δq T = nr ln(v B/V A )

Transcription

1 hysical Chemistry 007 Homework assignment, solutions roblem 1: An ideal gas undergoes the following reversible, cyclic rocess It first exands isothermally from state A to state B It is then comressed adiabatically to state C Finally, it is cooled at constant volume to its original state, A a Each state is characterized by the volume, ressure and temerature of the gas,, and Sketch the cyclic rocess in the, lane and in the, lane b Calculate the change in entroy of the gas in each one of the three rocesses and show that there is no net change in the cyclic rocess, consistent with the fact that entroy is a state function a On the - diagram, oint C is right above oint A but B is to the right and at lower ressure but larger volume he adiabatic curve C-B is steeer than the isothermal curve A-B In the - diagram, oint A is right above oint B and oint C is to the right and above b For simlicity, assume we are dealing with one mol of the gas Ste 1: Isothermal exansion from A to B S 1 B A δq nr ln B/ A Ste : Adiabatic comression from B to C S 0 since it is adiabatic But, we need to find the final temerature for later Use the relationshi between and changes during an adiabatic exansion/comression C B γ 1 B C where γ C /C, the heat caacity ratio For an ideal gas, C C + R, so γ 1 R/C ake the logarithm, to get the relationshi C C ln B R ln B C Ste 3: Cooling at constant volume from C to A S 3 A C δq A C C d C ln A C 1

2 Now use the fact that A B and the relationshi between B / C and B / A found for the adiabatic ste above to rewrite this as S 3 R ln A B Adding u all three contributions to the entroy change gives S tot S 1 + S + S 3 R ln B A R ln A B 0 consistent with the fact that entroy is a state function roblem : Exerimental measurements have been used to determine how the molar heat caacity of CO gas deends on temerature over the temerature range from 00 K to 1000 K and the results can be summarized by the following exression C a + b + c where the constants are a 6648 kj/molk, b kj/molk and c kj/molk 3 Calculate the enthaly and entroy change when one mole of the gas is heated from room temerature to 1000 K H 1 H d C d [a + b 3 ] 1 + c 3334kJ/mol 3 1 S 1 δq rev 1 C d [a ln + b + c ] 1 554J/molK roblem 3: a he basic roerties of gases are tyically described by an equation of state that gives the ressure as a function of volume and temerature,, In many cases, one is interested in evaluating the artial derivative, but this is not readily obtained from, Rewrite this artial derivative using Euler s chain rule in terms of artial derivatives that can be obtained directly from the equation of state, and b Give an exression for when the equation of state is of the van der Waals form

3 a Use Euler s chain rule: For a function zx, y we have hat is, y z z y y z z 1 y x z Making here the corresondance z, y and x, we get b For van der Waals gas, z y y x R m b a m where m /n So, the artial derivatives become and m m Inserting into the result from art a gives R/ m b 1 R n m b + a R NR m b a 1 m b m 3 3 m roblem 4: In this roblem, you will analyse the ressure and volume deendence of the entroy of a certain gas Exerimental measurements have been used to develo an equation of state written as a virial exansion u to second order m R 1 + B where the second virial coefficient is over the limited temerature interval of interest given by B a + b Also, the constant volume heat caacity has been measured and the results summarized by the exression C α + β + γ he arameters a, b, α, β and γ have been determined by the exerimental measurements Considering the entroy to be a function of and, a small change in the entroy can be written as ds S d + S d 3

4 a Obtain an exression for the artial derivative S in terms of the exerimentally determined arameters a, b, α, β and γ recall: q ds and q C d, see notes from lecture b Use a Maxwell relation derived from one of the thermodynamic state functions U, H, G or A to exress S in terms of a artial derivative that can be obtained readily from the exerimentally obtained information given above Evaluate the artial derivative in terms of the exerimentally determined arameters a, b, α, β and γ c Derive a Maxwell relationshi from the second derivative of the entroy in order to gain information about the ressure deendence of the heat caacity C and evaluate this artial derivative in terms of the exerimentally determined arameters arameters a, b, α, β and γ o summarize: Given m R1+B where B a+b and C α+β+γ, where a, b, α, β and γ are constants known from measurements a Since q ds for a reversible rocess by definition of S and q C d for heat flow at constant, we have ds C d and by dividing by d and setting constant get S C α + β + γ b Use the combined first and second law for the Gibbs free energy dg Sd + d to get Maxwell relation S which can be calculated from the virial exansion m R1 + a + b to give So, m R 1 + a + b + R b R S nr 1 + a + b + Ra + Rb c he Maxwell relationshi is obtained from the mixed second derivative of S S S aking the derivative of the result in art a with resect to and the derivative of the result in b with resect to, gives 1 C 4 nrb

5 roblem 5: For a certain chemical, the Gibbs free energy is found to vary with temerature according to G/ a + b/ + c/ over a given interval in temerature Here, a and b are constants determined exerimentally a Find how the enthaly varies with temerature over this temerature range b Find how the entroy varies with temerature c Demonstrate that the three equations are consistent a he Gibbs-Helmholts equation gives G/ 1/ H b + c b From the coefficient of d in the combined first and second law for G, one gets G S so S G a + c c Since G H S, the exressions given in a and b need to add u to the exression given for G in the statement of the roblem o test whether this is true, add a and b H S b + c a + c a + b + c as it should 5