Kinetic theory of gases

src: i.ytimg.com

The kinetic theory describes gas as a large number of submicroscopic particles (atoms or molecules), all of which are in constant fast motion that has randomness arising from their many collisions with one another and with the container walls.

Kinetic theory explains the macroscopic properties of gases, such as pressure, temperature, viscosity, thermal conductivity, and volume, taking into account the composition and movement of the molecule. This theory argues that gas pressure is caused by impact, on the walls of containers, molecules or atoms moving at different speeds.

The kinetic theory defines the temperature in its own way, in contrast to the definition of thermodynamics.

Under the microscope, the molecules that form the liquid are too small to be visible, but the movement of anxious pollen or dust particles can be seen. Known as Brownian motion, the direct result of a collision between a grain or a particle and a molecule of liquid. As analyzed by Albert Einstein in 1905, the experimental evidence for this kinetic theory is generally seen as a confirmation of the existence of concrete materials of atoms and molecules.

Video Kinetic theory of gases

Assumption

The ideal gas theory makes the following assumptions:

• The gas consists of very small particles known as molecules. Their small size is such that the total volume of individual gas molecules added upward is negligible compared to the smallest open-ball volume containing all the molecules. This is equivalent to stating that the average distance that separates large gas particles is compared to their size.
• These particles have the same mass.
• The number of molecules is so great that statistical treatment can be applied.
• These molecules move constantly, randomly, and quickly.
• Fast-moving particles continue to collide between themselves and with the walls of the container. All these collisions are very elastic. This means the molecules are considered to be perfectly spherical and elastic in nature.
• Except during the crash, interactions between molecules can be ignored. (That is, they do not use the power of each other.)

This means:

1. Relativistic effects can be ignored.

2. Quantum mechanical effects can be ignored. This means the inter-particle distance is much greater than the de Broglie wavelength and the molecule is treated as a classical object.

3. Because of the two above, their dynamics can be treated classically. This means that the molecular motion equation is time-reversible.

• The average kinetic energy of a gas particle depends only on the absolute temperature of the system. The kinetic theory has its own temperature definition, not identical with the definition of thermodynamics.
• The elapsed time of the collision between the molecule and the container wall is negligible when compared to the time between successive collisions.
• Since they have masses, the gas molecules will be affected by gravity.

More modern developments have relaxed these assumptions and are based on the Boltzmann equation. It can accurately describe the properties of solid gases, since they include the volume of molecules. The necessary assumptions are the absence of quantum effects, molecular chaos and small gradients in bulk properties. Expansion to higher orders in density is known as viria expansion.

An important book on kinetic theory is that by Chapman and Cowling. An important approach to this subject is called Chapman-Enskog's theory. There are many modern developments and there are alternative approaches developed by Grad based on moment expansion. At another limit, for highly purified gases, the gradients in bulk properties are not small compared to the average free path. This is known as the Knudsen regime and expansion can be carried out in Knudsen numbers.

Maps Kinetic theory of gases

The nature of equilibrium

Pressure and kinetic energy

In the kinetic model of the gas, the pressure is equal to the force given by the hitting and bouncing atoms from the surface area of ​​the gas container unit. Consider the molecular gas N , each mass m , enclosed in the volume cube V = L ³ . When a gas molecule collides with a container wall perpendicular to the x-axis and bounces in the opposite direction at the same rate (elastic collision), the momentum change is given by:

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\mathrm{?}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂpÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{   Â ·                          me         Â mo moan,            <Â> x                          -           Â ·                  Â          Â mo moan,            <Â> x                          =           Â ·                          me         Â mo moan,            <Â> x                          -        (         -           Â ·                          me         Â mo moan,            <Â> x                         )         =         2           Â ·                          me         Â mo moan,            <Â> x                          =         2          m            Â     v                       <Â> x                          ,               {\ displaystyle \ Delta p = p_ {i, x} -p_ {f, x} = p_ {i, x} - (- p_ {i, x}) = 2p_ {i, x} = 2mv_ {x},}  Â}

where p is the momentum, i and f shows the initial and final momentum (before and after the collision), x shows that only the direction of x is being considered, and x is the particle velocity (same before and after the collision).

Partikel ini berdampak pada satu sisi dinding tertentu sekali

${\ displaystyle \ Delta t = {\ frac {2L} {v_ {x}}},} Â Â$

where L is the distance between the opposite wall.

Gaya yang disebabkan oleh partikel ini

${\ displaystyle F = {\ frac {\ Delta p} {\ Delta t}} = {\ frac {mv_ {x} ^ 2}} { L}}. } Â Â$

Gaya total de dinding

${\ displaystyle F = {\ frac {Nm {\ overline {v_ {x} ^ 2}}} {L}},} Â Â$

where the bar shows the average above the particle N .

Karena gerakan partikel bersifat acak dan tidak ada bias yang diterapkan ke arah mana pun, rata-rata kecepatan kuadrat di setiap arah identik:

${\ displaystyle {\ overline {v_ {x} ^ 2}} = {\ overline {v_ {y} 2}} = \ overline {v_ {z} ^ 2}}}} Â Â$

Denominate the Pythagoras theorem of the dimensional dimension, totally approved v diberikan oleh

${\ displaystyle {\ overline {v ^ 2}} = {\ overline {v_ {x} ^ 2}} {\ overline {v_ {y} 2}}} {\ overline {v_ {z} ^ 2}},} Â Â$

${\ displaystyle {\ overline {v2}} = 3 {\ overline {v_ {x} ^ 2}}}} Â Â$

Karena itu:

${\ displaystyle {\ overline {v_ {x} ^ 2}} = {\ frac {\ overline {v2}} {3 }},} Â Â$

dan kekuatan dapat ditulis sebagai:

${\ displaystyle F = {\ frac {Nm {\ overline {v2}}} {3L}}.} Â Â$

Gaya ini diberikan pada area L ² . Karena itu, tea gasnya

${\ displaystyle P = {\ frac {F} {L2}} = {\ frac {Nm {\ overline {v2} }}} {3V}},} Â Â$

where V = L ³ is the volume of the box.

Dalam hal energi kinetik gas K :

${\ displaystyle PV = {\ frac {2} {3}} \ kali {K}.} Â Â$

Ini adalah hasil non-sepele pertama dari teori kinetik karena berhubungan tekanan, properti makroskopik, dengan energi kinetik (translate) dari molekul ${\ displaystyle N {\ frac {1} {2}} m {\ overline {v2}}} Â$ , yang merupakan properti microskopis.

Suhu dan energi kinetik

Tulis ulang hasil di atas untuk tekanan sebagai ${\ displaystyle PV = {Nm {\ overline {v2}} \ over 3}} Â$ , kita dapat menggabungkannya denotes hookum ideal gas

di mana ${\ displaystyle \ displaystyle k_ {B}} Â$ adalah constant Boltzmann dan ${\ displaystyle \ displaystyle T} Â$ suhu absolut yang ditentukan oleh hukum gas ideal, untuk mendapatkan

${\ displaystyle k_ {B} T = {m {\ overline {v2}} \ over 3}} Â Â$ ,

yang mengarah that emitting sederhana would give energy to kinetik rata-rata by molecule,

${\ displaystyle \ displaystyle {\ frac {1} {2}} m {\ overline {v2}} = {\ frac {3} {2}} k_ {B} T} Â Â$ .

Energi kinetik dari sistem adalah N kali lipat dari sebuah molekul, yaitu ${\ displaystyle K = {\ frac {1} {2}} Nm {\ overline {v2}}} Â Â$ . Kemudian suhu ${\ displaystyle \ displaystyle T} Â$ mengambil formulate

becoming

The equation ( 3 ) is one of the important results of kinetic theory: The average kinetic energy of molecules is proportional to the absolute temperature of the ideal gas law . From Equation ( 1 ) and Equation ( 3 ), we have

Thus, the product of pressure and volume per mole is proportional to the average molecular kinetic energy (translation).

The equations ( 1 ) and Equation ( 4 ) are called "classic results", which can also be derived from statistical mechanics; for more details see:

Karena ada ${\ displaystyle \ displaystyle 3N}  ÂÂ$ derajat kebebasan dalam sistem monatomik-gas dengan ${\ displaystyle \ displaystyle N}  ÂÂ$ partikel, energi kinetik per derajat kebebasan per molekul adalah

In the kinetic energy per degrees of freedom, the temperature proportionality constant is 1/2 times the Boltzmann constant or R/2 per mole. In addition, the temperature will decrease when the pressure drops to a certain point. This result is related to the equipartition theorem.

As noted in the article on heat capacity, diatomic gas must have 7 degrees of freedom, but a lighter diatomic gas acts as if it has only 5. Monatomic has 3 degrees of freedom.

