Page 88 - Physical chemistry eng

NUMERICAL PROBLEMS 65

P3.6 A vessel is filled completely with liquid water and P3.16 The Joule coefficient is defined by (0T>0V) =
U
sealed at 13.56°C and a pressure of 1.00 bar. What is the (1>C )[P - T(0P>0T) ] . Calculate the Joule coefficient for
V
V
pressure if the temperature of the system is raised to an ideal gas and for a van der Waals gas.
82.0°C? Under these conditions, b water = 2.04 * 10 -4 K -1 , P3.17 Using the result of Equation (3.8), (0P>0T) = b>k ,
V
-5
b vessel = 1.42 * 10 -4 K -1 , and k water = 4.59 * 10 bar -1 . express as a function of and V for an ideal gas, and as
k
b
b
m
P3.7 Integrate the expression b = 1>V(0V>0T) P assuming a function of b, , and V for a van der Waals gas.
k
m
that is independent of temperature. By doing so, obtain an P3.18 Show that the expression (0U>0V) =
b
T
expression for V as a function of T and at constant P. T(0P>0T) - P can be written in the form
b
V
P3.8 A mass of 32.0 g of H O(g) at 373 K is flowed into
2
1
295 g of H O(l) at 310. K and 1 atm. Calculate the final tem- a 0U b = T a0c P d n0Tb =- a0c P d n0c db
2
2
perature of the system once equilibrium has been reached. 0V T T V T T V
Assume that C P, m for H O is constant at its values for 298 K P3.19 Derive an expression for the internal pressure of a gas
2
throughout the temperature range of interest. Describe the that obeys the Bethelot equation of state,
state of the system.
RT a
P3.9 Because (0H>0P) =-C m , the change in P = - 2
P J-T
T
enthalpy of a gas expanded at constant temperature can be V m - b TV m
calculated. To do so, the functional dependence of m J-T on P P3.20 Because U is a state function, (0>0V (0U>0T) ) =
V T
must be known. Treating Ar as a van der Waals gas, calculate (0>0T (0U>0V) ) . Using this relationship, show that
T V
¢H when 1 mole of Ar is expanded from 325 bar to 1.75 bar (0C >0V) = 0 for an ideal gas.
T
V
at 375 K. Assume that m J-T is independent of pressure and is P3.21 Starting with the van der Waals equation of state, find
given by m J-T = [(2a>RT) - b]>C P,m , and C P,m = 5R>2 an expression for the total differential dP in terms of dV and
for Ar. What value would ¢H have if the gas exhibited ideal dT. By calculating the mixed partial derivatives
gas behavior? (0(0P>0V) >0T) V and (0(0P>0T) >0V) T , determine if dP is
V
T
P3.10 Derive the following expression for calculating the an exact differential.
isothermal change in the constant volume heat capacity: P3.22 Use (0U>0V) = (bT - kP)>k to calculate
T
2
2
(0C >0V) = T(0 P>0T ) V . (0U>0V) T for an ideal gas.
T
V
P3.11 A 75.0 g piece of gold at 650. K is dropped into 180. g P3.23 Derive the following relation,
of H O(l) at 310. K in an insulated container at 1 bar pres- 0U 3a
2
sure. Calculate the temperature of the system once equilib- a b =
rium has been reached. Assume that C P, m for Au and H O is 0V m T 22TV (V m + b)
m
2
constant at their values for 298 K throughout the temperature for the internal pressure of a gas that obeys the Redlich-Kwong
range of interest. equation of state,
P3.12 Calculate w, q, ¢H , and ¢U for the process in which RT a 1
1.75 moles of water undergoes the transition H O(l, 373 K) : P = -
2
m
H O(g, 610. K) at 1 bar of pressure. The volume of liquid water V m - b 2T V (V m + b)
2
3
-5
at 373 K is 1.89 * 10 m mol -1 and the molar volume of P3.24 A differential dz = f(x, y)dx + g(x, y)dy is exact
3
-2
steam at 373 K and 610. K is 3.03 and 5.06 * 10 m mol -1 , if the integral f(x, y)dx + g(x, y)dy is independent
respectively. For steam, C P, m can be considered constant over of the path. Demonstrate that the differential dz = 2xydx +
1
1
-1
the temperature interval of interest at 33.58 J mol -1 K . 2
x dy is exact by integrating dz along the paths
2
P3.13 Equation (3.38), C = C + TV(b >k) , links C P (1,1) : (1,8) : (6,8) and (1,1) : (1,3) : (4,3) :
V
P
and C with and . Use this equation to evaluate C – C V (4,8) : (6,8) . The first number in each set of parentheses is
k
b
V
P
for an ideal gas.
the x coordinate, and the second number is the y coordinate.
P3.14 Use the result of Problem P3.26 to derive a formula
P3.25 Show that dr>r =-bdT + kdP where is the
r
for (0C >0V) T for a gas that obeys the Redlich-Kwong equa- density r = m>V . Assume that the mass m is constant.
V
tion of state,
P3.26 For a gas that obeys the equation of state
RT a 1
P = - RT
V m - b 2T V (V m + b) V m = + B(T)
m
P3.15 The function f(x, y) is given by f(x, y) = P
2
xy sin 5x + x 2y ln y + 3e -2x 2 cos y . Determine derive the result
2
2
0f 0f 0 f 0 f 0 0f a 0H b dB(T)
a b , a b , a b , a b , ¢ a b ≤ = B(T) - T
0x y 0y x 0x 2 y 0y 2 x 0y 0x y x 0P T dT
0 0f 0 0f 0 0f P3.27 Because V is a state function, (0(0V>0T) >0P) =
P
T
and a a b b . Is a a b b = a a b b ? (0(0V>0P) >0T) . Using this relationship, show that the
0x 0y x y 0y 0x y x 0x 0y x y T P
isothermal compressibility and isobaric expansion coefficient
Obtain an expression for the total differential df. are related by (0b>0P) T =-(0k>0T) P .