化工热力学英文习题

Michigan State University

DEPARTMENT OF CHEMICAL ENGINEERING AND MATERIAL SCIENCE

ChE 821: Advanced Thermodynamics Fall 2008

1. (30) A thermodynamicist is attempting to model the process of balloon inflation by assuming

that the elastic casing behaves like a spring opposing the expansion (see below). The model

assumes that the piston/cylinder is adiabatic. As air (following the ideal gas law) is admitted, the

spring is compressed. The pressure on the spring side of the piston is zero, so that the spring

provides the only force opposing movement of the piston. The pressure in the tank is related to

the gas volume by Hooke’s law

P − P i = k (V – V i )

where k = 1E-5 MPa/cm 3, P i = 0.1 MPa, T i = 300K, and V i = 3000 cm 3, Cv = 20.9 J/mol K,

independent of temperature, and the reservoir is at 0.7 MPa and 300K.

Provide the balances needed to determine the gas temperature in the cylinder at volume V =

4000cm 3. Perform all integrations. Do not calculate the gas temperature, but provide all

equations and parameter values to demonstrate that you could determine the gas temperature.

2. (30) Consider two air tanks at the initial conditions shown below. We wish to obtain work

from them by exchanging heat and mass between the tanks. No gas may be vented to the

atmosphere, and no heat may be exchanged with the atmosphere. Reversible devices may be

used to connect the two tanks.

Provide the balances necessary to calculate the maximum work that may be obtained. Perform all

integrations. Do not calculate the work value, but provide all equations and parameter values to

demonstrate that you could determine the work value. C p = 29.3 J/molK. Use the ideal gas law.

Tank A 400 K 5 bar 6 m 3 Tank B 200 K 0.1 bar 10 m 3

3. (a) (15) It is desired to express the derivative ()

T

V ∂, which is related to isothermal compressibility in terms of ()

S V P ∂∂, which is related to adiabatic compressibility. Derive a relation by starting with ()T V ∂ and interposing P and S using the Jacobian method. Leave the answer in terms of derivatives involving S .

(b) (15) Express ((()S P P T V S ∂∂∂ in terms of measureable properties.

4. (10) Show ,,T V T N V μμ∂∂⎛⎞⎛⎞=−⎜⎟⎜⎟⎝⎠⎝⎠.

Equations

R = 8.3143 J/(molK) = 8.3143 cm 3MPa/(molK) = 83.143 cm 3bar/(molK)

dU = TdS --- PdV + μdN Æ -(∂P/∂S)V = (∂T/∂V)S dH = TdS + VdP + μdN Æ (∂V/∂S)P = (∂T/∂P)S dA = -SdT - PdV + μdN Æ (∂P/∂T)V = (∂S/∂V)T dG = -SdT + VdP + μdN Æ -(∂V/∂T)P = (∂S/∂P)T

Jacobian Formula

(),(,)Y X Y X X Y Y X

K K X Y K L K L K L X Y X Y Y X L L X Y ∂∂ʈʈÁ˜Á˜Ë¯Ë¯∂∂∂∂∂∂∂ʈʈʈʈ=-=Á˜Á˜Á˜Á˜Ë¯Ë¯Ë¯Ë¯∂∂∂∂∂∂∂ʈʈÁ˜Á˜Ë¯Ë¯∂∂

e e d c b c b ⎟⎠⎞⎜⎝⎛∂∂=⎟⎠⎞⎜⎝⎛∂∂, , c b c b e d =⎟⎟⎠⎞⎜⎜⎝⎛∂∂, , ,d e e e b b b d N c c d c ⎡⎤∂∂∂⎛⎞⎛⎞⎛⎞=−⎜⎟⎜⎟⎜⎟⎢⎥∂∂∂⎝⎠⎝⎠⎝⎠⎣⎦ , ,,,b e d e c e

d c b d c b ∂⎛⎞−⎜⎟∂∂⎝⎠⎛⎞=⎜⎟∂∂⎛⎞⎝⎠⎜⎟∂⎝⎠