MCP 1 Chapter 2 Statistical Mechanics

Questions about the lecture 'From Molecular to Continuum Physics 1' of the RWTH Aachen Chapter 2 Statistical Mechanics

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This flashcard set delves into the principles of statistical mechanics at a university level, focusing on key concepts like energy definitions, partition functions, and various ensembles. It covers the characteristics of different energy components such as translational, rotational, and vibrational energies, along with their respective partition functions. The set also explores fundamental equations of statistical mechanics, including entropy, internal energy, and bridge equations. It is particularly useful for students and researchers in computer science who need to understand the statistical behavior of molecular systems and their thermodynamic properties.

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18.02.2017 / 20.02.2017

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What are the fundamental equations of statistical mechanics? [8]

1. S = k_B*T*(d ln Q/dT)_V,N + k_B ln Q

2. p = k_B*T*(d ln Q/dV)_T,N

3. E = k_B*T²*(d ln Q/dT)_V,N

4. mye_i= -k_B*T*(d ln Q/dN_i)_T,V_a!=i

5. c_V = 2kT*(dlnQ/dT)_V + kT²*(d²lnQ/dT²)_V

6. H – H_0 = U + PV = kT²*(dlnQ/dT)_V + kTV(dlnQ/dV)_T

7. G – G_0 = H – TS = A + PV = -kTlnQ + kTV(dlnQ/dV)_T

8. G – G_0 = -nRT*ln(q_m/N_A) for monoatomic gas with indistinguishable independent molecules Q=q^N/N!

What are the components of the energy states eps? [4]

Sum of 1. translation eps^T 2. rotation eps^R 3. vibration eps^V and 4. electronic contribution eps^E

Which is the energy order of the energy states components?

-| translation, rotation, vibration, electronic contribution |+

What are the components of the partition regarding different energy components?

q = q^T*q^R*q^V*q^E

What is the basic definition of the sub partition q^X?

q^x = Sum_i exp(-\beta*eps_i^x)

What is the definition of eps^T? [3]

1. eps^T = n²h²/8ma² with n speed a length

2. peta = h²/8ma² // For n=1

3. eps_n^T = (n²-1)*peta

What is the definition of q^T? [3]

1. q^T = Sum_n exp(-\beta*(n²-1)*peta)

2. q^T = sqrt(2pi*mkT/h²)*a = a/Delta // For 1D container

3. q^T = q_x^T + q_y^T + q_z^T = V/Delta³ with V=abc

What is the definition of eps^R?

eps_J^R = hcB*J(J+1)

What is the definition of q^R for hetero- and homonuclear diatomic? [2]

1. q^R ~ T/sigma*Theta^R with Theta^R = hcB/k

2. sigma is symmetry number // If T>>Theta^R

What is the definition of q^R for nonlinear? [2]

1. q^R ~ pi^1/2*T^3/2 / sqrt(sigma*Theta^A*Theta^B*Theta^C)

2. Theta^X = h²/8pi²*X*k

What holds for nonlinear regarding q^R? [2]

1. Three moments of inertia

2. Three rotational constants

What is the definition of eps^V for 1D harmonic oscillators?

eps^V = (u+1/2)hV

What is the definition of eps^V for coupled harmonic oscillators (So-called polyatomic molecules)? [3]

1. eps^V = Sum_i^f (u_i+1/2)hV_i with degrees of freedom f

2. f = 3N-6 for nonlinear molecule

3. f = 3N-5 for linear molecule

What is the definition of q^V for 1D harmonic oscillators? [2]

1. q^V = exp(-Theta^V/2T)/ 1-exp(-Theta^V/T)

2. Theta^V=hv/k

What is the definition of q^V for coupled harmonic oscillators (So-called polyatomic molecules)?

q^V = T/Theta^V // If T>>Theta^V

What is the definition of q^E?

q^E ~ g_0 * exp(-\beta*eps_0^E) with g_0 = g^E // If >>T

What are the characteristics of the Sackur-Tetrode equation? [4]

1. For monoatomic gas with indistinguishable independent molecules Q=q^N/N!

2. S = [U(T)-U(0)]/T + nR*(ln q – ln N + 1) with N=n*N_A and k=R/N_A

3. Only mode of motion is translation

4. S(T) = nR*ln(e^5/2*V/nN_A*Delta³)

What is the general definition of mean energy <eps^M>?

<eps^M> = - 1/q^M * (dq^M/d\beta)_V with M:=T R V or E

What is the definition of the translation mean energy <eps^T>?

<eps^T> = 3/2 *kT

What is the definition of the rotational mean energy <eps^R>?

<eps^R> = kT

What is statistical mechanics?

Tastatur-Befehle:

= drehen,

= vor-/rückwärts,

= scrollen

Motion of mas objects with mass m under force

What describes statistical mechanics?

Tastatur-Befehle:

= drehen,

= vor-/rückwärts,

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Relationship of mis matter behavior and its mas properties

What does statistical mechanics connects?

Tastatur-Befehle:

= drehen,

= vor-/rückwärts,

= scrollen

Thermodynamics to mis behavior (Statistical thermodynamics)

What is statistical mechanics interested in?

Tastatur-Befehle:

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= vor-/rückwärts,

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Average motion of particles due to number in e.g. one mole

What happens if objects become sufficiently small?

