 We've established that in three dimensions, pressure P times volume V equals number of molecules N times one-third molecular mass M times mean square velocity, and this equals N times Boltzmann's constant K times temperature T. Also, KT equals molecular mass times mean square velocity in a single dimension, so that one-half KT is the kinetic energy per, quote, degree of freedom. It's instructive to consider if this is true for a mixture of molecules with different masses. If so, all molecules would tend to have the same kinetic energy, but not the same mean square velocity. There is a very elegant demonstration that collisions among balls of different masses tend to equalize kinetic energy. Details are given in the Feynman lectures, link at the bottom of the screen. Here's a quick outline. Two balls have masses M1 and M2, and velocities V1 and V2. The center of mass velocity V0 is the total momentum M1 V1 plus M2 V2 divided by the total mass M1 plus M2. The relative velocity Vr is V1 minus V2. This is the velocity of particle one as seen by particle two. This is also U1 minus U2, where the U's are the velocities in the center of mass frame. In the center of mass frame, the velocities are anti-parallel. We can take them to be in the right and left directions as shown. After an elastic collision, the velocity magnitudes are unchanged, but the directions are rotated by an angle theta, which depends on the vertical offset of the particles before the collision. Over many collisions, we expect that the angle alpha between the relative velocity Vr and the center of mass velocity V0 will take on all possible values. So averaged over all collisions, the so-called inner product between these two velocities will be zero. Writing this out in terms of V1 and V2, we get two terms. The first is the average of twice the difference in kinetic energy. The second is the difference in masses times the average inner product of V1 and V2. Over many collisions, we also expect that the angle between these two velocities will take on all possible values. So this second term will vanish. But then the first term must vanish, which requires, on average, the two particles to have equal kinetic energy. Let's verify this with a simulation using 16 red balls of mass M1 and 16 blue balls with four times this mass. The lighter balls are initially at rest, and the heavier balls are given random initial velocities. As time goes on, it does appear that, on average, the lighter red balls end up moving faster than the heavier blue balls. Here's a plot of the total kinetic energy versus time in green, and the fractions due to the heavy and light balls in blue and red. Initially the blue balls contain all the kinetic energy. As time goes on, energy spreads to the red balls until each type accounts for roughly half the total energy. There are only 16 of each type of balls, so the fractions fluctuate quite a bit. But on average, over a long time period, the kinetic energy of each ball is the same. The plots have many very rapid dips. During a collision, when balls are in contact, they compress against each other. This briefly converts kinetic energy into elastic potential energy, and results in a dip in the kinetic energy curves. For perfectly hard balls, these dips would have zero duration, and the total kinetic energy curve would be a flat line. We can now summarize our results in the ideal gas law. Pressure times volume equals number of molecules times Boltzmann's constant times absolute temperature. For n and t constant, the right side is constant, so the left side is also. This is Boyle's law. Dividing by p and t, we get v over t equals nk over p. For constant n and p, v over t is constant. This is Charles' law. Dividing by v and t, we get p over t equals nk over v. For constant n and v, p over t is constant. This is the law of Gay-Luzak. And dividing by kt, we get n equals pv over kt. For constant p, v, and t, n is constant. This is Avogadro's law. In a macroscopic sample of gas, the number of molecules is huge. It's convenient to quantify this using Avogadro's constant. This constant, also called Avogadro's number, was, until recently, defined as the number of atoms in 12 grams of carbon-12. In the revised international system of units, Avogadro's constant is defined as exactly 6.02214076 times 10 to the 23. This number of molecules is called one mole of a substance. The molar mass of an element is the mass in grams of one mole of atoms of that element. In this periodic table, atomic number, the number of protons in the nucleus, is shown above the element's chemical symbol, and molar mass is shown below. Mole mass essentially equals the number of protons plus the number of neutrons in the nucleus. Note that the molar mass of carbon is not 12. This is because naturally occurring carbon is about 98.9% carbon-12, which has 6 protons and 6 neutrons, and about 1.1% carbon-13, which has 6 protons and 7 neutrons. The resulting molar mass is 12.011 grams. Analogous isotope blends occur for other elements, with the result that their molar mass values are also not integers. In the revised international system of units, Boltzmann's constant is defined as exactly 1.38069 times 10 to the minus 23 joules per Kelvin. It's convenient to combine the Avogadro and Boltzmann constants into a new constant, R, called the gas constant. To four digits, this is 8.314 joules per mole Kelvin. This gives us a new form of the ideal gas law. If we express the number of molecules as the number of moles, lower case N, times Avogadro's constant, then pressure times volume equals number of moles times the gas constant times temperature in Kelvin. If we use this to determine the volume of one mole of gas molecules at 20 degrees Celsius in standard atmospheric pressure, we find a volume of about 24 liters. In the previous video, we introduced the concept of heat capacity, and looked at the heat capacity program, the so-called specific heat, of several substances. We saw no discernible relation between the specific heats of different materials. Let's consider the heat capacity of a monatomic ideal gas. Heat capacity is defined as the added heat energy Q required to raise the temperature by one Kelvin. It has units of joules per Kelvin, or alternately calories per Kelvin. For a monatomic gas, the ideal gas law reads P times V equals N times one-third M times the mean square velocity, which equals N k T. The internal kinetic energy of the gas is N times one-half M times mean square velocity. Multiplying the expression above by three-halves, we can express the internal energy as three-halves N k T. The heat capacity is the change in energy when temperature increases by one Kelvin, which is three-halves N k. More precisely, this is the heat capacity at constant volume, denoted with a subscript V. We could also define a heat capacity at constant pressure, and obtain a different value. We'll come back to this idea in a later video. If we set N to Avogadro's constant, we find the molar heat capacity at constant volume is three-halves the gas constant, or 12.47 joules per mole Kelvin. This predicts that the heat capacity per mole of all monatomic gases will be the same. Let's see if this is true. Helium, which is monatomic, has a molar mass of 4.003 grams, and a specific heat of 3.1156 joules per gram Kelvin. Multiplying these, we obtain a molar heat capacity of 12.47 joules per mole Kelvin, agreeing with our theory to four digits. Doing the same for neon and argon, we obtain the same molar heat capacity, even though they have very different molar masses and specific heats. This is an important insight. Heat capacity, at least for a monatomic gas, depends only on the number of molecules. It's independent of the particular substance. Looking at the diatomic gases, nitrogen and oxygen, each composed of two atom molecules, we find different molar heat capacities near 21 joules per mole Kelvin. So, in general, gases do not all have the same molar heat capacity. The value is dependent, in some way, on molecular structure. The monatomic value is 3.5R, while the diatomic value is close to 5.5R. We might have guessed that molecules with two atoms would either have the same or twice the heat capacity of single atom molecules. But that is not the case. And in contrast to the highly regular monatomic case, the molar heat capacities of these two diatomic gases don't even agree to three digits. We'll treat the heat capacity of non-monatomic gases along with liquids and solids in future videos. For now, we'll limit consideration to monatomic gases, which we seem to have a good model for.