Lecture-21-2023

pdf

School

Simon Fraser University *

*We aren’t endorsed by this school

Course

309

Subject

Chemistry

Date

Oct 30, 2023

Type

pdf

Pages

Uploaded by MateHyenaMaster225

Review of statistical mechanics • Thermodynamics is ‘a science of heat and temperature and entropy’. • Thermodynamics is powerful but it is purely phenomenological . • This means that thermodynamics is based on empirical observations. • It allows us to calculate macroscopic properties given other macro- scopic properties and make predictions about engines, reactions, etc. • However, it lacks: ⇒ Fundamental justification space Why should entropy increase? space Why is entropy at zero absolute temperature zero? Or is it? space Why are work and heat related by the change of internal energy? ⇒ Insights into the thermodynamic variables space What exactly is entropy? What is temperature? Thermodynamics doesn’t really answer such questions. 171

• There are many phenomena thermodynamics can’t explain , in- cluding: space The dependence of heat capacity on T for various materials. space The intensity and color of light emitted by heated objects space (black-body radiation spectrum). space Properties of a small number of atoms or molecules. According to Steven Weinberg (2002) ‘ To find out whether the laws of thermodynamics apply to a par- ticular physical system, you have to ask whether the laws of ther- modynamics can be deduced from what you know about that sys- tem. Sometimes they can, sometimes they can’t. Thermodynam- ics itself is never the explanation of anything - you always have to ask why thermodynamics applies to whatever system you are studying, and you do this by deducing the laws of thermodynamics from whatever more fundamental principles happen to be relevant to that system. ’ 172

• So – one might argue – Thermodynamics is not a fundamental theory and the laws of thermodynamics are not unversal laws. • Can we build a theory that would be based on more fundamental (universally applicable) principles and yield all the thermodynamics equations? • We know that all materials consist of atoms and molecules. • The properties of atoms and molecules are determined by quantum mechanics (which is fundamental). • We know how to calculate the properties of individual atoms and molecules using quantum mechanics. • Statistical mechanics allows us to use this knowledge of the quantum properties of the individual particles to obtain the thermo- dynamic properties: Quantum description of individual particles -→ Thermodynamics 173

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

• Statistical mechanics thus achieves three goals: ⇒ Yields thermodynamics equations based on fundamental, universally applicable principles . ⇒ Offers a way to calculate thermodynamic variables from first principles, i.e. without empirical observations . ⇒ Offers an insight into the thermodynamic variables (entropy, temperature, pressure, etc.) • Let us try to understand conceptually what we have learned so far. 174

• How do we apply statistical mechanics? • First, we say that our system (and surroundings) consist of elemen- tary particles. • Second, we ask: what are the conditions appropriate for the partic- ular experiment to be analyzed? space ⇒ Is the number of particles constant? space ⇒ Is temperature constant? space ⇒ Is volume constant? . . . 175

• Depending on the conditions, we choose the type of the ensem- ble – for example, the canonical ensemble ( NV T ). • Having fixed the ensemble, we need to determine the possible mi- crostates (labelled by i ) that belong to this ensemble and the prob- abilities P i for each microstate. • Having this information, we can calculate any thermodynamic prop- erty, as ¯ Γ = X i P i Γ i , where Γ i is a macroscopic property. 176

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

• For example, to calculate internal energy of the system, we must determine the energy E i or each microstate i and evaluate the sum: ¯ E = X i P i E i . • Therefore, the probabilities P i of microstates play a cen- tral role. • For the canonical ensemble, we have obtained P i = e - βE i Z where β = 1 k B T and Z = X i e - βE i 177

• Evaluating the sum ∑ i Γ i P i is not always the easiest way to calcu- late the thermodynamic properties. • In many cases, it will be easier to relate the thermodynamic prop- erties to the partition function Z . • In fact, if we know Z , we can calculate any thermodynamic prop- erties from the derivatives of Z . 178

• For example, for the canonical ensemble , we have the following relations: A = - k B T ln Z ¯ E = - ∂ ln Z ∂β V,N P = 1 β ∂ ln Z ∂V β,N μ i = - RT ∂ ln Z ∂N i T,V,N j C V = 1 k B T 2 ∂ 2 ln Z ∂β 2 V,N G = k B TV 2 ∂ ∂V [ V - 1 ln Z ] T,N S = k B ln Z + ¯ E T Over the next few classes, we will learn to compute the partition functions Z for different systems. Thus, we will learn to compute the thermodynamic properties from first principles. 179

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

How does statistical mechanics help us understand the thermodynamic properties and laws? • For example, what is entropy? The microscopic definition of entropy is S = - k B X i P i ln P i The minimum of this quantity ( S = 0) occurs when only one state is populated: P some index = 1 and P all other indexes = 0. The maximum of this quantity happens when all P i are the same. The entropy is thus a measure of how many states have sig- nificant probabilities Second law thus says that the probability will spread over accessible states as evenly as possible . 180

• We stated that First Law = Conservation of Energy. Why? How does work change internal energy? In quantum mechanics, there is a theorem – called adiabatic theorem – saying that an external action, if applied slowly, changes the energies ( E i ) of the system, but does not change the probabilities ( P i ) The microscopic definition of work is therefore δW = X i P i dE i This is why work alone does not change entropy ! Heat, on the other hand, changes the populations of different mi- crostates, but does not change their energy, so δQ = X i E i dP i This is why heat does change entropy ! 181

Thus, we have: δW + δQ = X i P i dE i + X i E i dP i = X i d ( P i E i ) = d   X i P i E i   or δW + δQ = d ¯ E This is conservation of energy!! 182

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

• Third Law follows directly from S = - k B X i P i ln P i and the expression for the probabilities P i = e - βE i Z When T → 0, P 0 → 1 and P i> 0 → 0, where 0 is the lowest-energy state of the system. Thus S → 0 (provided the lowest energy state is non-degenerate). 183

Exercise : Consider a gas of particles moving along one dimension at a temper- ature T . 1) What is the average speed of the particles? 2) What is the average kinetic energy of the particles? 184