Let's start with the definition of a mole. A mole is a number, just like a dozen is a number. A dozen means 12, whereas, a mole means 6.022 x 1023. A mole is used to define the number of atoms in an element. One mole of any element, be it aluminum, iron, carbon, etc, contains 6.022 x 1023 molecules. This (6.022 x 1023) is Avogadro's number (NA). We are going to find that this number is pretty important.
In our previous lesson, we talked about the thermal expansion and
contraction of solids due to a change in temperature. It turns out, however, that for gases, these laws don't hold very well. The volume of a gas depends on both the temperature and the pressure (and
how much gas) there is a in a system. These
relations are called an equation of state, and in this course, we will consider only gases that have reached an
equilibrium condition (pressure, temperature, and volume are not changing) when we examine their properties.
Which brings us to a gentleman named Robert Boyle and Boyle's Law. Boyle found that the volume of a gas is inversely proportional to the absolute pressure of the gas when the temperature of the system is kept constant. Mathematically, we can state that as
A graph of pressure vs. volume at a constant temperature looks like the figure at the right. (By the way, this is only a very good approximation.) Changing either pressure or volume in a system will cause the other to change as well. As long as the temperature remains constant! (But you know that it won't, so read on!) About a hundred years after Boyle did his work, a Frenchman by the name of Jacques Charles found that, for a gas kept at a constant pressure, and if the pressure is not too high, then there is a direct and proportional relationship between the volume of the gas and the temperature of the gas. We state this as Charles' Law:
BUT, we need to be aware of something very important here. The temperature is measured on an absolute scale called the Kelvin scale. In the Kelvin scale, (degrees Kelvin, or K without the o sign) 0K equates to a temperature condition called absolute zero. This temperature has not quite been reached, but interesting things start to happen to matter when it approaches absolute zero. 0K = -273.15 oC, and an increment in K equals an increment in Celsius. We change K to oC by adding 273 to the number, or
T(K) = T(oC) + 273.15
This number was determined by plotting a graph of volume vs. temperature and extrapolating the data back to the point where the volume of the gas would be 0. The final gas law we will take a look at was developed by Joseph Gay-Lussac. He found a relationship between pressure and temperature of a gas when the volume is held constant:
All three of these laws are really approximations and only hold under
certain conditions. Again, the gases must be in equilibrium and the temperatures and pressures cannot be at extremes. (In other words, the gas cannot be close to becoming a liquid, or
condensing.) But for our purposes, these are going to be very useful. Boyle's, Charles', and Gay-Lussac's laws were all derived experimentally and may all be
combined into one general equation. The mass of
gas also contributes to the equations, and it was found that pressure, volume, temperature, and the mass of the gas (expressed in moles) are all proportional, or, in a closed system:
where n is the number of moles. Further experimentation allows us to write an equation:
This equation is called the ideal gas law. R is called the universal gas constant and is valid for all gases (thus universal). It can be expressed in several SI terms:
R = 8.315 Joules/(mol-K)
R = 0.0821 (Liters-Atm)/(mol-K)
R = 1.99 calories/(mol-K)
Note that the differences relate to how we are expressing our pressures. Mass will always be in moles, and temperature will always be in Kelvin. Alternatively, the ideal gas law may be expressed in terms of N (number of molecules), rather than n (number of moles). Avogadro said that equal volumes of gas at the same temperature and pressure have the same number of molecules. Let N = the total number of molecules in a gas, and NA equal the number of molecules in a mole (6.02 x 1023). Then n, the number of moles, = N/NA, and the ideal gas law may be written as:
But R/NA is a constant known as Boltzmann’s constant (k). k = 1.38 x 10-23 J/K, and the ideal gas law may also be written:
An Ideal Gas is assumed to have these properties:
- An ideal gas consists of a large number of identical molecules.
- The volume occupied by the molecules themselves is negligible compared to the volume occupied by the gas.
- The molecules move in random motion (and obey Newton’s laws of motion).
- The molecules experience forces only during collisions.
- Collisions by molecules are completely elastic, and take a negligible amount of time.
We use these properties to explain the behavior of ideal gases. For example, if we halve the volume of the gas and hold the temperature constant, we can assume that the molecules have the same kinetic energy, but are now colliding twice as frequently with the walls of the container. This is what pressure is - a measure of the force of the collisions of molecules striking an area of the container. We use a kinetic theory of the movement of molecules and atoms to explain the behavior of gases. Temperature is actually a measure of the average kinetic energy of the molecules in an ideal gas. We can express the average kinetic energy in terms of Boltzman's constant and the temperature (in Kelvin):
We can then use this to figure the average velocity of our gas molecules by using our laws of kinematics.
For more on the Gas Laws, try: http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/idegas.html
For Practice Problems, Try: