Merge pull request #91 from pndineshwaran/main

kinetic-theory-of-gases

content.en/kinetic-theory-of-gases/brownian-motion/index.md

@@ -19,78 +19,13 @@ According to kinetic theory, any particle suspended in a liquid or gas is contin
 
 2. Brownian motion decreases with bigger particle size, high viscosity, and density of the liquid (or) gas.
 
-**EXAMPLE 9.6**
-
-An oxygen molecule is traveling in air at 300 K and 1 atm, and the diameter of the oxygen molecule $$ (1.2 \times 10^{-10}) m. $$ Calculate the mean free path of the oxygen molecule.
-
-**Solution:**
-From equation (9.26), $$(lambda = \frac{2\pi d}{N})$$.
-
-We have to find the number density \(n\). By using the ideal gas law,
-
-$$
-n = \frac{N}{V} = \frac{P}{kT}
-$$
-
-$$
-n = \frac{101.31 \times 10^3}{1.381 \times 10^{-23} \times 300}
-$$
-
-$$
-n \approx 2.449 \times 10^{25} \text{ molecules/m}^3
-$$
-
-$$
-\lambda = \frac{2 \times \pi \times 1.2 \times 10^{-10}}{2.449 \times 10^{25}}
-$$
-
-$$
-\lambda \approx 0.63 \times 10^{-6} \, \text{m}
-$$
-
----
-
-| Figure 9.9 Particles in Brownian motion |
-
-- Kinetic theory explains the microscopic behavior of gases in terms of temperature, pressure.
-
-- The pressure exerted on the walls of a gas container is due to the collisions of gas molecules on the walls.
-
-$$
-P = \frac{1}{3} \cdot n \cdot m \cdot \overline{v^2}
-$$
-
-The pressure is related to the mass of the molecule and the mean square speed.
-
-- The temperature of a gas is a measure of the average kinetic energy of a molecule of the gas. The average kinetic energy is directly proportional to absolute temperature of gas and independent of the nature of molecules.
-
-- The pressure is also equal to $$( \frac{2}{3} )$$ of the internal kinetic energy per unit volume.
-
-- The rms speed of gas molecules: $$(v_{\text{rms}} = \sqrt{\frac{3kT}{m}})$$
-
-- The average speed of gas molecules: $$( \overline{v} = \frac{8}{\pi} \sqrt{\frac{kT}{m}} )$$
-
-- The most probable speed of gas molecules: $$(v_{\text{mp}} = \sqrt{\frac{2kT}{m}})$$
-
-- Among the speeds, $$(v_{\text{rms}}) $$is the largest and $$(v_{\text{mp}})$$is the least.
-
-- The number of gas molecules in the range $$(v)$$ to $$(v + dv)$$ follows the Boltzmann distribution:
-
-$$
-N(v) \, dv = 4 \pi \left( \frac{m}{2 \pi kT} \right)^{3/2} v^2 e^{-\frac{mv^2}{2kT}} \, dv
-$$
-
-- The minimum number of independent coordinates needed to specify the state and configuration of a thermodynamic system is known as the degrees of freedom of the system. If a sample of gas has $$(N)$$ molecules, then the total number of degrees of freedom $$(f)$$ is given by $$(f = 3N)$$. If there are $$(q)$$ number of constraints, then $$(f = 3N-q)$$.
-
-- For a monoatomic molecule, $$(f = 3)$$. For a diatomic molecule (at normal temperatures), $$(f = 5)$$.
-
-
-
-**S U M M A R Y**
-
-The experimental verification of Brownian motion was conducted by Jean Perrin in the year 1908. This motion provided direct evidence of the existence of atoms and molecules.
-
+
 **Note:**
+
+The experimental verification of Brownian motion was conducted by Jean Perrin in the year 1908. This motion provided direct evidence of the existence of atoms and molecules.
+
+**S U M M A R Y**
+
 The microscopic origin of macroscopic parameters like pressure and temperature is explained by kinetic theory.
 
 - The pressure in a gas container is due to the momentum imparted by gas molecules on the walls, directly proportional to the number density, average translational kinetic energy per molecule, and independent of the nature of molecules.
@@ -188,6 +123,71 @@ The microscopic origin of macroscopic parameters like pressure and temperature i
   - Observe the movement of particles and explore variants like energy, size ratio.
   - Understand Brownian motion's microscopic behavior.
 
