A cylinder made of copper with a mass of 0.80 kg is heated to 800C, then dropped into 6.00 kg of water at 11C. What is the total change in entropy (in J/K) of the cylinder-water system, assuming no energy is lost by heat from this system to the surroundings? The specific heat of copper is 387 J/(kg - K), and the specic heat of water is 4,186 J/(kg ~ K). (Hint: note that do = mch.) _x First, consider conservation of energy. Using the specific heats, can you find the final equilibrium temperature of the combined system? Next consider the infinitesimal change in entropy as. How does it depend on do and temperature? Note the expression given for dQ in the hint. Can you write an infinitesimal expression for each object that can be integrated from the initial to final temperature? J/K Entropy is defined in terms of microstates. We often illustrate how this works using a small number of discrete objects like coins or dice. When rolling three dice, 1+2+6 and 1+6+2 and 3+2+4 are different microstates, even though each adds up to 9. There is only one microstate that adds to 3, but there are 25 that add to 9. That is why you are much more likely to roll a 9 than a 3. With coin-like objects that have two choices, counting the number of possible microstates is illustrated by Pascal's triangle. The branch of mathematics that deals with counting such things is called combinatorics. In any system of more than a handful of atoms the number of microstates becomes larger than the number of protons in the universe. So we typically calculate changes in entropy, with an eye toward whether it is positive or negative. But in this problem we'll count microstates. You flip 6 coins. How many microstates are there? 4 W W What is the number of heads that has the highest number of microstates? If two or more have the same number, give the one with the lowest number of heads. W H How many microstates are there that have exactly one head? w W How many times more likely is it that you get the most likely number of heads than that you get one head? X A triatomic molecule can have a linear conguration, as does (:02 (Figure a), or it can be nonlinear, like H20 (Figure b). Suppose the temperature of a gas of triatomic molecules is sufficiently low that vibrational motion is negligible. 3//" \\\\H D (a (a) What is the molar specific heat at constant volume, expressed as a multiple of the universal gas constant (R) if the molecules are linear? EintT= (b) What is the molar specific heat at constant volume, expressed as a multiple of the universal gas constant (R) if the molecules are nonlinear? EintT= At high temperatures, a triatomic molecule has two modes of vibration, and each contributes %R to the molar specific heat for its kinetic energy and another %R for its potential energy. (c) Identify the high~temperature molar specific heat at constant volume for a triatomic ideal gas of the linear molecules. (Use the following as necessary: R.) EintT = (d) Identify the high-temperature molar specific heat at constant volume for a triatomic ideal gas of the nonlinear molecules. (Use the following as necessary: R.) Eint-r= Are the data in table below sufficient to make this determination? O Yes O No Figure Monatomic Gases Diatomic Gases Polyatomic Gases Molar Specific Heats of Various Gases Molar Specific Heat (J/mol . K) Gas Cp Cy Cp - Cy y = Cp/ Cy Monatomic gases He 20.8 12.5 8.38 1.67 Ar 20.8 12.5 8.38 1.67 Ne 20.8 12.7 8.12 1.64 Kr 20.8 12.3 8.49 1.69 Diatomic gases H2 28.8 20.4 8.38 1.41 Ng 29.1 20.8 8.38 1.40 29.4 21.1 8.38 1.40 CO 29.3 21.0 8.38 1.40 Cl, 34.7 25.7 8.96 1.85 Polyatomic gases CO2 37.0 28.5 8.50 1.30 SO2 40.4 81.4 9.00 1.29 H,O 35.4 27.0 8.37 1.30 CHA 35.5 27.1 8.41 1.81 * All values except that for water were obtained at 300 K. CiGas Cp CV Cp - Cv Y = Cp / CV He 20.8 12.5 8.33 1.67 Ar 20.8 12.5 8.33 1.67 Ne 20.8 12.7 8.12 1.64 Kr 20.8 12.3 8.49 1.69 Read ItGas Cp CV Cp - Cv Y = Cp / Cv H 2 28.8 20.4 8.33 1.41 N2 29.1 20.8 8.33 1.40 02 29.4 21.1 8.33 1.40 CO 29.3 21.0 8.33 1.40 C2 34.7 25.7 8.96 1.35\fAs shown in the figure, a cylinder with a moveable piston and containing a monatomic ideal gas in an initial stateA undergoes an isovolumetric, then an isothermal, and finally an isobaric process to complete the cycle. I'lnlml I} \\'(I.l (B When the gas is in the initial state, the volume is 3.00 L, the pressure is 1.00 atm, and the temperature is 300 K. The gas is first warmed at constant volume to a pressure of 2 times the initial value (state B). The gas is then allowed to expand isothermally to some new volume (state C). Finally it is compressed isobarically to its initial state. (Due to the nature of this problem, do not use rounded intermediate values in your calculationsincluding answers submitted in WebAssign.) (a) Find the number of moles of the gas. U moles (b) Find the temperature of the gas at state B (in K). K H (c) Find the temperature of the gas at state C (in K). K H (d) Find the volume of the gas at state C (in L). L H (e) Determine values (in U) for Q, W, and AE for the process A > B. int o=:}w w=:}w AE...=:lw (f) Determine values (in kj) for Q, W, and AE , for the process B - C. Q = KJ W = KJ AE int = KJ (g) Determine values (in kj) for Q, W, and AE; , for the process C - A. Q = KJ W = KJ AE int KJ (h) Determine values (in kj) for Q, W, and AE; , for the complete cycle A -> B - C - A. Q kJ W= KJ AE int KJ