 This is Thermodynamics 2, Ideal Gases. Welcome! In the first video, we experimented with a simple gas thermometer, consisting of a 100 milliliter syringe, sealed at both ends. We measured its volume at three different temperatures and saw that they were well represented by a linear relation between volume and temperature. Extrapolating this line to zero volume leads to the concept of the lowest possible temperature. This occurs at minus 273.15 degrees Celsius. We use this to define the absolute or Kelvin temperature scale. Temperature in Kelvin equals temperature in degrees Celsius plus 273.15. Unless otherwise stated, in all that follows we assume this absolute temperature scale. Over the centuries, through experiments of the type we performed, although of much greater vigor, various gas laws were discovered. In 1662, Boyle's law was presented. For a given quantity of gas at constant temperature, the product of pressure and volume remains constant. If the pressure and volume of a given quantity of gas are in one state, P1 and V1, and in a second state, P2 and V2, then P1 times V1 equals P2 times V2. Pressure is inversely proportional to volume. If volume increases, pressure decreases so as to keep their product constant. This was before Fahrenheit's development of the mercury thermometer, and temperature was not a well-defined quantity. However, constant temperature was implicit in the experiments as they were performed at room temperature with the apparatus returning to thermal equilibrium for each measurement. With the development of accurate thermometers, the explicit effects of temperature could be studied. In 1787, Charles's law was announced. This states that for a given quantity of gas at constant pressure, the ratio of volume to temperature is constant. That is, volume is proportional to temperature. This is the principle of the gas thermometer. In 1809, the law of Gay-Luzak was presented. This states that for a given quantity of gas at constant volume, the ratio of pressure to temperature is constant. Pressure is proportional to temperature. This law has application to heat engines. By raising or lowering temperature, we can increase or decrease pressure, and hence the force on a piston. Finally, in 1811, Avogadro published his law. At the same pressure and temperature, two equal volumes of different gases contain the same number of molecules. He came to this conclusion from considering how the volumes of two gases reacting to form a volume of a third gas had the same relationship as the presumed number of molecules responsible for the reaction. These gas laws were empirical, derived from observation, but they can be understood in terms of molecular theory as arising very simply from the mechanical properties of the vast number of molecules that make up the gas. This is the so-called kinetic theory of gases. Let's explore this, starting with the simplest possible gas, one consisting of a single atom moving in a single dimension. Let's treat the atom as a tiny, perfectly hard sphere of mass m, moving in the horizontal direction with velocity vx, enclosed by walls separated by distance lx. Some of the walls have area, ax. Here we show atoms traveling at the same speed but in different volumes. Because the atoms travel with the same speed, they all exert the same force when they strike a wall. But the atoms in the larger volume take longer to travel between the walls, so they exert this force fewer times per second. From this we expect that pressure should decrease with increasing volume. The atom's momentum is mvx. When it bounces off a wall, its momentum goes from plus this amount to minus this amount, or vice versa. Therefore, the total change in momentum, call it delta px, is 2mvx. Call the round trip time through the volume delta t round trip. This is the total round trip distance, 2lx, divided by the velocity vx. In writing this, we are treating the atom's diameter as negligibly small. Force is the rate of change of momentum with respect to time. So the average force on a wall is delta px over delta t round trip, which is 2mvx over 2lx over vx, which is mvx squared over lx. Pressure is force divided by area, which is mvx squared over axlx. Axlx is the enclosure volume, capital V. So we end up with pressure times volume equals mvx squared. Assuming higher temperature corresponds to faster moving atoms, the right-hand side should be related to temperature. In practice, of course, we are interested in systems with huge numbers of atoms, generally moving with different speeds. Each would make an mvx squared contribution. For n atoms, the result is pv equals nm times the mean, or average, square velocity, which we denote by angled brackets. Now we set this proportional to temperature. Specifically, we write that this equals n times a constant k times temperature in Kelvin. In doing so, we are guided by the laws of Charles and Gay-Lussac. For constant pressure, volume is proportional to temperature. For constant volume, pressure is proportional to temperature. If we didn't already have a definition of temperature, we could leave out the constant k and write this as n times t. This would define temperature as the mass of a single molecule times its mean square velocity in one dimension, with units of energy, or joules. But then, room temperature would be about 4 times 10 to the minus 21 joules, which isn't very practical for day-to-day use. In order to keep the Kelvin and Celsius scales, we add the constant k, called Boltzmann's constant. It has units of joules per Kelvin. Now consider atoms traveling at different speeds, but in the same volume. Higher velocity has two effects. First, at higher velocity, more force is exerted when an atom bounces off a wall. Second, at higher velocity, an atom travels the round trip through the volume more times per second. So for a given volume, we would expect pressure to vary as two factors of velocity, that is, the square of velocity, which is consistent with our previous result. For a two-dimensional gas, we add top and bottom walls of area AY separated by distance LY. The atom's velocity has x component vx and y component vy. Applying our one-dimensional result to each dimension, pressure in the horizontal or x dimension, call it px, times volume equals mvx squared. For the vertical or y dimension, py times volume equals mvy squared. The square of total velocity, v squared, equals vx squared plus vy squared. Assume that, on average, vx squared and vy squared are equal, with the value one-half v squared. This will be true when taken over the huge number of atoms present in any macroscopic sample of a gas. Then, in both dimensions, we have pv equals one-half mv squared. For n atoms, this becomes pv equals n one-half m times the mean square velocity. It's still true that kt equals the mass of a molecule times its mean square velocity in one dimension. Finally, we come to the real three-dimensional world. An atom has three velocity components, vx, vy, and vz. The square of total velocity, v squared, equals vx squared plus vy squared plus vz squared. And, on average, the square of any velocity component will be one-third the square of total velocity. So, we have pv equals one-third mv squared. For n atoms, this becomes pv equals n one-third m times the mean square velocity, which equals nkt. Once again, kt equals the mass of a molecule times its mean square velocity in one dimension. Since kinetic energy is one-half mass times velocity squared, we can say that one-half kt is the kinetic energy of an atom associated with a single dimension, or what we call a degree of freedom.