-\* Pictures are indicative only.
-\* If the browser requires, allow Flash Player or Java Script.
-
+- Kinetic theory explains the microscopic origin of macroscopic parameters like
+temperature, pressure.
+
+- The pressure exerted on the walls of gas container is due to the momentum imparted
+by the gas molecules on the walls.
+
+- The pressure \\(P = \frac{1}{3}nm\overline{v^2}\\). The pressure is directly proportional to the number density,
+mass of molecule and mean square speed.
+
+- The temperature of a gas is a measure of the average translational kinetic energy per
+molecule of the gas. The average kinetic energy per molecule is directly proportional
+to absolute temperature of gas and independent of nature of molecules.
+
+- The pressure is also equal to 2/3 of internal energy per unit volume.
+
+- The rms speed of gas molecules = \\(v_{rms} = \sqrt{\frac{3kT}{m}} = 1.73\sqrt{\frac{kT}{m}}\\)
+
+- The average speed of gas molecules \\(\overline{v} = \sqrt{\frac{8kT}{\pi m}} = 1.60\sqrt{\frac{kT}{m}}\\)
+
+- The most probable speed of gas molecules \\(v_{mp} = \sqrt{\frac{2kT}{m}} = 1.41\sqrt{\frac{kT}{m}}\\)
+
+- Among the speeds \\(v_{rms}\\) is the largest and \\(v_{mp}\\) is the least
+
+$$v_{rms} > \overline{v} > v_{mp}$$
+
+- The number of gas molecules in the range of speed v to v+dv is given by MaxwellBoltzmann distribution 
+
+$$N d\nu = 4\pi N \left(\frac{m}{2\pi kT}\right)^{\frac{3}{2}} v^2 e^{-\frac{mv^2}{2kT}} dv$$
+
+- The minimum number of independent coordinates needed to specify the position
+and configuration of a thermodynamical system in space is called the degrees of
+freedom of the system. If a sample of gas has N molecules, then the total degrees of
+freedom f = 3N. If there are q number of constraints then total degrees of freedom
+f = 3N-q.
+
+- For a monoatomic molecule, f = 3
+
+    For a diatomic molecule (at normal temperature), f = 5
+
+    For a diatomic molecule (at high temperature), f = 7
+
+    For a triatomic molecule (linear type), f = 7
+
+    For a triatomic molecule (non-linear type), f = 6
+
+- The average kinetic energy of sample of gas is equally distributed to all the degrees of
+freedom. It is called law of equipartition of energy. Each degree of freedom will get 1/2 kT energy.
+
+- The ratio of molar specific heat at constant pressure and constant volume of a gas
+
+$$\gamma = \left[\frac{C_p}{C_v}\right]$$
+
+For Monoatomic molecule: 1.67
+
+Diatomic molecule (Normal temperature) : 1.40
+
+Diatomic molecule (High temperature): 1.28
+
+Triatomic molecule (Linear type): 1.28.
+
+Triatomic molecule (Non-linear type): 1.33
+
+- The mean free path \\(\lambda = \frac{kT}{\sqrt{2}\pi d^{2}P}\\).The mean free path is directly proportional to
+temperature and inversely proportional to size of the molecule and pressure of the
+molecule
+
+- The Brownian motion explained by Albert Einstein is based on kinetic theory. It
+proves the reality of atoms and molecules.

content.en/kinetic-theory-of-gases/degrees-of-freedom/index.md

@@ -20,11 +20,20 @@ _The minimum number of independent coordinates needed to specify the position an
 
 3. A particle moving in space has three degrees of freedom.
 
-If there are $$(N)$$ number of gas molecules in the container, then the total number of degrees of freedom is $$(f = 3N)$$. However, if the system has $$(q)$$ number of constraints (restrictions in motion), then the degrees of freedom decrease, and it is equal to $$(f = 3N-q)$$, where $$(N)$$ is the number of particles.
+Suppose if we have N number of gas molecules in the container, then the total number of degrees of freedom is f = 3N.
+
+But, if the system has q number of constraints
+(restrictions in motion) then the degrees of
+freedom decreases and it is equal to f = 3N-q
+where N is the number of particles.
 
 ## Monoatomic molecule
 
-A monoatomic molecule, by virtue of its nature, has only three translational degrees of freedom. Therefore, $$(f = 3)$$.
+A monoatomic molecule by virtue of its
+nature has only three translational degrees
+of freedom.
+
+Therefore f = 3
 
 **Examples:** Helium, Neon, Argon
 
@@ -34,9 +43,32 @@ There are two cases.
 
 **1. At Normal Temperature**
 
-A diatomic gas molecule consists of two atoms bound by a force of attraction. Physically, the molecule can be regarded as a system of two point masses fixed at the ends of a massless elastic spring. The center of mass lies in the center of the diatomic molecule. So, the motion of the center of mass requires three translational degrees of freedom (Figure 9.5a). In addition, the diatomic molecule can rotate about three mutually perpendicular axes (Figure 9.5b). But the moment of inertia about its own axis of rotation is negligible (about the y-axis in Figure 9.5). Therefore, it has only two rotational degrees of freedom (one rotation is about the Z-axis, and another rotation is about the X-axis). Therefore, there are five degrees of freedom.
-
-$$[ f = 5 ]$$
+A molecule of a diatomic gas consists of
+two atoms bound to each other by a force
+of attraction. Physically the molecule can
+be regarded as a system of two point masses
+fixed at the ends of a massless elastic spring.
+
+The center of mass lies in the center of the
+diatomic molecule. So, the motion of the
+center of mass requires three translational
+degrees of freedom (figure 9.5 a). In
+addition, the diatomic molecule can rotate 
+
+![Alt text](figure9.5.png) **Figure 9.5** Degree of freedom of
+diatomic molecule
+
+
+about three mutually perpendicular axes
+(figure 9.5 b). But the moment of inertia
+about its own axis of rotation is negligible
+(about y axis in the figure 9.5). Therefore, it
+has only two rotational degrees of freedom
+(one rotation is about Z axis and another
+rotation is about X axis). Therefore totally
+there are five degrees of freedom.
+
+$$(f=5)$$
 
 **2. At High Temperature**
 
@@ -52,19 +84,38 @@ There are two cases.
 
 **Linear Triatomic Molecule**
 
-In this type, two atoms lie on either side of the central atom. A linear triatomic molecule has three translational degrees of freedom. It has two rotational degrees of freedom because it is similar to a diatomic molecule, except there is an additional atom at the center. At normal temperature, a linear triatomic molecule will have five degrees of freedom. At high temperature, it has two additional vibrational degrees of freedom. So, a linear triatomic molecule has seven degrees of freedom.
+In this type, two atoms lie on either side of
+the central atom as shown in the Figure 9.6
+
+![Alt text](figure9.6.png)**Figure 9.6** A linear triatomic molecule.
+
+Linear triatomic molecule has three
+translational degrees of freedom. It has two
+rotational degrees of freedom because it is
+similar to diatomic molecule except there is
+an additional atom at the center. At normal
+temperature, linear triatomic molecule
+will have five degrees of freedom. At high
+temperature it has two additional vibrational
+degrees of freedom. So a linear triatomic
+molecule has seven degrees of freedom.
+
 
 **Example:** Carbon dioxide.
 
-![images](figure9.5.png)
-
 **Non-linear Triatomic Molecule**
 
-In this case, the three atoms lie at the vertices of a triangle. A non-linear triatomic molecule has three translational degrees of freedom and three rotational degrees of freedom about three mutually orthogonal axes. The total degrees of freedom, $$( f = 6 )$$.
+In this case, the three atoms lie at the vertices
+of a triangle as shown in the Figure 9.7
+
+![Alt text](figure9.7.png)**Figure 9.7** A non-linear triatomic
+molecule
+
+It has three translational degrees of freedom
+and three rotational degrees of freedom
+about three mutually orthogonal axes. The
+total degrees of freedom, f = 6
 
 **Example:** Water, Sulphur dioxide.
 
-![images](figure9.6.png)
-
-![images](figure9.7.png)
 

content.en/kinetic-theory-of-gases/kinetic-theory/index.md

@@ -1,29 +1,42 @@
 ---
 title: 'kinetic theory'
 weight: 1
+extensions:
+    - katex
 ---
-
+
+{{< katex diaplay >}}  {{< /katex >}}
+
+**9.1**
+**Learning Objectives**
+**In this unit, the student is exposed to**
+
+• necessity of kinetic theory of gases
+
+• the microscopic origin of pressure and temper
+
+• correlate the internal energy of the gas and translational kinetic energy of gas molecules
+
+• meaning of degrees of freedom
+
+• calculate the total degrees of freedom for mono atomic, diatomic and triatomic molecules
+
+• law of equipartition of energy
+
+• calculation of the ratio of CP and CV
+
+• mean free path and its dependence with pressure, temperature and number density
+
+• Brownian motion and its microscopic origin
+
+# KINETIC THEORY
+
+## Introduction
+
 Thermodynamics is basically a macroscopic science. We discussed macroscopic parameters like pressure, temperature and volume of thermodynamical systems in unit 8. In this unit we discuss the microscopic origin of pressure and temperature by considering a thermodynamic system as collection of particles or molecules. Kinetic theory relates pressure and temperature to molecular motion of sample of a gas and it is a bridge between Newtonian mechanics and thermodynamics. The present chapter introduces the kinetic nature of gas molecules.
-
-
-
-
-
-_“With thermodynamics one can calculate alm calculate fe_  
-
-ature slational kinetic energy of gas molecules
-
-atomic, diatomic and triatomic molecules
-
-re, temperature and number density
-
-**C THEORY OF GASES**
-
-_ost everything crudely; with kinetic theory, one can wer things, but more accurately.”_ \- Eugene Wigner
-
+
 ### Postulates of kinetic theory of gases
-
-
+
 Kinetic theory is based on certain assumptions which makes the mathematical treatment simple. None of these assumptions are strictly true yet the model based on these assumptions can be applied to all gases.
 
 1\. All the molecules of a gas are identical, elastic spheres.
@@ -36,13 +49,6 @@ Kinetic theory is based on certain assumptions which makes the mathematical trea
 
 5\. The molecules collide with one another and also with the walls of the container.
 
-
-
-
-
-
-**9.2**
-
 6\. These collisions are perfectly elastic so that there is no loss of kinetic energy during collisions.
 
 7\. Between two successive collisions, a molecule moves with uniform velocity.