Thermal Conductivity And Heat Flux
Following a similar logic as above, one can derive the kinetic model for thermal conductivity of a dilute gas:
Consider two parallel plates separated by a gas layer. Both plates have uniform temperatures, and are so massive compared to the gas layer that they can be treated as thermal reservoirs. The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy Îµ which increases uniformly with distance y above the lower plate. The non-equilibrium energy flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.
Let 0 } be the molecular kinetic energy of the gas at an imaginary horizontal surface inside the gas layer. The number of molecules arriving at an area d
Safety Procedures And Precautions
Students should wear goggles and aprons or lab coats during the activity and exercise due caution around Bunsen burners or hot plates. Inform students that cooking oil boils at a higher temperature than water , and have them cover their beakers with aluminum foil to contain the popping corn and boiling oil.
Note: Remind students not to eat any of the popcorn produced in the lab.
The Kinetic Molecular Theory Of Gases: Why Gases Do What They Do
As mentioned before, gases are harder to visualize than other phases of matter. This is true not only because it’s difficult to see and study them, but because all the molecules in a gas behave independently of one another. As a result, studying the behavior of a gas containing one mole of molecules that are flying all over the place is a lot harder than studying the behavior of a crystal containing a mole of stationary molecules.
Because it’s tough to study all of the particles in a gas, scientists have come up with a variety of theories to simplify gases’ behavior so they can be more easily understood. Probably the most important of these theories is referred to as the Kinetic Molecular Theory .
All theoretical models only approximate the behavior of the thing being modeled. The approximations that define each model are designed to make the real phenomenon easier to understand and predict. However, no model is perfect, which explains why weather forecasting models usually get the five-day forecast wrong.
The KMT makes the following assumptions about the behavior of the particles in a gas. Again, these assumptions are not always true, but allow us to understand gases more easily.
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Molecular Speeds And Kinetic Energy
The previous discussion showed that the KMT qualitatively explains the behaviors described by the various gas laws. The postulates of this theory may be applied in a more quantitative fashion to derive these individual laws. To do this, we must first look at speeds and kinetic energies of gas molecules, and the temperature of a gas sample.
In a gas sample, individual molecules have widely varying speeds; however, because of the vast number of molecules and collisions involved, the molecular speed distribution and average speed are constant. This molecular speed distribution is known as a Maxwell-Boltzmann distribution, and it depicts the relative numbers of molecules in a bulk sample of gas that possesses a given speed .
The kinetic energy of a particle of mass and speed is given by:
The Kinetic Energies Of Gas Molecules Are Directly Proportional To Their Temperatures In Kelvin
The definition of kinetic energy might sound complicated, but what it really means is that when you heat a gas, the molecules move more quickly. Temperature, as it turns out, is a measurement of how much motion the particles in a material have, so it makes sense that increasing the temperature will increase the motion.
Kinetic energy refers to energy caused by the motion of an object. The faster an object moves, the more kinetic energy it has.
The Kelvin scale of temperature is the same as the Celsius scale we’ve used so far, but it’s higher by 273.15 . We can easily convert degrees Celsius to Kelvin using the following conversion.
- K = C + 273
For example, if the temperature outside is 20 C, we can convert this to 293 K. Note that the temperature is properly written simply as “Kelvin” and not “degrees Kelvin.”
The reason we use Kelvin when working with gases instead of degrees Celsius is that gases often exist at temperatures below 0 C. As a result, if we said that the kinetic energy of a gas was proportional to the temperature in degrees Celsius, the kinetic energy of any gas cooled below the freezing point of water would be negative.
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What Is The Definition Of Kinetic Molecular Theory
The kinetic molecular theory is a collection of several rules that describe the behavior of gases. The nature of gas molecules was examined by scientists, such as Robert Boyle and Jacques Charles, who outlined their observations in several laws that eventually became the Kinetic Molecular Theory. Volume, temperature and pressure are all taken into account when observing and understanding the behavior of gases.
The kinetic molecular theory is composed of five postulates, one of which states that gas particles are in constant random motion and, according to Newtons laws, remain in random motion unless acted upon by an outside force. Another rule is that the volume gas molecules occupy is negligible when compared to the volume of the container in which they exist. A third rule posits that when gas molecules collide, there is no loss or gain of energy. Another rule states that there is no notable force of attraction between gas particles. The last rule explains that kinetic energy can be determined by using the equation 3kT/2, in which k is a constant and T is the temperature.