Thus kinetic energy per kelvin (monatomic ideal gas) is 3 [R/2] = 3R/2:

• per mole: 12.47 J
• per molecule: 20.7 yJ = 129? eV.

At the standard temperature (273.15 K), we get:

• per mole: 3406 J
• per molecule: 5.65 zJ = 35.2 meV.

Collision with container

One can calculate the number of collisions of atoms or molecules with the walls of the container per unit area per unit time.

They deny that they have ideal gas, hasil derivasi dalam persamaan untuk jumlah total tabrakan per satuan waktu per area:

${\ displaystyle N ^ {*} = {\ frac {1} {4}} {\ frac {N} {V}} v _ {\ text {avg}} = {\ frac {n} {4}} {\ sqrt {\ frac {8k_ {B} T} {\ pi m}}}.} Â Â$

This quantity is also known as "the rate of impingement" in vacuum physics.

The molecular speed

di mana v ada di m/s, T ada di kelvin, dan m adalah massa satu molekul gas. Kecepatan yang paling mungkin (atau mode) ${\ displaystyle v _ {\ text {p}}}  ÂÂ$ adalah 81,6% dari kecepatan rms ${\ displaystyle v _ {\ text {rms}}}  ÂÂ$ , dan mean (rata-rata aritmetik, atau rata-rata) ${\ displaystyle {\ bar {v}}}  ÂÂ$ adalah 92,1% dari kecepatan rms (distribusi kecepatan isotropik).

See:

• Average,
• Average root speed
• Arithmetic means
• Means
• Mode (stats)

src: s1-ssl.dmcdn.net

Transport properties

The kinetic theory of gases not only deals with gases in thermodynamic equilibrium, but is also very important with gases that are not in thermodynamic equilibrium. This means considering what is known as "transport properties", such as viscosity and thermal conductivity.

Viscosity and kinetic momentum

In books on basic kinetic theory one can find results for the modeling of dilute gases that have been widely used. The decrease in the kinetic model for shear viscosity usually begins by considering the flow of Couette in which two parallel plates are separated by the gas layer. The top plate moves with constant velocity to the right due to force F. The stationary bottom plate, and the same force and opposite should therefore act on it to keep it silent. The molecules in the gas layer have advanced velocity components ${\ displaystyle u}$ which increases uniformly with distance ${\ displaystyle y}$ above the bottom plate. The non-equilibrium flow is superimposed on the Maxwell-Boltzmann equilibrium distribution of molecular motion.

Biarkan ${\ displaystyle \ sigma}  ÂÂ$ menjadi penampang tabrakan dari satu molekul yang bertabrakan dengan satu molekul lain. Kepadatan angka ${\ displaystyle C}  ÂÂ$ didefinisikan sebagai jumlah molekul per (ekstensif) volume ${\ displaystyle C = N/V}  ÂÂ$ . Penampang tabrakan per volume atau kepadatan benturan tabrakan adalah ${\ displaystyle C \ sigma}  ÂÂ$ , dan itu terkait dengan jalur bebas rata-rata ${\ displaystyle l}  ÂÂ$ oleh

${\ displaystyle \ quad l = {\ frac {1} {{\ sqrt {2}} C \ sigma}}} Â Â$

Note that cross-sectional units per volume ${\ displaystyle C \ sigma}$ is the opposite of length. The average free path is the average distance traveled by a molecule, or a number of molecules per volume, before they make the first collision.

Biarkan ${\ displaystyle u_ {0}}  ÂÂ$ menjadi kecepatan maju gas pada permukaan horizontal imajiner di dalam lapisan gas. Rata-rata, sebuah molekul yang melintasi permukaan membuat tabrakan terakhir sebelum menyeberang pada jarak yang sama dengan dua pertiga dari jalur bebas rata-rata (yaitu ${\ displaystyle 2l/3}  ÂÂ$ ) jauh dari permukaan. Pada jarak ini di atas dan di bawah permukaan, momentum ke depan dari molekul adalah masing-masing

${\ displaystyle \ quad p_ {x} ^ {\ pm} = m \ kiri (u_ {0} \ pm {\ frac {2} {3} } l {du \ over dy} \ right)} Â Â$

where m is the mass of the molecule. Flux molecule ${\ displaystyle \ Phi}$ includes all molecules that arrive at one side of the surface element inside the gas layer. The incoming molecule comes from all directions on one side of the surface and at all speeds. These molecular fluxes (ie the flux of numbers) are related to the average molecular velocity ${\ displaystyle {\ bar {v}}}$ by