(4-98) The van der Waals equation may be used for the calculation of μ and it gives moderately good agreement with experiment.
The Joule-Thomson effect is generally larger than the Joule effect. For C 02 at STP, μ = 1.30 Κ a t m- 1 (as compared to 0.73 Κ a t m- 1 calculated from the van der Waals equation), and for 02, μ = 0.31 Κ a t m- 1. The Joule-Thomson effect may be produced in a cyclic operation, and machines that do this are widely used in the liquefaction of gases. In practice, the expansion may be through a throttle rather than a porous plug.
SPECIAL TOPICS, SECTION 2 139 Tyflvib $ which is the same as that shown in Fig. 4-10 except that the heat capacity scale is now from zero to 3R. The high-temperature limit is 3R, or the equipartition value (and that given by the Dulong-Petit law). Experimental results agree fairly well with the Einstein equation, as shown by the broken lines in Fig. 4-13(a). In each case 0Vib is empirically determined so as to give the best lit. Significant deviation sets in at low temperatures, however, and the theory was considerably improved by Debye. He assumed a range of possible v0 values up to a maximum vm j i x , using the
θ = 1 0 6 0 K G r a p h i t e D i a m o n d
Γ , Κ (a)
0 0.5 1.0 1.5 2 . 0 2.5 Τ/ΘΌ
(b)
F I G . 4-13. Heat capacities of various crystalline solids as a function of temperature, (a) The best-fitting curves according to the Einstein equation (4-98), dashed lines, (b) Scaled by the best choice of 6D to give a fit to the theoretical Debye curve (solid line). [Adapted from S. M. Blinder,
"Advanced Physical Chemistry." Copyright 1969, Macmillan, New York.]
G E N E R A L R E F E R E N C E S
BLINDER, S . M . ( 1 9 6 9 ) . "Advanced Physical Chemistry." Macmillan, N e w York.
HIRSCHFELDER, J . O . , CURTISS, C. F . , A N D B I R D , R. B . ( 1 9 5 4 ) . "Molecular Theory of G a s e s and Liquids." Wiley, N e w York.
EXERCISES
4-1 Given the function y = u(u + v), evaluate (a) (dy/du)v and (b) (dy/dv)u . Calculate Ay, J (dy/du)v du, and J (dy/dv)u dv for (c) path ( «x, i7x) (u2, vx) followed by ( w2, vx) («2, v2) a n d (d) path (ux, -> (ux, v2) followed by ( « ι , v2) (u2, v2), where ux = 2 , u2 = 5, vx = 3 , and v2 = 6 .
Ans. (a) 2u + v, (b) u, (c) Ay = 45, f (dy/du)v du = 30, J (dy/dv)u dv = 15, (d) Ay = 45, J (Byldv)u dv = 6 , J (Byldu)v du = 39.
4-2 For a right circular cylinder o f radius r and length /, the area is s/ = Itrrl and the volume is V = TtrH. (a) Evaluate the coefficients (drf/dr^ , (d^/dl)r, (Sr/dl)^, {?V\dr\ , (dV/dl)r. (b) Evaluate (ds//dV)i by expressing as a function o f / and Κ only and then carry out the indicated partial differentiation, (c) Using Eqs. (4-13) and (4-15), express (ds/jdV)i in terms involving only partial derivatives listed in (a) and thus evaluate (ds//dV)i indirectly.
Ans. (a) 2 π / , 2ΤΓΓ, - ^ / 2 π /2, 2ΤΓΓ/, nr2. (b) a n d (c) 1/r.
4-3 Express (dE/dT)P in terms o f derivatives which y o u can evaluate from the ideal gas law.
Ans. (ΘΕ/ΒΤ)Ρ = (dE/dT)v + (dEldV)T(dV/dT)P = Cv + 0 = Cv
4-4 A certain gas obeys the equation P(V - b) = RT. (a) Evaluate (dV/dT)P and {jdV\dP)T. Obtain (dP/dT)v by (b) use o f Eq. (4-12) and (c) direct differentiation.