Other properties of gases can be inferred from these rules. For example, if the temperature of a gas remains constant, then the kinetic energy of a sample of molecules does not change, no matter how much time passes. Furthermore, the shape and mass of gas particles doesn’t determine their kinetic energy; only temperature does this.
Chemistry End Of Chapter Exercises
Using the postulates of the kinetic molecular theory, explain why a gas uniformly fills a container of any shape.
Can the speed of a given molecule in a gas double at constant temperature? Explain your answer.
Yes. At any given instant, there are a range of values of molecular speeds in a sample of gas. Any single molecule can speed up or slow down as it collides with other molecules. The average velocity of all the molecules is constant at constant temperature.
Describe what happens to the average kinetic energy of ideal gas molecules when the conditions are changed as follows:
The pressure of the gas is increased by reducing the volume at constant temperature.
The pressure of the gas is increased by increasing the temperature at constant volume.
The average velocity of the molecules is increased by a factor of 2.
The distribution of molecular velocities in a sample of helium is shown in . If the sample is cooled, will the distribution of velocities look more like that of H2 or of H2O? Explain your answer.
H2O. Cooling slows the velocities of the He atoms, causing them to behave as though they were heavier.
What is the ratio of the average kinetic energy of a SO2 molecule to that of an O2 molecule in a mixture of two gases? What is the ratio of the root mean square speeds, urms, of the two gases?
A 1-L sample of CO initially at STP is heated to 546 K, and its volume is increased to 2 L.
What is the effect on the average kinetic energy of the molecules?
Diffusion Coefficient And Diffusion Flux
Following a similar logic as above, one can derive the kinetic model for mass diffusivity of a dilute gas:
Consider a steady diffusion between two regions of the same gas with perfectly flat and parallel boundaries separated by a layer of the same gas. Both regions have uniform number densities, but the upper region has a higher number density than the lower region. In the steady state, the number density at any point is constant . However, the number density n in the layer increases uniformly with distance y above the lower plate. The non-equilibrium molecular flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.
Let 0 } be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area d
Kinetic Theory Of Gases
The kinetic theory of gases is a simple, historically significant classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic particles , all of which are in constant, rapid, randommotion. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas, and considers no other interactions between the particles.
The kinetic theory of gases explains the macroscopic properties of gases, such as volume, pressure, and temperature, as well as transport properties such as viscosity, thermal conductivity and mass diffusivity. The model also accounts for related phenomena, such as Brownian motion.
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The Fundamentals Of Kinetic Molecular Theory
The molecules of a gas are in a state of perpetual motion in which the velocity of each molecule is completely random and independent of that of the other molecules. This fundamental assumption of the kinetic-molecular model helps us understand a wide range of commonly-observed phenomena. According to this model, most of the volume occupied by a gas is empty space; this is the main feature that distinguishes gases from condensed states of matter in which neighboring molecules are constantly in contact. Gas molecules are in rapid and continuous motion; at ordinary temperatures and pressures their velocities are of the order of 0.1-1 km/sec and each molecule experiences approximately 1010collisions with other molecules every second.
The five basic tenets of the kinetic-molecular theory are as follows:
The Kinetic-Molecular Theory is “the theory of moving molecules.” -Rudolf Clausius, 1857
Clausius Incorporates Energy Into Kinetic Theory
Unlike Lavoisier and Dalton, the 19th century German physicist Rudolf Clausius rejected caloric theory. Instead of regarding heat as a substance that surrounds molecules, Clausius proposed that heat is a form of energy that affects the temperature of matter by changing the motion of molecules in matter. This kinetic theory of heat enabled Clausius to study and predict the flow of heata field we now call thermodynamics .
In his 1857 paper, On the nature of the motion which we call heat, Clausius speculated on how heat energy, temperature, and molecule motion could explain gas behavior. In doing this, he proposed several ideas about the molecules of gases. These ideas have come to be accepted for ideal gasestheoretical gases that perfectly obey the ideal gas equation . Clausius proposed that the space taken up by ideal gas molecules should be regarded as infinitesimal when compared to the space occupied by the whole gas in other words, a gas consists mostly of empty space. Second, he suggested that the intermolecular forces between molecules should be treated as infinitesimal.
