Temperature and the Boltzmann factor

Question 1

The _________ ensemble is an idealized collection of systems, each of which can exchange its energy with a large reservoir of heat. An ensemble of snapshots of a system with the same N, V, and T.

Answer 1

canonical.

Question 2

As a function of time, systems in thermal contact tend towards thermal equilibrium, rather than away from it. The process that leads to thermal equilibrium is called _____________.

Answer 2

thermalization.

Question 3

The energy content and the temperatures of the two bodies will no longer be changing with time.

Answer 3

Thermal equilibrium.

Question 4

All the parts of a thing taken together, so that each part is considered only in relation to the whole.

ex. 1. A group of soliders that make up a squadron.
ex. 2. The particles that make up a fluid.

Answer 4

Ensemble.

Question 5

The system will most likely be found in a configuration that is represented by the most/least microstates. What is this based on?

Answer 5

most.

1. Each one of the possible microstates of the system is equally likely to occur. 2. The system’s internal dynamics are such that the microstates of the system are continually changing. 3. Given enough time, the system will explore all possible microstates and spend equal time in each of them.

Question 6

For a thermometer to work well, its heat capacity must be much higher/lower than that of the object whose temperature one wants to measure. If this is not the case, the action of measurement (placing the thermometer in thermal contact with the object) could alter the temperature of the object.

Answer 6

lower.

Heat capacity is a weighted average of the temperature measured. Thus you would want the thermometer to have a smaller heat capacity so not to exchange enough energy with the system to change the temperature of the system.

Question 7

___________ _______ is a boundary that freely allows heat to be exchanged (i.e., very short relaxation time for systems separated by this).

Answer 7

Diathermal wall

dia- a prefix occurring in loanwords from Greek (diabetes; dialect) and used, in the formation of compound words, to mean “passing through”.

Question 8

Specifying a _______ is like determining the r and v of each and every particle in a system say a gas.

Answer 8

microstate

Question 9

The temperature T of a system is given by what?

Answer 9

β = 1 /k_BT = d(ln(Ω)) / dE.

k_B is the boltzmann constant, E is its energy, and Ω is the number of microstates (i.e., the number of ways of arraging the quanta of energy in the system).

Question 10

In thermodynamics, a _____ ________ is an independent variable of a state function like internal energy, enthalpy, and entropy. Examples include temperature, pressure, and volume. Heat and work are not state functions, but process functions.

Answer 10

state variable.

Question 11

Define the Zeroth law of Thermodynamics.

Answer 11

Two systems, each separately in thermal equilibrium with a third, are in equilibrium with each other.

Question 12

The bodies in thermal equilibrium have the same ____________.

Answer 12

Temperature.

Question 13

The system could be described by a way large number of equally likely ____________.

Answer 13

microstates.

Question 14

Which thermometer is perfect and gives the real thing, against which all other thermometers should be judged?

Answer 14

The hypothetical machine called a Carnot engine.

Question 15

What you measure is a property of the ____________ of the system. The ____________ are not equally likely, because different ___________ correspond to different numbers of microstate.

Answer 15

macrostate.

Question 16

Which law gives us justification for how to actually measure temperature?

Answer 16

The Zeroth Law.

Question 17

The _____________ ensemble is an idealized collection of systems that each have the same fixed energy, number of particles, and volume.

Answer 17

microcanonical.

Question 18

The ____________ ensemble is an idealized collection of systems, each of which can exchange both energy and particles with a large reservoir.

Answer 18

grand canonical.

Question 19

Which ensemble gives us the Boltzmann factor & the Partition function which describes exactly how a small system behaves when coupled to a large reservoir at temperature T?

Answer 19

Canonical ensemble.

Question 20

A _____ ________ is a function defined for a system relating several state variables or state quantities that depends only on the current equilibrium state of the system, for example a gas, a liquid, a solid, crystal, or emulsion. ______ ________ do not depend on the path by which the system arrived at its present state. A ______ ________ describes the equilibrium state of a system and thus also describes the type of system. For example, a _____ _______ could describe an atom or molecule in a gaseous, liquid, or solid form; a heterogeneous or homogeneous mixture; and the amounts of energy required to create such systems or change them into a different equilibrium state.

Answer 20

state function.

Question 21

In statistical mechanics, a ___________ is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations. In contrast, the ____________ of a system refers to its macroscopic properties, such as its temperature, pressure, volume and density. ___________ appear as different possible ways the system can achieve a particular ____________.

Answer 21

microstate, macrostate; Microstates, macrostate.

Question 22

In a canonical ensemble, give the probability of a system having a microstate of E_r out of E_i microstates.

Answer 22

P(E_r[energy E in microstate r]) = e^{-E_r / k_BT} / Σe^{-E_i / k_BT}, where Z = Σe^{-E_i / k_BT}​ is called the partition function.

Question 23

Bodies in which are able to exchange energy.

Answer 23

Thermal contact.

Question 24

_____________ is a measure of the tendency of an object to spontaneously give up/absorb energy to/from its surroundings.

Answer 24

Temperature.

Question 25

Time required to reach thermal equilibrium is the __________ _____.

Answer 25

relaxation time.

Question 26

____________ wall is a system boundary that freely allows heat to be exchanged, i.e. very short relaxation time for systems separated by a ___________ wall.

Our skin.

Answer 26

Diathermal.

Dia-, in Latin, means to pass through.

Question 27

____________ wall is a system boundary that does not allow heat to be exchanged, i.e. very long relatation time for systems separated by a _________ wall.

Answer 27

Adiabatic.

Question 28

Define the partition function Z.

Answer 28

Z = Σ e^-nɛ​β , where β = 1 / k_BT and ɛ is the energy.

Question 29

What will these two converge to under the correct assumptions?

What does the power n represent in the finite series?

Answer 29

Finite geometric series = a₁ [1 - rⁿ] / [1 - r]

n represents the number of terms.

Infinite geometric series = a₁ / [1 -r]

Question 30

A ____________ for a system is all about time and the energy of the molecules in that system. “In a system its energy is constantly being redistributed among its particles. In liquids and gases, the particles themselves are constantly redistributing in location as well as changing in the quanta (the individual amount of energy that each molecule has) due to their incessantly colliding, bouncing off each other with (usually) a different amount of energy for each molecule after the collision.. Each specific way, each arrangement of the energy of each molecule in the whole system at one instant is called a ___________.”

Answer 30

microstate.

Question 31

A __________ is the thermodynamic state of any system that is exactly characterized by measurement of the system’s properties such as P, V, T, H and number of moles of each constituent. Thus, a __________ does not change over time if its observable properties do not change.

Answer 31

macrostate.

Question 32

_______________ appear as different possible ways the system can achieve a particular ______________.

Answer 32

Microstate: Macrostate

Question 33

Define configuration.

Answer 33

noun;

an arrangement of elements in a particular form, figure, or combination.

Question 34

In probability theory and statistics, the ___________ distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

Answer 34

exponential

Question 35

For the canonical ensemble, the probability that a particular system has energy ε is given by P(ε) which is proporitonal to …..

Answer 35

The Boltzmann distribution,

where e^-β​ε is known as the Boltzmann factor.

Question 36

Define the electron volt.

Derive it’s value.

Answer 36

In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacuum. When used as a unit of energy, the numerical value of 1 eV in joules (symbol J) is equivalent to the numerical value of the charge of an electron in coulombs (symbol C).

1 eV = 1.602176634×10⁻¹⁹ J