Kinetic+Molecular+Theory

=KINETIC MOLECULAR THEORY= By: Alex Ghaben and Matt McSherry = = == [|http://www.lmm.jussieu.fr/~ameur/Translational_motion.gif]

=__Introduction__= Kinetic Molecular Theory or Collision Theory seeks to explain the macroscopic properties of gases such a pressure, temperature, and volume by examining their molecular composition and motion. The gas described under the Kinetic Molecular Theory is an "Ideal Gas". In essence such a gas, would follow the following five Tenants:


 * Tenants**
 * 1) Gases consist of large numbers of molecules that are in continuous, random motion.
 * 2) The volume of all the molecules of the gas in negligible compared to the total volume in which the gas is contained.
 * 3) Attractive and repulsive forces between gas molecules are negligible.
 * 4) Energy can be transferred between molecules during collisions, but the //average// kinetic energy of the molecule does not change with time, as long as the temperature of the gas remains constant; the collisions are perfectly elastic.
 * 5) The average kinetic energy of the molecules is proportional to the absolute temperature. At any given temperature the molecules of all gasses have the same average kinetic energy

=__Derivations and Explanations__= Tenants 2-3 provide for perfectly elastic collisions of molecules, a concept which is used to explain pressure and temperature at a molecular level. An elastic collision is a collision in which no momentum is lost, although the individual speeds of the particles may collide. As the particles are colliding in constant random fashion, some particles strike the walls of their container. The magnitude of the force and the frequency of the collisions with the wall of a container determine the pressure of a gas. The Kinetic-Molecular Theory explains temperature in terms of heat released by the kinetic energy of the molecules. In fact, temperature is a measure of the **average** kinetic energy of the molecules in a gas. According to the Kinetic Molecular Theory, temperature and kinetic energy are directly related. This means that when you double the average kinetic energy of a gas, its pressure will double. Again, it is emphasized that this is the average kinetic energy because if a gas consisted of particles possessing uniform kinetic energy, the first tenant of the theory, providing for constant random motion would be violated.

The following is a graphic that provides a visual representation of pressure according to the Kinetic Molecular Theory http://www.emsb.qc.ca/laurenhill/science/pressure1.jpg

The Kinetic Molecular Theory can also be used to explain the Gas Laws. A few are paraphrased here, but for a complete list, [|visit this site].

If a gas held at constant temperature and pressure is compressed, the magnitude of the collisions do not increase, but their frequency increases at a rate inversely proportional to the volume. http://www.emsb.qc.ca/laurenhill/science/pressure1.jpg
 * Boyle's Law:**

If a gas is heated, its particles will move faster because of their increase in kinetic energy, and thereby exert a greater force on their container. If their container is pliable, it will expand until the internal pressure of the container is equal to the external force of air pressure acting on the container.
 * Charles' Law:**

As the number of gas particles increase, the frequency of collisions will increase which results in an increase in pressure. Just as in Charles' Law, if the walls of the container of the gas are pliable, they will expand to reach equilibrium with respect to pressure.
 * Avogadro's Law:**

=__Kinetic Molecular Theory and Root Mean Square Theorem__= The Root Mean Square Theorem is used to find the speed of a molecule possessing average kinetic energy (expressed as //u//). Just as the arithmetic mean of a set of n numbers is : and is based on a polynomial of base 1 ([|http://en.wikipedia.org/wiki/Arithmetic_mean)]

The Root Mean Square is a different statistical average of the magnitude of a constantly varying function. It is formula is the square root of an arithmetic of a set of values. The formula for finding the root mean square of a set of numbers is: Note, for class purposes, substitute Xrms with //u// http://en.wikipedia.org/wiki/Root_mean_square It is important to use the Root Mean Square Theorem when calculating the average speed of a particle in a gaseous medium because it does not have a consistent speed at all times or under all conditions. The Root Mean Square Theorem takes this into account and gives a slightly more accurate response given these conditions. The Root Mean Square Theorem would be used when the speeds of many particles of a gas were given in a problem statement, and the average particle velocity was required as an answer. __//**IN SIMPLE TERMS**//__ to find the root mean square value for a set of terms, add the squares of all the terms, divide by the number terms, and then take the square root of that value.


 * Example:**
 * Find the root mean square of 4, 6, 8, and 10.
 * Square each term and find the sum (16+36+64+100 = 216)
 * Divide by the number of terms, in this case, 4 (216/4 = 54)
 * Find the square root of the resulting number (sqrt(54) = 7.35)
 * The resulting number is the root mean square value (7.35)

The average speed of a particle in a gas, //u,// can also be used to calculate the average kinetic energy of a gas (E). E and //u// are related through the following formula where E and //u// are as given, and m is the mass of gas.

[|http://www.chem.ufl.edu/~itl/2045/lectures/lec_d.html] It can be seen from this equation that the average kinetic energy of the gas is proportional to the square of the average speed of an individual particle of the gas.

