Lecture 7 Kinetic Theory of Gases Part 2 Last lecture we began to explore the behavior of an ideal gas in terms of the molecules in it We found that the pressure of the gas was 2 N mv 2 mN v x P x i V i 1 V Since there s nothing special about the x direction we should write this in terms of the total velocity of the molecules N v2 2 v i i 1 N N i 1 vx2 v 2y vz2 2 vx i v 2y i vz2 i N 2 2 2 v v and v Since x is not a special direction x y z must all have the same value therefore we can write v 2 3vx2 which means mN v 2 P 3V We can rewrite this as 2 N 1 2 P mv 3V 2 We do this because we recognize the quantity inside the parentheses it s the average kinetic energy of a gas molecule Temperature So according to Newtonian physics 2 1 2 PV N mv 3 2 But the ideal gas law which is experimentally verified tells us that PV Nk BT The only way both these can be true is if 2 1 2 N mv Nk BT 3 2 2 1 2 T mv 3k B 2 So we see that the temperature depends only on the average kinetic energy of the molecules We can use this fact to find a typical speed of the molecules v rms 3k BT 3RT v m M 2 Molecular mass Be careful with the interpretation of this it s not the average velocity which is 0 unless the gas as a whole is moving in some direction For this distribution the average is 0 but the RMS is as shown by the arrow Assume we have a monatomic only one atom per molecule ideal gas Then the only internal energy associated with the gas is the kinetic energy of the atoms which we ve already shown is related to the temperature 2 1 2 T mv 3k B 2 1 2 3 mv k BT 2 2 We could also write this as 1 1 3 2 2 2 2 2 2 m v x v y v z m v x v x v x k BT 2 2 2 3 2 3 mvx k BT 2 2 1 2 1 2 1 2 1 mvx mv y mvz k BT 2 2 2 2 What this says is that for each possible direction of motion the gas has internal energy equal to 1 k T per B 2 molecule For multi atomic molecules there are other ways that energy can be stored in the gas the molecules can rotate or vibrate Any way of storing energy is called a degree of freedom for the molecule The theorem of equipartition of energy says that For every degree of freedom in a system the system stores 1 2kBT of energy Note also that the internal energy of an ideal gas depends only on the temperature Specific Heat What can kinetic theory tell us about the specific heat of a gas to be precise we ll talk about the heat capacity per mole or molar specific heat defined as Q n T Let s first assume that the gas is held at constant volume The 1st Law of Thermodynamics tells us that Eint Q W Since the volume is constant there s no work being done on the gas so Eint Q But we ve already shown that Eint is proportional to the temperature of the gas Assuming the gas is monatomic we have 3 Eint nRT 2 3 3 Eint nRT nR T Q 2 2 Since we also know that Q nCV T we find that 3 nR T 2 3 J CV R 12 5 2 mol K nCV T Table 21 2 in the text shows that this values agrees well with what s observed for real monatomic gases Molar Specific Heat at Constant Pressure Now let s assume that the pressure on the gas is constant as the temperature changes Then the work being done on it is W P Vi V f nRT nRT f i P P P nR Ti T f nR T Using the 1st Law we have Eint Q W Q nR T But we already know that Eint nCV T So we can write nCV T Q nR T Q n T CV R Q CV R n T The quantity in red is what we defined as molar specific heat to remind us that this equation applies when pressure is held constant we write this one as Cp This means that for a monatomic gas 5 J C P CV R R 20 8 2 mol K And for any gas C P CV R CV for More Complex Gases If a gas is not monatomic then it has additional ways to store internal energy or additional degrees of freedom Consider a diatomic molecule In addition to moving in any of the three dimensions of space this molecule can store energy by Vibrating Has both kinetic and potential energy 2 degrees of freedom Rotating or 2 more degrees of freedom Recall that the theorem of equipartition of energy says that for every degree of freedom the gas has internal energy of 1 Eint nRT 2 So that our diatomic gas should have Eint 7 nRT 2 Repeating our derivation of CV for this gas we find 7 J CV R 29 1 2 mol K But this turns out to be a poor approximation none of the diatomic gases listed in Table 21 1 have CV this large What Went Wrong All our derivation relied on was 1 counting degrees of freedom 2 assuming that each degree of freedom could store any amount of energy Step 1 is pretty safe And so is step 2 as long as the molecules behave the way Newton says they should What we observe then is a failure of Newtonian physics The rotational and vibrational degrees of freedom can t take on any energy they please there s a minimum amount of energy needed to change them In other words the energy is quantized This was one of several hints that led to the development of quantum mechanics Adiabatic Processes We already have derived the relationship between pressure and volume of a gas for isothermal isobaric and isovolumetric processes But what about for adiabatic ones i e processes where no heat flows into or out of the gas Since no heat is involved the change in internal energy of the gas is equal to the work done on it dEint nCV dT dW PdV Of course the gas follows the ideal gas law so PV nRT d PV d nRT PdV VdP nRdT This means we have two expressions for dT PdV PdV VdP dT nCV nR We ve shown before that CV R Cp RPdV CV PdV VdP C R dV C CV R PdV CV VdP dP V V P Note that the left side depends only on …