Ans. (a) R/P, -RT/P2, (b) and (c) Rl(V-b).
4-5 T h e coefficient o f thermal expansion for A l is ( 1 / Κ ) ( δ Κ / δ Γ )Ρ = 2.4 χ 10"5 ''C"1, and (dE/dV)T is estimated at 1.0 χ 1 05 atm. Calculate CP — Cv for Al at 25°C and atmospheric pressure.
Ans. 0.58 cal m o l e "1.
4-6 O n e mole o f an ideal, monatomic gas at S T P undergoes an isochoric heating t o 25°C.
Calculate (a) P, (b) q, (c) w, (d) AE, and (e) AH.
Ans. (a) 1.09 atm, (b) 74.5 cal, (c) 0, (d) 74.5 cal, (e) 124 cal.
same distribution function as for blackbody radiation (Section 16-ST-l). This
^max is simply a cutoff at the point where a total of 3N0 frequencies is reached and is an empirical parameter of the theory. The characteristic temperature is now defined as hvm&JkTand is thus also an empirical quantity. For the low-temperature region the Debye treatment predicts that the heat capacity should be proportional to Tz, and this provides a valuable limiting law. It is used, for example, in extrap
olating experimental heat capacity values toward 0 Κ when calculating thermo
dynamic quantities.
PROBLEMS 141 4-7 T h e same gas as in Exercise 4-6 undergoes an isobaric heating t o 25°C. Calculate the same
quantities as before, plus (f) the final volume.
Ans. (a) 1 atm, (b) 124 cal, (c) 49.7 cal, (d) 74.5 cal, (e) 124 cal, (f) 24.5 liter.
4-8 The same gas as in Exercises 4-6 and 4-7 undergoes a reversible adiabatic compression such that the final temperature is 25°C. Calculate quantities (a)-(f).
Ans. (a) 1.24 atm, (b) 0 , (c) - 7 4 . 5 cal, (d) 74.5 cal, (e) 124 cal, (f) 19.7 liter.
4-9 T e n m o l e s o f an ideal g a s o f Cv = 3.5 cal K "1 m o l e "1 undergoes a reversible isothermal compression from 0.2 t o 5 a t m at 100°C. Calculate (a) the initial a n d final volumes, (b) q, (c) w, (d) ΔΕ, and (e) ΔΗ.
Ans. (a) 1531 a n d 61.24 liter, (b) - 2 3 , 8 7 0 cal, (c) - 2 3 , 8 7 0 cal, (d) 0 , (e) 0.
4-10 Calculate t h e equipartition heat capacity CP for (a) Os , (b) X e , (c) HC1, a n d (d) CaH4 r assuming ideal gas behavior, and Cv for (e) diamond and (f) N a C l (solid).
Ans. (a) 1R, (b) 2iR, (c) 4$R, (d) 1 6 * , (e) 3R, (f) 6R.
4-11 A foam plastic has cells of 0.1 m m dimension. Calculate (a) the quantum number η for the one-dimensional translational state whose energy would be equal t o Α: Γ at 25°C in the case of argon gas and (b) Qtranete-dim) at 25°C assuming the cells t o be spheres of radius 0.1 m m .
Ans. (a) 7.1 χ 10β, (b) 1.02 χ 1 02 1 c m8. 4-12 T h e rotational constant Be o f a linear molecule is defined as Be = A /8TT2C/, where c is the velocity o f light; Be = 1.93 c m- 1 for C O . Calculate (a) the m o m e n t o f inertia / for C O , (b) the quantum number / for the rotational state w h o s e energy is equal t o kT at 25°C, (c) the characteristic rotational temperature, a n d (d) Qr o t at 25°C.
Ans. (a) 1.45 χ 1 0 "3 9 g c m2, (b) about 10, (c) 2.78 K , (d) 107.
4-13 T h e vibrational characteristic temperature is 3084 Κ for C O . Calculate (a) hv0, (b) the force constant / , ( c ) the energy of the t> = 2 state, (d) CV(vn>) at 25°C, (e) Qv l b at 25°C, a n d (f) Evl b at 2 5 ° C .
Ans. (a) 4.257 x 1 0 "1 8 erg, (b) 1.86 x 10* d y n c m "1, (c) 1.06 x 1 0 "1 2 erg (or 2 5 . 8 A T a t 25°C), (d) 3 . 4 4 x 1 0 "8 R9
(e) 5.675 χ 1 0 "8, (f) 2.13 χ 1 0 ~1 8 erg (or 5.2kT).
P R O B L E M S
4-1 T h e area of a right c o n e is given by wr(r2 -f Λ2)1/2, where r is the radius o f the base and h is the altitude, and the volume is (n/3)r2h. Evaluate (ds/ldr)hi (ds//dh)r, (dr/dh)^, (dV[dr)ht
(dV/dh)r, and {dr\dK)v. Evaluate {?si\dV\ by expressing si as a function of h and V only and then carrying out the indicated partial differentiation. Finally, evaluate (ds//d V)h in terms o f the differentials given here only, using the various partial differential relationships given in the chapter.
4-2 Derive the equation CP - Cv = -(dP/dT)v[(dH/dP)T - V].
V
4-8 One m o l e of an ideal monatomic gas may be taken from the initial condition Px = 3 atm, νΎ = 10 liter to the final condition P2 = 0.5 atm, K2 = 2 liter by either one of the following paths: Path 1. (a) decrease in volume at constant pressure followed by (b) decrease in pressure at constant volume. Path 2. (a) decrease in pressure at constant volume followed by (b) decrease in volume at constant pressure.
F o r each path calculate ΔΕ, q, and w. If the gas is taken from Px, Vx to P2, V% by path 1 and returned to the initial state by path 2, what are the values of ΔΕ, q, and w for the cycle ? 4-9 Calculate ΔΕ, ΔΗ, q, and w when 1 m o l e of an ideal monatomic gas initially at 0°C and 2 atm is taken to a final pressure of 15 atm by the reversible path defined by the equation PV2 = constant. Calculate (by means of a derivation) the heat capacity along this path, that is dqldT for the path.
4-10 One mole of an ideal monatomic gas may be taken from the initial condition Px = 2 atm, Vx = 15 liter to the final condition P2 = 4 atm, K2 = 4 0 liter by either of the following paths: Path 1: (a) increase in volume at constant pressure followed by (b) an increase in pressure at constant volume. Path 2: (a) increase in pressure at constant volume followed by (b) an increase in volume at constant pressure.
For each path calculate ΔΕ, q, and w. If the gas is taken from Ρτ, Vx to P2, K2 by path 1 and returned to the initial state by path 2, what are the values of ΔΕ, q, and w for the cycle ? 4-11 One hundred grams of nitrogen at 25°C are held by a piston under 30 atm pressure. The pressure is suddenly released to 10 atm and the gas expands adiabatically. If Cv for nitrogen is 4.95 cal ° C_ 1 m o l e- 1, calculate the final temperature of the gas. What are ΔΕ and ΔΗ for the process? Assume the gas is ideal.
4-12 Consider the hypothetical experiment of Fig. 4-5 in which a gas e x p a n d s against a fixed weight of 0.1 atm. S u p p o s e the gas to b e o n e m o l e of a n ideal m o n a t o m i c o n e initially at S T P , and that there is e n o u g h friction that the piston m o v e s slowly, that is, negligible 4-3 Suppose that for a certain gas (dE/dV)T = 0, but P(V - b) = RT. Calculate (dHldV)T
and Cp — Cy.
4-4 D e r i v e the relationship (duldy)v(dv/du)y(dy/dv)u = — 1. This is a useful alternative form to o n e o f the equations in the text. It n e e d n o t b e m e m o r i z e d ; just notice that u, v, a n d y occur o n c e in each numerator, d e n o m i n a t o r , a n d subscript. Verify the relationship for y =f(V, T), where the f u n c t i o n / i s for a n ideal gas.
4-5 O n e m o l e o f water is vaporized reversibly at 5 0 ° C . T a k e ΔΗν t o b e 10.0 kcal m o l e "1. Calculate ΔΕν a n d the reversible work.
4-6 T h e heat capacity ratio, y, is 1.20 for a certain ideal gas. B y what factor d o e s the pressure c h a n g e if the v o l u m e is do u bl e d in a reversible adiabatic e x p a n s i o n ?
4-7 One m o l e of an ideal monatomic gas undergoes the following processes: 1. Adiabatic expansion from Pl t Vl9 7 \ to P2, ^ 2 , T2. 2. Return t o initial state by the straight line path s h o w n in the accompanying diagram. Calculate Δ £ , AH, q, and w for each step and for the cycle if Λ = 2 atm, Tx = 0°C, and V2 = 2VX.
ρ >v Step 2
•P2>Vi>T2
SPECIAL TOPICS PROBLEMS 143
SPECIAL T O P I C S P R O B L E M S
4-1 D e r i v e t h e equation
'['-*(»κΜ©,·
4-2 G a s e s such as H2, Oa, or N2 are termed " n o n c o n d e n s a b l e " because they cannot b e liquefied at ordinary temperatures, although, o f course, liquefaction does occur if they are c o o l e d at t h e proper pressure t o a temperature b e l o w the critical temperature. If the J o u l e - T h o m s o n coefficient is positive, t h e c o o l i n g m a y b e accomplished b y allowing the compressed g a s at r o o m temperature t o expand t o 1 a t m pressure, part o f t h e g a s liquefying if the conditions are right. I n t h e commercial process based o n these facts the liquefied g a s is collected (at its boiling point at 1 a t m pressure), a n d t h e residual u n vibrational contribution t o the heat capacity, (b) A n ideal gas has an equipartition value o f Cp/Cy of 1.083.
One mole o f C 02 at 100°C at 8 0 a t m is compressed isothermally t o 100 atm. The pressure is then reduced t o 80 a t m by cooling at constant volume. Finally, the original state is regained b y warming at constant pressure. A s s u m i n g that C 02 o b e y s the v a n der Waals equation with constants as given in Table 1-5, calculate q a n d w for each step a n d for t h e cycle.
Referring t o Exercise 4-16, calculate the heat capacity Cv for C O for various temperatures between 25°C a n d 1000°C a n d plot t h e results a s Cv versus t. independent contribution t o the heat capacity.
T h e rotational temperature is 66 Κ for H D . Calculate t h e average rotational energy at 30 Κ a n d at 4 0 Κ a n d estimate t h e rotational contribution t o t h e heat capacity o f H D at 35 K .
Oa at 1 atm, 248 K ) . 2. CP is 6.7 cal K "1 m o l e "1 (assume to be independent of tem
perature). 3. T h e heat of vaporization of liquid o x y g e n is 1600 cal m o l e- 1.
4 - 3 Calculate the Joule coefficient / and the J o u l e - T h o m s o n coefficient μ for C H4 assuming it to be a van der Waals gas. A s s u m e STP.
4-4 T h e D e b y e characteristic temperature ΘΌ = 86 Κ for P b . Estimate the heat capacity of P b at - 1 0 0 ° C and at 25°C. Calculate 0v l b for P b for the best-fitting Einstein equation.
4-5 T h e heat capacity of Pt is (in calories per gram) 0.00123 at - 2 5 5 ° C , 0.0261 at - 1 5 2 ° C , 0.0324 at 2 0 ° C , and 0.0365 at 750°C. Estimate the D e b y e characteristic temperature 0D and 0vib for the best-fitting Einstein equation. Plot the data as heat capacity versus tem
perature along with the t w o theoretical curves.
4-6 A n alternative J o u l e - T h o m s o n type of experiment is to measure the quantity of heat that must be supplied when the expansion occurs at constant temperature. This gives the isothermal J o u l e - T h o m s o n coefficient φ = (dH/dP)T . Relate φ to μ and calculate φ for C H4 at S T P assuming it to be a van der Waals gas and looking up any additional data needed.