(a) How much work is done by the gas? (b) Assuming the gas behaves as an ideal gas, what is the final temperature? V γ+1 2 V γ+1 ) pdv = K 1 γ + 1

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1 P340: hermodynamics and Statistical Physics, Exam#, Solution. (0 point) When gasoline explodes in an automobile cylinder, the temperature is about 2000 K, the pressure is is Pa, and the volume is 00 cm 3. he piston has cross sectional area of 80 cm 2. he gas then expand adiabatically as the piston is pushed downward until its volume incrases by a factor of 0. he gasoline has the parameter of γ = C p /C v =.4. (a) How much work is done by the gas? (b) Assuming the gas behaves as an ideal gas, what is the final temperature? (a) (b). W = hus 2 = 796 K. pdv = K γ + ( V γ+ 2 V γ+ ) = 20 J V γ = 2 V γ 2 2. (0 point) Let p be the pressure and V be the volume for a real gas of n moles. If the equation of state is given by pv = φ(), where is the temperature of the system. he internal energy of the system U is also a function of only, i.e. ( U/ V ) = 0. Show that the function φ() must be linear function of the temperature, i.e. φ() = c, where c is a constant. Since ( ) U V = 0, we find du = ( ) ( ) U U d + dv = C V d. V V Since) du is an exact differential, we find C V is a function of temperature only, i.e. = 0. Now, we candider the first law of thermodynamics: ( CV V ds = du + pdv = C V d + pdv ds = C V φ() d + V dv Since ds is an exact differential, we obtain ( ) ( ) (φ()/v ) (CV /) = V V hus φ()/ must be a constant, and we find φ() = c, where c is a constant. = 0

2 3. (5 point) A room, filled with air (ideal gas) at SP condition ( atm, 0 C), has a dimension of (0m) (0m) (3m). [Properties of the air: ρ =.29 kg/m 3, C V = J/(kg K), γ = C p /C V =.4]. (a) Find the amount heat Q needed to raise the temperature of the room, with insulated rigid walls, from 0 C to 20 C. (b) Calculate the change of the internal energy of the air in this room. (c) Calculate the change of entropy of the system. he total mass of the air is m = ρv =.29 (0 0 3) = 387 kg. (a) he amount of heat needed is Q = mc V = J (b) he change of internal energy is U = Q W = Q = J. (c) he change of entropy is 2 C V S = m d = mc V ln 2 = J/K. 4. (0 point) An ideal gas in a container with an initial condition of p = Pa, =.0 m 3 and = 300 K. Calculate the change of the entropy of the system and its environment in the following processes. (a) he system is in thermal contact with a reservoir, so that the volume is increased isothermally from.0 m 3 to 2.0 m 3 in a reversible process. (b) he volume of the gas is free-expanded by the Gay-Lussac-Joule free expansion process to a volume of 2.0 m 3. Using pv = nr, we find nr = J/K. (a) When the system is isothermally expanded from to V 2, the change of entropy is P ( S) system = dv = nr ln V 2 = nr ln 2 = 23 J/K. he change of the entropy for the environment is ( S) environment = Q/ = W/ (isothermal). We need to calculate the work done by the system! W = hus ( S) environment = 23 J/K. PdV = nr ln V 2 = nr ln 2. ( S) universe = ( S) system + ( S) environment = 23 + ( 23) = 0 J/K. 2