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Applying Kinetic Theory To Gas Laws
Charles Law states that at constant pressure, the volume of a gas increases or decreases by the same factor as its temperature. This can be written as:
According to Kinetic Molecular Theory, an increase in temperature will increase the average kinetic energy of the molecules. As the particles move faster, they will likely hit the edge of the container more often. If the reaction is kept at constant pressure, they must stay farther apart, and an increase in volume will compensate for the increase in particle collision with the surface of the container.
Boyles Law states that at constant temperature, the absolute pressure and volume of a given mass of confined gas are inversely proportional. This relationship is shown by the following equation:
At a given temperature, the pressure of a container is determined by the number of times gas molecules strike the container walls. If the gas is compressed to a smaller volume, then the same number of molecules will strike against a smaller surface area; the number of collisions against the container will increase, and, by extension, the pressure will increase as well. Increasing the kinetic energy of the particles will increase the pressure of the gas.
The Kinetic Molecular Theory of Gas YouTube: Reviews kinetic energy and phases of matter, and explains the kinetic-molecular theory of gases.
The Kinetic Molecular Theory of Gas YouTube: Uses the kinetic theory of gases to explain properties of gases
The Particles In A Gas Are In Constant Random Motion
The KMT assumes that the particles in a gas, like very small children, constantly move from place to place in an unpredictable fashion. This assumption is correct. Furthermore, the KMT assumes that these particles travel in straight lines until they bash into something, at which point they turn around and go somewhere else. As is also the case with very small children, this assumption is true.
The assumption that gas molecules are in constant random motion explains why they have no fixed shape or volume.
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Gas Molecules Undergo Perfectly Elastic Collisions
Elastic collisions are collisions in which kinetic energy is transferred from one thing to another without a loss. If you’ve ever played pool, you know that the balls slow down just a little bit when they hit the bumpers on the sides of the table. If these collisions were elastic, the balls would bounce off the bumpers moving exactly as quickly as they hit them, never stopping until they either hit a pocket or you got tired of watching them bounce around the table.
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Pressure And Kinetic Energy
In the kinetic theory of gases, the pressure is assumed to be equal to the force exerted by the atoms hitting and rebounding from the gas container’s surface. Consider a gas of a large number N of molecules, each of mass m, enclosed in a cube of volume V = L3. When a gas molecule collides with the wall of the container perpendicular to the x axis and bounces off in the opposite direction with the same speed , the change in momentum is given by:
Thus the kinetic energy per Kelvin of one mole of is 3 = 3R/2. Thus the kinetic energy per Kelvin can be calculated easily:
- per mole: 12.47 J / K
- per molecule: 20.7 yJ / K = 129 Î¼eV / K
At standard temperature , the kinetic energy can also be obtained:
- per mole: 3406 J
- per molecule: 5.65 zJ = 35.2 meV.
Although monatomic gases have 3 degrees of freedom per atom, diatomic gases should have 6 degrees of freedom per molecule . However, the lighter diatomic gases may act as if they have only 5 due to the strongly quantum-mechanical nature of their vibrations and the large gaps between successive vibrational energy levels. Quantum statistical mechanics is needed to accurately compute these contributions.
Kinetic Interpretation Of Absolute Temperature
According to the kinetic molecular theory, the average kinetic energy of an ideal gas is directly proportional to the absolute temperature. Kinetic energy is the energy a body has by virtue of its motion:
- \ is the kinetic energy of a molecule,
- \ is the mass of the molecule, and
- \ is the magnitude of the velocity of a molecule.
As the temperature of a gas rises, the average velocity of the molecules will increase; a doubling of the temperature will increase this velocity by a factor of four. Collisions with the walls of the container will transfer more momentum, and thus more kinetic energy, to the walls.
If the walls are cooler than the gas, they will get warmer, returning less kinetic energy to the gas, and causing it to cool until thermal equilibrium is reached. Because temperature depends on the average kinetic energy, the concept of temperature only applies to a statistically meaningful sample of molecules. We will have more to say about molecular velocities and kinetic energies farther on.