General Chemistry--Unit 2
�Before You Begin:
To master this material, you must know the gas laws.
A law is a summary of observations, and a theory is an explanation of those observations. The individual gas laws give us a set of mathematical tools to help predict the behavior of gases under specific conditions of pressure, temperature, volume and number of moles of gas. They do not, however, explain why gases behave the way they do. Kinetic molecular theory is an attempt to explain some of the bulk properties of matter by describing how particles interact with one another. Kinetic molecular theory can help us understand how and why the gas laws work and to predict when the gas laws won’t work.
Daniel Bernoulli started kinetic molecular theory in 1738 when he proposed a thought model consistent with Boyle’s Law in an attempt to explain how gases exert pressure. Clausius refined the theory in the mid-1800s.
The Assumptions of Kinetic Molecular Theory:
In order to explain how gases behave we can make the following assumptions:
The KMT Assumptions and the Gas Law Variables:
We can connect these assumptions with the four variables from the individual gas laws.
The root mean square speed, u, is the square root of the average of the squares of the speeds, which is pretty close to the average for this type of distribution. The higher the temperature of the gas is, the higher the root mean square of the speed. To put it more simply, the hotter the temperature is, the faster the particles are moving. The root mean square of the temperature is related to the kinetic energy of the particles:
where ε is the kinetic energy, m is the mass of the particle and u is the root mean square speed.
The KMT and the Individual Gas Laws:
Boyle’s Law states that, at constant number of moles and temperature, pressure and volume are inversely proportional. “Constant number of moles” implies that the number of gas particles remains the same. “Constant temperature” implies that the average speed of the particles remains the same. If the pressure and volume are inversely proportional, an increase in volume will lead to a decrease in pressure. If volume increases, the distance each particle travels before it hits the wall increases. If the same number of particles is traveling the same speed and they have to travel farther to hit the wall of the container, they must not hit the wall as often. The frequency with which particles collide with the wall is the same as the gas pressure; if the collision rate drops, so does the pressure. Therefore, volume and pressure are inversely proportional.
Charles’ Law states that, at constant number of moles and pressure, the volume and the temperature are directly proportional. “Constant number of moles” implies that means that the number of gas particles remains the same. “Constant pressure” implies that the rate at which the particles collide with the wall of the container remains the same. If volume and temperature are directly proportional, an increase in temperature will lead to an increase in volume. If temperature increases, the average speed of the gas particles increases. If the same number of particles is colliding at the same rate even though they are moving faster, they must be traveling farther. The distance between the particles and the wall is the same as the volume; if the distance increases, the volume increases. Therefore, volume and temperature are directly proportional.
The Third Law:
At constant number of moles and volume, the temperature and pressure are directly proportional. “Constant number of moles” implies that the number of gas particles remains the same. “Constant volume” implies that the distance between particles and the wall remains the same. If the temperature and pressure are directly proportional, an increase in temperature will lead to an increase in pressure. If the temperature increases, the average speed of the gas particles increases. If the same number of particles is traveling the same distance and moving faster, they must strike the wall more often. An increase in the collision rate is an increase in pressure. Therefore, temperature and pressure are directly proportional.
Avogadro’s Law states that at constant temperature and pressure, the number of moles of gas and the volume are directly proportional. “Constant temperature” means that the average speed stays the same. “Constant pressure” means that the rate at which the particles strike the wall stays the same. If the number of moles and the volume are directly proportional, an increase in the number of moles will lead to an increase in the volume. If the particles travel the same speed and hit the wall at the same rate, yet there are more particles, they must spread out more and traveling farther to reach the wall. A greater distance is the same as an increase in volume. Therefore number of particles and volume must be directly proportional.
The Combined Gas Law and Ideal Gas Law:
The combined gas law is a statement of the relationships among pressure, volume and temperature at constant amount of gas. If the kinetic molecular theory describes each pair of variables, it must also describe the inter-relationship among pressure, volume and temperature for a set amount of gas undergoing a change in conditions. The ideal gas law is a statement of the relationships among pressure, volume, amount of gas, and temperature for a gas at a set condition. These variables are inter-related as collisions, speed and distance. If a gas obeys the assumptions of kinetic molecular theory, it must be “ideal.”