Tastatur-Befehle:

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= vor-/rückwärts,

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Covered by quantum mechanics?

What are the characteristics of the ensemble concept? [4]

Tastatur-Befehle:

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1. Mas observables

2. Collection of systems with mis interactions and mas properties

3. Every system changes due to mis laws of motion from its initial condition to its own unique mis state

4. All systems share at least one extensive property

What are the characteristics of mas observables? [3]

Tastatur-Befehle:

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1. Not sensitive to precise mis details

2. Averages A (so-called equilibrium ensembles)

3. Connected to a mis function

What holds for the mas observables connected with a mis function? [3]

Tastatur-Befehle:

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= vor-/rückwärts,

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1. A=A(r^N,q^N) with coordinates r and momenta q

2. Probability density p=p(r^N,q^N)

3. <A>ens = int int dr^N dq^N A(r^N,q^N)*p(r^N,q^N)

What is the ergodic hypothesis concerning mas observables? [3]

Tastatur-Befehle:

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= vor-/rückwärts,

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1. Time average equals ensemble average over long times

2. <A>ens = <A>time for t→inf

3. Study via molecular dynamics simulation

What are the characteristics of population p_i concerning mas observables? [2]

Tastatur-Befehle:

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= vor-/rückwärts,

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1. <A>=Sum_i Ai*pi with population p_i

2. p_i = n_i/N = e^(-\beta*eps_i)/q

What are the characteristics of partition q concerning mas observables? [5]

1. q = Sum_i e^(-\beta*eps_i)

2. q = 1/(1-e^(-\beta*eps)) for equally energy levels

3. q = Sum_l g_l*e^(-\beta*eps_l) for degeneracy g_l

4. lim_T→0 q = g_0 // One surviving term

5. lim_T→inf q = number of molecular states

Which types of ensemble exist? [4]

1. Micro-canonical (N,V,E) 2. canonical (N,V,T) 3. grand-canonical and 4. isobaric-isothermal (N,P,T)

What are the characteristics of a micro-canonical ensemble? [3]

1. Fix energy

2. Fix composition

3. Equal probability for each possible state

What are the characteristics of a canonical ensemble? [4]

1. Varying energy

2. Fix composition

3. Thermal equilibrium with temperature T

4. Probability depends on energy

What are the characteristics of a grand-canonical ensemble? [5]

1. Varying energy

2. Varying composition

3. Thermal equilibrium with temperature T

4. Chemical equilibrium with chemical potentials

5. Probability depends on energy and composition

What are the characteristics of a system in statistical mechanics? [5]

1. Composition of N particles

2. Population of states n_0 … n_n // {n_0, …, n_n} is called instantaneous configuration

3. Degeneracy g_i max amount in n_i // State_i is seen as a container with max capacity

4. State of energies eps_0 … eps_n // Total energy E

5. “Number of different ways” or “weight of a configuration” W

What is the principle of equal a priori probabilities?

“All possible energy distribution are equally probable”

What are the characteristics of composition? [3]

1. Distinguishable

2. Independent

3. Not interacting

What holds for the Boltzmann distribution? [3]

1. Dominant configuration

2. n_i/N = e^(-\beta*eps_i) / Sum_i e^(-\beta*eps_i) forall i

3. \beta = 1/kT

What is the definition of W for states? [4]

1. W=N!/Prod_i n_i!

2. ln W = ln N! - Sum_i ln n_i!

3. Sterling approximation with ln x! = x*lnx-x for << x

4. ln W = N ln N – Sum_i n_i ln n_i

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Lernkarten

What is the definition of eps^V for 1D harmonic oscillators?

eps^V = (u+1/2)hV

What is the definition of eps^V for coupled harmonic oscillators (So-called polyatomic molecules)? [3]

1. eps^V = Sum_i^f (u_i+1/2)hV_i with degrees of freedom f

2. f = 3N-6 for nonlinear molecule

3. f = 3N-5 for linear molecule

What is the definition of q^V for 1D harmonic oscillators? [2]

1. q^V = exp(-Theta^V/2T)/ 1-exp(-Theta^V/T)

2. Theta^V=hv/k

What is the definition of q^V for coupled harmonic oscillators (So-called polyatomic molecules)?

q^V = T/Theta^V // If T>>Theta^V

What is the definition of q^E?

q^E ~ g_0 * exp(-\beta*eps_0^E) with g_0 = g^E // If >>T

What are the characteristics of the Sackur-Tetrode equation? [4]

1. For monoatomic gas with indistinguishable independent molecules Q=q^N/N!

2. S = [U(T)-U(0)]/T + nR*(ln q – ln N + 1) with N=n*N_A and k=R/N_A

3. Only mode of motion is translation

4. S(T) = nR*ln(e^5/2*V/nN_A*Delta³)

What is the general definition of mean energy <eps^M>?

<eps^M> = - 1/q^M * (dq^M/d\beta)_V with M:=T R V or E

What is the definition of the translation mean energy <eps^T>?

<eps^T> = 3/2 *kT

What is the definition of the rotational mean energy <eps^R>?

<eps^R> = kT

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MCP 1 Chapter 2 Statistical Mechanics

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