Kinetic Theory of Gases Flashcards
What is the primary goal of Kinetic theory?
To relate molecular level behavior to macroscopic gas dynamics.
Boyd, NEGD and Molecular Sim., Pg. 3
What two processes take place at the molecular level of a gas?
- Inter-molecular collisions
- Translational motion
Boyd, NEGD and Molecular Sim., Pg. 3
Particles have what four general properties?
- Mass
- Size
- Position
- Internal energy
Boyd, NEGD and Molecular Sim., Pg. 4
What is a “simple gas?”
A collection of gas in which all molecules are of the same chemical species.
Boyd, NEGD and Molecular Sim., Pg. 4
True or False
Weak forces causing attraction are important only at very low gas temperatures (e.g., less than 100 K).
True
Boyd, NEGD and Molecular Sim., Pg. 5
Define “number density.”
The number of particles per unit volume.
Note: Defined by “n” in the reference below.
Boyd, NEGD and Molecular Sim., Pg. 6
Define “mass density.”
The mass per unit volume.
Note: Defined by ρ in the reference below.
Boyd, NEGD and Molecular Sim., Pg. 7
Describe the Velocity Distribution Function (VDF).
The VDF is a function that outputs the probability (0 to 100%) of finding a particle at a given velocity.
VFD(particle i, x, y, z, V) = x | x ∊ [0,1]
Boyd, NEGD and Molecular Sim., Pg. 7
True or False
Translational energy arises due to kinetic motion.
True
Boyd, NEGD and Molecular Sim., Pg. 9
What is the universal gas constant in units of J/(kg-mol-K)?
8314 J/(kg-mol-K)
Note: Defined by and R with a hat over it in the reference below.
Boyd, NEGD and Molecular Sim., Pg. 10
What is Boltzmann’s constant in units of J/K?
k = 1.38e-23 J/K
Boyd, NEGD and Molecular Sim., Pg. 10
What is the physical mechanism that pushes a gas toward equilibrium?
Molecular Collisions
Boyd, NEGD and Molecular Sim., Pg. 12
What two concepts are usually used to quantify the degree of nonequilibrium in a gas?
- Mean free path, λ
- Collision frequency, Θ
Boyd, NEGD and Molecular Sim., Pg. 12
Define “mean free path.”
The average distance traveled by each particle between successive collisions.
Boyd, NEGD and Molecular Sim., Pg. 12
Define “collision frequency.”
The number of collisions per unit time experienced by each particle.
Note: Given by Θ in the reference below.
Boyd, NEGD and Molecular Sim., Pg. 12
Define “mean free time.”
The average time between successive collisions of each particle.
Tau_(mft) = 1/Θ
Boyd, NEGD and Molecular Sim., Pg. 12
True or False
As λ increases, n decreases.
n = number density
lambda = mean free path
True. This is why a rarfied gas has a large mean free path.
Boyd, NEGD and Molecular Sim., Pg. 14
True or False
λ does not depend on temperature so long as only hard sphere particles are being considered.
True
Boyd, NEGD and Molecular Sim., Pg. 14
Explain how the Knudsen number relates to equilibrium.
For a high Knudsen number (e.g., Kn >0.01), there are not enough particle collisions per characteristic length of the flow to establish equilibrium.
Note: Kn = λ / L
where L is the characteristic length and λ is the mean free path.
Boyd, NEGD and Molecular Sim., Pg. 14
True or False
Thermodynamic equillibrium can be thought of as when all macroscopic properties are uniform in space and time.
True.
Boyd, NEGD and Molecular Sim., Pg. 14
Explain what a probability density function (PDF) is.
A function that tells you what the probability of a certain outcome, based on the inputs.
For example, if we have a jar of blue, red, and gree marbles, the PDF would tell you the probability of drawing each color.
ChatGPT
Diffusion concerns the transport of _________.
Chemical species (or just mass for a non-reacting flow).
Boyd, NEGD and Molecular Sim., Pg. 15
The rate of diffusion is described by _______ law.
Fick’s
Boyd, NEGD and Molecular Sim., Pg. 15
Viscosity involves the transport of _______.
Momentum
Boyd, NEGD and Molecular Sim., Pg. 16
Thermal conductivity involves the transport of _________.
It is described by ________.
Energy (e.g., heat)
Fourier’s Law of heat conduction (q = - k dT/dx)
Boyd, NEGD and Molecular Sim., Pg. 16
Define the “normalized velocity distribution function.”
A function that provides the probability of finding a particle with a velocity within a small range of velocities.