Graham’s Law of effusion states that the rate at which a gas effuses is inversely proportional to the square root of its molar mass. In other words, the smaller the gas particle the faster it effuses. This works because the molar mass of a gas increases as the effective diameter of the gas particle increases. The average speed of an ideal gas depends on temperature rather than particle size, but this is relative to empty space between particles. As a gas effuses through a small hole, the size of the particle is no longer negligible. A gas particle is more likely to pass through any microscopic hole in the wall of the container if it is small.
Graham’s law has its mathematical form because the root mean square speed is related to the mass of the particle. Different gases at the same temperature have the same kinetic energy. But, if their masses are different, they must be moving at different speeds. The heavier the particle is, the faster its speed must be. For two gases at the same temperature, we get
where m is mass and u is root mean square speed for particles one and two. The rate of effusion is proportional to the root mean square speed, and the mass of the particle is proportional to the molar mass. Substituting and rearranging gives Graham’s Law.
The KMT and Non-Ideal Behavior:
As experimental apparatus improved in the late 1800s, chemists recognized that gas laws were only approximate. The best experimental agreement with the mathematical predictions occurs when the gas is under relatively low pressure and high temperature. Some gases obey the laws better than others, even under the same sets of conditions. Gases with smaller molar masses and ones that are relatively inert obey better than larger, more reactive gases. Whether by nature or by conditions such as pressure, gases that do not obey the gas laws very well are called “real” gases and those that do obey are called “ideal” gases. If an ideal gas is one who obeys the assumptions of kinetic molecular theory, a real gas must be one that violates one or more of these assumptions.
High Molar Mass and Reactivity:
One assumption of the kinetic molecular theory is that the gas particles are neither attracted to nor repelled by one another. When the particles of a gas are very large, they have higher induced dipole attractions, so they are more attracted to one another. When the gas particles collide, they stick together and the average kinetic energy drops. These sticky gas particles hit the walls of the container less frequently and with less force than ideal gas particles do. Polar molecules have greater intermolecular attractions, too, so a molecule like water vapor is much less ideal than one like helium. The upshot is that some substances make better gases than others. Under the same conditions of temperature and pressure, helium is an ideal gas and water vapor is a real gas.
One assumption of the kinetic molecular theory is that the volume of the container is large enough that the volume of the particles is negligible. When a gas is under very high pressure (and/or the volume for that mass of gas is very low), the volume of the particles themselves can no longer be ignored, and calculated volumes are lower than real volumes. To make matters worse, as the distance between the particles drops, the attractions between particles increase. As particles stick together, they are less likely to strike the wall of the container, so calculated pressures are higher than real pressures.
Even very small, very inert gases like helium have induced dipole attraction for one another. Hot gas particles have a lot of kinetic energy to overcome these weak attractions. When a gas is very cold, the average molecular speeds and kinetic energy are low. The kinetic energy is no longer able to supply the energy needed to overcome the attractions between particles. Particles stick together and thus average kinetic energy drops. The upshot is that really cold gases are so non-ideal that they become liquids or even solids and the particles are not free to move throughout the container.
The van der Waals Equation
Atoms and molecules are never truly ideal because they all interact with other gas particles; weak attractions between separate gas particles are known as intermolecular attractions or van der Waals forces after the chemist who proposed a correction to the ideal gas law to calculate pressure of a real gas. Van der Waals proposed that the ideal gas equation could be corrected for real gas behavior by subtracting the effective gas particle volume from the volume of the container and by correcting for intermolecular attractions:
where b is the gas particle volume and a is the relative attractive force of the gas particles. These values are unique for each gas.
4Concept Check: Water is not a very good ideal gas, which is why it tends to be a liquid at STP. For water vapor, the van der Waals constants are a = 5.46 L2atm/mol2 and b = 0.0305 L/mol. If water were an ideal gas, one mole of water vapor would occupy 22.41 L of volume at 273.2 K and 1.000 atm. What pressure does one mole of water vapor actually exert at this temperature, if the volume is 22.41 L?
This is pretty close to 1 atm, even if water is a real gas!
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This site was last updated 05/13/05