical Monatomic Ideal Gas
Equipartition Principle
• Statement: The equipartition principle is a statement about classical systems with quadratic Hamiltonians.
It states that any canonical variable ξ (a q or a p) that only appears in a quadratic additive term of the form
κ ξ 2 in H, where κ is a constant, and whose range of values can be considered to be (−∞, ∞), contributes
a term 21 kB T to the mean energy of the system, or 12 kB to the heat capacity. In particular, the mean energy
of a system of N non-interacting particles (distinguishable or not), each of which contributes f quadratic
terms in the positions and/or momenta to the total Hamiltonian is Ē = 21 f N kB T .
• Proof: Suppose the Hamiltonian is of the form H = H̃ + κ ξ 2 , where κ and ξ are as described above and
H̃ does not involve ξ. Then the partition function and mean energy can be calculated as
Z
Zc =
dΩ e−βH(q,p) = Z̃
Z
+∞
2
dξ e−βκ ξ = Z̃
−∞
Γ
r
π
,
βκ
Ē = −
∂
ln Zc = Ẽ +
∂β
1
2
∂
ln β = Ẽ + 12 kB T ,
∂β
where Z̃ and Ẽ are the parts of Zc and Ē that come from the variables other than ξ, and we have used the
p
R +∞
2
well-known Gaussian integral −∞ dx e−ax = π/a.
Classical Monatomic Ideal Gas Thermodynamics
• Setup: As a first example of application of the equipartition principle, consider a monatomic ideal gas.
The Hamiltonian is a sum of terms from individual non-interacting particles; if the particles are single atoms
and T is low enough that their internal, electronic or nuclear degrees of freedom areP
not excited, then each
N
term in the Hamiltonian is just the particle’s translational kinetic energy, H(q, p) = i=1 p~i2 /2m.
• Canonical partition function: For a single particle, using again the Gaussian integral,
Z1 =
1
h3
Z
V
d3 x
Z
+∞
dp e−βp
−∞
2
/2m
3
=
V
h3
r
π
β/2m
3
=
V
,
λ3
p
where λ := h2 β/2πm is the thermal wavelength. For N particles, since the particles are non-interacting
(H = H1 + ... + HN ) the partition function is the product of N single-particle ones. If the particles are
identical, however, one must take into account the fact that any permutation of the N particles leads to an
indistinguishable state, even classically. This Boltzmann statistics leads to the N -particle partition function
Zc =
1 N
VN
Z1 =
.
N!
N ! λ3N
• Mean energy: We can obtain the mean energy directly from the equipartition principle, or as a result of a
short calculation,
3N/2
∂
V N 2πm
Ē = −
ln
= 32 N kB T ,
∂β
N ! h2 β
an expression that agrees with what we expect from the kinetic theory of ideal gases.
• Helmholtz free energy: Using the partition function Zc above for a monatomic ideal gas, we find
F = −kB T ln Zc = −kB T ln
VN
= −kB T (ln V N − ln N ! − ln λ3N ) .
N ! λ3N
This expression can now be used to obtain the pressure p, entropy S and chemical potential µ using definitions
obtained from the fundamental identity of thermodynamics for F .
29 jul 2016
classical monatomic ideal gas . L10–2
• Pressure: The ideal gas pressure equation of state can be obtained from the free energy,
d
∂F 1
= kB T
p=−
ln V N = N kB T
,
or
pV = N kB T .
∂V T,N
dV
V
(Alternatively, from S(E, V ) = kB ln Ω = kB ln(V N f (E)) and dS = δQ/T = (dE + pdV )/T we get that
p/T = ∂S/∂V |E = N kB /V .)
• Entropy: From the general expression for the entropy given by S = T −1 (Ē − F ), we get
n h V 2πm 3/2 i
o
5
+
.
S = 23 N kB + kB (ln V N − ln N ! − ln λ3N ) ≈ N kB ln
2
N h2 β
This last expression, obtained using the Stirling approximation ln N ! ≈ N ln N − N , is the Sackur-Tetrode
formula. Notice that, had we not had the N ! factor in the denominator of Z, the ln N ! term would have
been absent in the first expression above (Ē would not have changed, nor the pressure equation of state).
As a result, the entropy would not have been additive (extensive), leading to the Gibbs paradox.
• Chemical potential: Using the expression of µ as a derivative of the Helmholtz free energy, we get
∂ ∂F V
=
,
− kB T (ln V N − ln N ! − ln λ3N ) ≈ −kB T ln
µ :=
∂N T,V
∂N
N λ3
where we have again approximated ln N ! using the Stirling formula. Alternatively, we can use
G
1
µ=
=
(E − T S + pV ) .
N
N
Second-Order Quantities and Fluctuations
• Remark: As our system is in a canonical state, we were able to derive expressions for all thermodynamic
variables above just using the partition function Zc instead of having to use the full ρ(q, p). Second-order
thermodynamic quantities can also be calculated using ony Zc , because they are derivatives of first-order
ones, and some fluctuations of phase-space functions can be reduced to second-order quantities.
• Heat capacity: There is a simple expression for any system in a canonical state. Using (∂/∂β)Zc = −Zc Ē,
i (∆E)2
1
∂ Ē dβ ∂ 1
1 h
1
2 −βH
Ē
−
CV =
Ē)
Z
=
tr (H e−βH ) = −
−
(−Z
tr
(H
)
e
=
.
c
c
∂T V,N
dT ∂β Zc
kB T 2
Zc2
Zc
kB T 2
This is a special case of a general result known as the fluctuation-dissipation theorem. Notice that it is always
positive, CV > 0. For the classical monatomic ideal gas in particular, we can calculate CV = ∂ Ē/∂T = 32 N kB ,
as we already knew from kinetic theory.
• Compressibility: As we also knew already, the isothermal compressibility is κt = −(1/v) (∂V /∂p)T,N = 1/p.
(And again, like other response functions for any system in a canonical ensemble, it is always positive.)
• Energy fluctuations: The variance of the distribution of values of the energy can be obtained from the
general relation (∆E)2 = kB T 2 CV for any system in thermal equilibrium. Then, in our case of the
p monatomic
ideal gas, (∆E)2 = 23 N (kB T )2 = Ē 2 /( 32 N ), and the relative energy fluctuation is (∆E)/Ē = 2/(3N ).
The Maxwell Speed Distribution
A quantity for which we actually need to know the full distribution function ρ(q, p) is the Maxwell probability
distribution
of particle speeds. From the fact that the probability density that a particle have momentum p
R
is h−3 V dr ρ(r, p), we get that the probability density that the magnitude of the velocity be v is
3/2
Z
Z
Z
2
4π m3 v 2
m
m3 v 2
−βH1 (r,p)
sin θ dθ dφ
dr ρ(r, p) =
dr e
=
4πv 2 e−mv /2kB T .
f (v) =
3
3
h
h Z1
2π kB T
V
V
Reading
• Pathria & Beale: Secs 3.9 (classical) and 8.2 (quantum).
• Other textbooks: Halley, pp 150–154; Mattis & Swendsen, Sec 2.2;
Plischke & Bergersen, —; Reif, Secs 6.3 (classical), 7.8 (quantum); Schwabl, Secs 6.3–6.4.