Dr. W. Pezzaglia Physics 8C, Spring 014 Page 1 Las Positas College Lecture #11 & 1: Temperature 014Mar08 Lecture Notes 014Feb 5 and 7 on Temperature A. Temperature: What is temperature? Distinguish it from heat. Rest of this section covered in powerpoint slides. B. Thermal Expansion: Most materials get bigger when hotter. 1. Linear Expansion (in 1 dimension). Area Expansion (in dimensions) Note is coefficient of thermal expansion.. Thermal Stress, etc discussed on powerpoint slides Rest of this topic covered in powerpoint slides. 1
Dr. W. Pezzaglia Physics 8C, Spring 014 Page Las Positas College Lecture #11 & 1: Temperature 014Mar08 C. Gas Laws: 1. Kinetic Theory of Pressure (in 1 dimension) Note, this first part is probably more coherent in the powerpoint slides. This derivation is for motion only in 1 dimension. The formula in the bottom right on the slide below is misleading. It would be better to simply stop with the statement that: P K, where K is the total Kinetic Energy of the gas. Jumping ahead, the kinetic theory of temperature is that temperature is a 1 measure of the average kinetic energy of molecules, or: K N kt, where N is the number of molecules and k is Boltzmann s constant. Substituting, we arrive at the ideal gas law: P NkT
Dr. W. Pezzaglia Physics 8C, Spring 014 Page Las Positas College Lecture #11 & 1: Temperature 014Mar08 ii. Equipartition Theorem Now we extend the concept to dimensions. The energy is shared between the motion along the three axes. The equipartition theorem states that (on average) each direction gets an equal share of the total energy. iii. The ideal Gas law (in dimensions): A three dimensional treatment would have us replace the result of board with P=(/)K. Again substituting the relationship between average kinetic energy and temperature (now in dimensions from board ) we end up back with the ideal gas law.
Dr. W. Pezzaglia Physics 8C, Spring 014 Page 4 Las Positas College Lecture #11 & 1: Temperature 014Mar08. Distribution of Molecular Speeds In the above derivations we used the average kinetic energy of the molecules. In the powerpoint slides we discussed that Maxwell derived the speed distribution of molecules (which was experimentally measured in 190 by Stern). Here we discussed some of the mathematical details of dealing with that formula to calculate the rms velocity. (i) The Gamma Function: For integer n, the Gamma function yields the factorial function: n x n 1 x e dx nn n! 0 Recall that 1!=0!=1. However, the integral can be evaluated for n that is any real number. In particular, it can be shown: 1. Hence for example you can show: 1!. The types of integrals we encounter in statistical mechanics are even more complicated, they involve an exponential to a power p of x. But by doing some tricks, you can show the integral can be reduced to something involving a gamma function. Maxwell s distribution for molecular speeds is given to be: f ( v) Below showed that this distribution function is normalized, i.e. f ( v) dv 1. 0 m kt mv v exp. kt Note if you wanted to calculate the number molecules with speed above a certain value, the integral cannot be done in closed form. You have to use a series approximation. 4
Dr. W. Pezzaglia Physics 8C, Spring 014 Page 5 Las Positas College Lecture #11 & 1: Temperature 014Mar08 Next we showed how to calculate the average speed, and the average of the squared speed. In particular, we note that the average of the squared speed is not the square of the average speed. In the derivation of the ideal gas law, we found a relationship between P and the average kinetic energy of molecules. Since Kinetic Energy depends upon the square of the speed, we need the average of the squared speed, not the square of the average speed. The class participated in trying to do these integrals. Unfortunately, the board photo is somewhat blurred. The results are: v v rms kt m v 8, where the average speed is: 8kT v m Its also useful to know the most probable speed, which is obtained by solving: 0. This gives: kt v p v. m So, we have shown that given this distribution, the formula 1 K N m v NkT df dv on average is valid. 5
Dr. W. Pezzaglia Physics 8C, Spring 014 Page 6 Las Positas College Lecture #11 & 1: Temperature 014Mar08. an der Waals Gas Law (derived 187, Nobel prize 1910) This topic is not covered in the powerpoint slides nor in the book. However it is routinely covered in texts (c.g. Tipler, Physics for Scientists and Engineers, rd edition, 1990, pp. 506-507). Here we give a brief coverage. (a) Excluded volume: The free volume is less than because molecules have volume. For Nitrogen b=9.1 cc/mole (corresponds to atomic diameter of about 0.4 nm). This correction is only important when the density is very high. For example at standard temperature and pressure the density of 1 mole of Nitrogen gas is a much bigger,400 cc/mole which means the correction is less than 0.% hence ignorable. (b) an der Waal s Forces (i.e. how geckos climb): Here we have to handwave a bit. In analogy to gravity, the forces between electric charges follows an inverse square law, and the electrostatic energy is inversely proportional to distance. Molecules have electric dipole moments for which the simplistic model is a stick with plus charge at one end and negative charge at the other end. [Magnets have similar topology, with North pole at one end, south pole at other.] The force law between dipoles goes like the inverse fourth power of distance, while potential energy goes like inverse cube power. The average distance cubed between molecules would be proportional to the total volume divided by the number of moles: r. The potential energy of one molecule with its neighbor would hence be on n the order of : n a, where a is some constant and the / factor will make the final result prettier. Note for nitrogen gas: a=0.14 Pa-m 6 /mol. Board 10: This had so many errors, I replace it with below: n The total potential energy of n moles of molecules would go like: PE a. The minus sign is because the forces are attractive. The Kinetic energy of the molecules would be the internal energy of the gas: U nrt, minus the potential energy due to the attraction of the dipoles. Hence we have that the (total) Kinetic n Energy K of the gas is: K nrt a. Combine this with the Kinetic Theory of Pressure (board 4): n P K, we arrive at P nrt a. 6
Dr. W. Pezzaglia Physics 8C, Spring 014 Page 7 Las Positas College Lecture #11 & 1: Temperature 014Mar08 n a Moving the last term to the other side: P nrt. an der Waal s Formula: Finally, putting back in the excluded volume, we arrive an an der n a Waal s gas formula: P nb nrt. (c) Liquid-apor Isotherms: Recall that isotherms (constant temperature) of the ideal gas are simply P=constant. If you plot the isotherms (constant temperature) for an der Waal s gas, then above the critical temperature approaches an ideal gas. Below that however the Pressure is a cubic function of the volume with three possible volume states for a given pressure. The middle one is unstable and does not represent a true state. Hence we artificially connect the two other roots with a straight line which represents the gas condensating between a high volume state (gas) to a small volume state (liquid). For Nitrogen at its condensation temperature the change in volume is a factor of nearly 700. With some messy algebra solving cubic equations you can show that the an der Waal s coefficients a and b can be related to the critical temperature, pressure and volume as: c nb a Pc 7b 8a Tc 7bR For Nitrogen, T c =16.K (-147C) and P c =.5 atm (,90 kpa) -end of temperature- 7