Sources __Bursten, Brown, & LeMay.__ Chemistry: The Central Science, 9th Edition.__ (The book used in Mr. Williams' AP Chem Class in the School Year 2007-2008) http://itl.chem.ufl.edu/2045_s00/lectures/lec_d.html http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch4/kinetic4.html http://en.wikipedia.org/wiki/Root_mean_square =KINETIC MOLECULAR THEORY= By: Alex Ghaben and Matt McSherry = = == [|http://www.lmm.jussieu.fr/~ameur/Translational_motion.gif]

=__Introduction__= Kinetic Molecular Theory or Collision Theory seeks to explain the macroscopic properties of gases such a pressure, temperature, and volume by examining their molecular composition and motion. The gas described under the Kinetic Molecular Theory is an "Ideal Gas". In essence such a gas, would follow the following five Tenants:


 * Tenants**
 * 1) Gases consist of large numbers of molecules that are in continuous, random motion.
 * 2) The volume of all the molecules of the gas in negligible compared to the total volume in which the gas is contained.
 * 3) Attractive and repulsive forces between gas molecules are negligible.
 * 4) Energy can be transferred between molecules during collisions, but the //average// kinetic energy of the molecule does not change with time, as long as the temperature of the gas remains constant; the collisions are perfectly elastic.
 * 5) The average kinetic energy of the molecules is proportional to the absolute temperature. At any given temperature the molecules of all gasses have the same average kinetic energy

=__Derivations and Explanations__= Tenants 2-3 provide for perfectly elastic collisions of molecules, a concept which is used to explain pressure and temperature at a molecular level. An elastic collision is a collision in which no momentum is lost, although the individual speeds of the particles may collide. As the particles are colliding in constant random fashion, some particles strike the walls of their container. The magnitude of the force and the frequency of the collisions with the wall of a container determine the pressure of a gas. The Kinetic-Molecular Theory explains temperature in terms of heat released by the kinetic energy of the molecules. In fact, temperature is a measure of the **average** kinetic energy of the molecules in a gas. According to the Kinetic Molecular Theory, temperature and kinetic energy are directly related. This means that when you double the average kinetic energy of a gas, its pressure will double. Again, it is emphasized that this is the average kinetic energy because if a gas consisted of particles possessing uniform kinetic energy, the first tenant of the theory, providing for constant random motion would be violated.

The following is a graphic that provides a visual representation of pressure according to the Kinetic Molecular Theory http://www.emsb.qc.ca/laurenhill/science/pressure1.jpg

The Kinetic Molecular Theory can also be used to explain the Gas Laws. A few are paraphrased here, but for a complete list, [|visit this site].

If a gas held at constant temperature and pressure is compressed, the magnitude of the collisions do not increase, but their frequency increases at a rate inversely proportional to the volume. http://www.emsb.qc.ca/laurenhill/science/pressure1.jpg
 * Boyle's Law:**

If a gas is heated, its particles will move faster because of their increase in kinetic energy, and thereby exert a greater force on their container. If their container is pliable, it will expand until the internal pressure of the container is equal to the external force of air pressure acting on the container.
 * Charles' Law:**

As the number of gas particles increase, the frequency of collisions will increase which results in an increase in pressure. Just as in Charles' Law, if the walls of the container of the gas are pliable, they will expand to reach equilibrium with respect to pressure.
 * Avogadro's Law:**

=__Kinetic Molecular Theory and Root Mean Square Theorem__= The Root Mean Square Theorem is used to find the speed of a molecule possessing average kinetic energy (expressed as //u//). Just as the arithmetic mean of a set of n numbers is : and is based on a polynomial of base 1 ([|http://en.wikipedia.org/wiki/Arithmetic_mean)]

The Root Mean Square is a different statistical average of the magnitude of a constantly varying function. It is formula is the square root of an arithmetic of a set of values. The formula for finding the root mean square of a set of numbers is: Note, for class purposes, substitute Xrms with //u// http://en.wikipedia.org/wiki/Root_mean_square It is important to use the Root Mean Square Theorem when calculating the average speed of a particle in a gaseous medium because it does not have a consistent speed at all times or under all conditions. The Root Mean Square Theorem takes this into account and gives a slightly more accurate response given these conditions. The Root Mean Square Theorem would be used when the speeds of many particles of a gas were given in a problem statement, and the average particle velocity was required as an answer. __//**IN SIMPLE TERMS**//__ to find the root mean square value for a set of terms, add the squares of all the terms, divide by the number terms, and then take the square root of that value.


 * Example:**
 * Find the root mean square of 4, 6, 8, and 10.
 * Square each term and find the sum (16+36+64+100 = 216)
 * Divide by the number of terms, in this case, 4 (216/4 = 54)
 * Find the square root of the resulting number (sqrt(54) = 7.35)
 * The resulting number is the root mean square value (7.35)

The average speed of a particle in a gas, //u,// can also be used to calculate the average kinetic energy of a gas (E). E and //u// are related through the following formula where E and //u// are as given, and m is the mass of gas.

[|http://www.chem.ufl.edu/~itl/2045/lectures/lec_d.html] It can be seen from this equation that the average kinetic energy of the gas is proportional to the square of the average speed of an individual particle of the gas.

Sources __Bursten, Brown, & LeMay.__ Chemistry: The Central Science, 9th Edition.__ (The book used in Mr. Williams' AP Chem Class in the School Year 2007-2008) http://itl.chem.ufl.edu/2045_s00/lectures/lec_d.html http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch4/kinetic4.html http://en.wikipedia.org/wiki/Root_mean_square