3 (b) For free expansion, the expansion occurs very fast. Gay-Lussac-Joule s experiment shows that ( / V ) u = 0, i.e. the process occurs isothermally. hus we find P ( S) system = dv = nr ln V 2 = nr ln 2 = 23 J/K. Since the expansion is very fast, there is no change of entropy from the environment, i.e. ( S) environment = 0, and we find ( S) universe = ( S) system + ( S) environment = = 23 J/K. 5. (0 point) A vessel with insulating walls of negligible heat capacity contains 0 kg of ice at 20 C. Now we pour 2 kg of water at +40 C into the vessel and seal it. [Specific heat of ice: 2090 J/(kg K); Specific heat of water: 480 J/(kg K); he latent heat of ice-water is l 2 = J/kg.] (a) How much ice (in kg) will remain in the vessel when the system is in thermal equilibrium again? (b) Find the entropy change that takes place inside the insulated vessel. he net heat content of 40 C water and 20 C ice relative to 0 C is Q = 2(kg) 480(J/kg K) 40(K)+0(kg) 2090(J/kg K) ( 20K) = J. he negative sign in the amount of heat shows that some water will become ice. he amount is m freeze = Q /l 2 = 0.25 kg. (a) he amount of ice is the vessel is = 0.25 kg, and amount of water is =.75 kg. (b) he change of entropy is d S = ds = C V = C V ln 2 S = (2kg)(480J/(kg K)) ln( 273 ) + (0kg)(2090J/(kg K)) ln( ) (0.25kg) l 2 = 4 J/K (0 points) Show that the heat conduction coefficient of an ideal gas is proportional to, where is the temperature. 3

4 Let the radius of each gas molecule be r. he cross-section of the the collision is π(2r) 2. hus the mean free path l is related to the average volume occupied by each molecule V/N by π(2r) 2 l = V N, l = V 4πr 2 N = 4πr 2 k p. Consider two adjacent slabs of length l and area A with temperature and 2 respectively. Molecules in each slab can move from the slab # to the slab #2 in one mean free path. he amount of heat exchange is Q = 2 (U U 2 ) = 2 C v( 2 ) = 2 C vl d dx, Q t = C vl 2 ta Ad dx. By definition, the heat conduction coefficient is k t = C vl 2A t = C vl 2A(l/ v) = C vl v 2V = f k 6πr 2 3k m, where we use C v = f 2 Nk = f 2 (pv/) and C vl/2v = fk/(6πr 2 ). 7. (20 points) We consider an ideal gas with N monatomic molecules contained in a volume V at the temperature. Let the total energy of the gas be E, i.e. E = 2m (p 2 ix + p 2 iy + p 2 iz). i We define the amplitude of the momentum as p = 2mE. (a) Find the available phase space states, Σ(P), for the particle to occupy from p = 0 to P(E) = 2mE. (b) Find the integral Z(, V ) = (c) Use Sterling formula to Evaluate ln(z). 0 dσ(p) dp dp de e E/k de. (d) Use the etrode-sackur Equation to evaluate U S, and compare with ln(z). 4

5 (a) he volume of 3N dimensional sphere of radius p and the number of available states are π3n/2 V p (p) = Γ( 3N + 2 )p3n Σ(p) = N! h V N V 3N p (p) where p = 2mE, V is the volume in the coordinate space, h is the Planck constant, and Γ is the gamma-function. (b) Carrying out the integration, we obtain Z(, V ) = = dσ dp exp E/kdE 0 dp de N! h V N (2πmk) 3N/2 3N (c) Using the stirling formula, we find ln(z) = N ln V N ( ) 3/2 2πmk +. h 2 (d) Using the etrode-sackur equation, we find F U S = k ln(z). We note that F is the Helmholtz free energy. 8. (5 points) Consider an Einstein solid of N oscillators and q quanta, where the energy of the system is U = qǫ. (a) Find the entropy in terms of U and N. (b) Find the U as a function of the temperature. (c) Find the heat capacity as a function of. (d) Find the chemical potential. Plot your chemical potential vs emperature. (a) he number of states and the entropy are (N + q)! Ω(N, q) =, N! q! S = k ln Ω = k[(n + q) ln(n + q) N ln N q ln q]. 5

6 (b) (c) (d) = S U = k ( ǫ ln + N ) q Nǫ U = e ǫ/k. C = U ( ) ǫ 2 e ǫ/k = Nk k (e ǫ/k ) 2. µ = S N = k ln( e ǫ/k ). 6