Boyd, NEGD and Molecular Sim., Pg. 20
What is a “normalization condition?”
In mathematics and especially in the context of probability and quantum mechanics, the term “normalization condition” refers to a requirement that certain mathematical functions have a total “amount” or “value” of 1 when integrated (or summed up) over all possible values or outcomes.
chatGPT
Describe “Phase Space” as it relates to particles.
Phase space is a multidimensional space where every possible state of a system can be represented as a unique point.
Additional Notes: For a single particle, its state is defined by its position and momentum (or equivalently, velocity). So, for a single particle moving in three-dimensional space, its phase space would be six-dimensional: three dimensions for its position (x, y, z) and three for its momentum (p_x, p_y, p_z) or velocity (v_x, v_y, v_z).
chatGPT
True or False
There is no variation in the VDF in time or space when the gas is in equilibrium.
True.
Boyd, NEGD and Molecular Sim., Pg. 26
What is the “principle of detailed balance?”
This principle states that at equilibrium, the rate of change in the forward direction is equal to the rate of change in the backward direction.
Extra Notes: The process could be collisions, chemical reactions, etc.
Boyd, NEGD and Molecular Sim., Pg. 26
Describe the H-Theorm as it relates to progression towards equilibrium.
Similar to the concept of entropy and thermodynamics, the H theorem specifies the direction that the collision process must take in order for a gas to reach equilibrium.
Boyd, NEGD and Molecular Sim., Pg. 26
What is the main takeaway from the H-Theorm is taken to it’s full conclusion?
The H-theorem tells us that the average fractional rate of change in the velocity distribution function due to collisions can only decrease and therefore the effect of collisions is to always move the system closer to equilibrium.
Boyd, NEGD and Molecular Sim., Pg. 29
The number of collisions Z that a molecule makes with other molecules provides an _______ limit for a reaction rate.
upper
Chem Kinetics and Rxn Dynamics, Houston, Pg. 2
True or False
The product of pressure and volume is directly proportional to the squared mean velocity of the gas.
True
pV = (1/3)-mol-N_A-m-\bar{v}^2
mol = number of moles
N_A = Avagodros number
m = mass of the molecules in the gas
\bar{v} = average velocity, m/s
Chem Kinetics and Rxn Dynamics, Houston, Pg. 3
True or False:
A “distribution” function is the same thing as a “probability” function.
True.
Chem Kinetics and Rxn Dynamics, Houston, Pg. 7
The pressure of a gas in a container is related to the number of particles N through what equation (give the actual equation).
p = Nm\bar{v}^2/(3V)
N = number of particles
m = mass of a single particle
\bar{v}^2 = Square of the mean velocity
V = Volume of the container.
Molecular Physical Chemistry for Engineers, Yates, Pg. 295
Give the equation for the average translational energy of N molocules.
(U - U_0)_{tr}= (1/2)Nm\bar{v}^2 = 3pV/2 := J
Molecular Physical Chemistry for Engineers, Yates, Pg. 295
True or False:
One of the general assumptions used in kinnetic theory is that the size of molocules is negligible compared to the distance between collisions.
True.
MTK, Lecture 20, Slide 2
Define the “Collision Frequency.”
The number of collisions experienced by a molecule per unit time.
MTK, Lecture 21, Slide 2
Define “Collision Number.”
Number of collisions in a unit volume per unit time.
MTK, Lecture 21, Slide 2
Define “Mean Free Path.”
The average distance a molecule travels between collisions with other molecules.
MTK, Lecture 21, Slide 2
Describe the Knudesen Number (Kn).
The Knudsen number (Kn) is a dimensionless number defined as the ratio of the mean free path of a particle to a characteristic physical length scale of the system (Kn = λ / L).
It is used to characterize the relative importance of molecular mean free path effects in a given physical situation, particularly in gas dynamics and fluid mechanics.
ChatGPT
True or False:
The mean free path is proportional to the transport of heat, momentum, and matter.
True.
MTK Slides
True or False:
Collision frequency and collision number are related to the rate at which chemical reactions occur (in the gas phase).
True
MTK Slides
In the simple collision model, the collision cross section is assumed to be _________ and equal to ________ for all E.
constant
pi d^2
In the simple collision model, the activation energy is ___________.
Zero
In the reactive hard sphere’s model, what energy is considered to be conducive to a reaction?
The line of centers energy.
True or False:
In general, the activation energy is not the same as the threshold energy for the reactive hard sphere model.
True.
Ea and E^* (Activation energy and threshold energy) differ by what amount?
kT/2
Where k is the Boltzmann constant.
