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12–1C
Consider the function z(x, y). Plot a differential surface on x-y-z
coordinates and indicate dx, dx, dy, dy,( dz)x, (dz)y, and dz.
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12–2C What is the difference between partial differentials and ordinary differentials?
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12–3C
Consider the function z(x, y), its partial derivatives (dz/dx)y and
(dz/dy)x, and the total derivative dz/dx. (a) How do the magnitudes
(dx)y and dx compare? (b) How do the magnitudes (dz)y and dz compare?
(c) Is there any relation among dz,( dz)x, and (dz)y?
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12–4C
Consider a function z(x, y) and its partial derivative (dz/dy)x. Under
what conditions is this partial derivative equal to the total derivative
dz/dy?
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12–5C
Consider a function z(x, y) and its partial derivative (dz/dy)x. If
this partial derivative is equal to zero for all values of x, what does
it indicate?
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12–6C Consider a function z(x, y) and its partial derivative (dz/dy)x. Can this partial derivative still be a function of x?
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12–7C
Consider a function f(x) and its derivative df/dx. Can this derivative
be determined by evaluating dx/df and taking its inverse?
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12–8
Consider air at 400 K and 0.90 m3/kg. Using Eq. 12–3, determine the
change in pressure corresponding to an increase of (a) 1 percent in
temperature at constant specific volume, (b) 1 percent in specific
volume at constant temperature, and (c) 1 percent in both the
temperature and specific volume.
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12–9 Repeat Problem 12–8 for helium.
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12–10
Prove for an ideal gas that (a) the P = constant lines on a T-v diagram
are straight lines and (b) the high-pressure lines are steeper than the
low-pressure lines.
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12–11
Derive a relation for the slope of the v = constant lines on a T-P
diagram for a gas that obeys the van der Waals equation of state.
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12–12
Nitrogen gas at 400 K and 300 kPa behaves as an ideal gas. Estimate the
cp and cv of the nitrogen at this state, using enthalpy and internal
energy data from Table A–18, and compare them to the values listed in
Table A–2b.
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12–13E
Nitrogen gas at 600 R and 30 psia behaves as an ideal gas. Estimate the
cp and cv of the nitrogen at this state, using enthalpy and internal
energy data from Table A–18E, and compare them to the values listed in
Table A–2Eb.
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12–14
Consider an ideal gas at 400 K and 100 kPa. As a result of some
disturbance, the conditions of the gas change to 404 K and 96 kPa.
Estimate the change in the specific volume of the gas using (a) Eq. 12–3
and (b) the ideal-gas relation at each state.
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12–15 Using the equation of state P(v - a) = RT, verify (a) the cyclic relation and (b) the reciprocity relation at constant v.
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12–16 Verify the validity of the last Maxwell relation (Eq. 12–19) for refrigerant-134a at 80°C and 1.2 MPa.
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12–17
Reconsider Prob. 12–16. Using EES (or other) software, verify the
validity of the last Maxwell relation for refrigerant-134a at the
specified state.
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12–18E Verify the validity of the last Maxwell relation (Eq. 12–19) for steam at 800°F and 400 psia.
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12–19 Using the Maxwell relations, determine a relation for (ds/dP)T for a gas whose equation of state is P(v - b) = RT.
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12–20
Using the Maxwell relations, determine a relation for (ds/dv)T for a
gas whose equation of state is (P - a/v2) (v - b) = RT.
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12–21 Using the Maxwell relations and the ideal-gas equation of state, determine a relation for (ds/dv)T for an ideal gas.
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12–22C What is the value of the Clapeyron equation in thermodynamics?
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12–23C Does the Clapeyron equation involve any approximations, or is it exact?
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12–24C What approximations are involved in the ClapeyronClausius equation?
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12–25
Using the Clapeyron equation, estimate the enthalpy of vaporization of
refrigerant-134a at 40°C, and compare it to the tabulated value.
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12–26
Reconsider Prob. 12–25. Using EES (or other) software, plot the
enthalpy of vaporization of refrigerant-134a as a function of
temperature over the temperature range -20 to 80°C by using the
Clapeyron equation and the refrigerant-134a data in EES. Discuss your
results.
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12–27
Using the Clapeyron equation, estimate the enthalpy of vaporization of
steam at 300 kPa, and compare it to the tabulated value.
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12–28 Calculate the hfg and sfg of steam at 120°C from the Clapeyron equation, and compare them to the tabulated values.
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12–29E
Determine the hfg of refrigerant-134a at 50°F on the basis of (a) the
Clapeyron equation and (b) the Clapeyron-Clausius equation. Compare your
results to the tabulated hfg value.
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12–30
Plot the enthalpy of vaporization of steam as a function of temperature
over the temperature range 10 to 200°C by using the Clapeyron equation
and steam data in EES.
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12–31
Using the Clapeyron-Clausius equation and the triplepoint data of
water, estimate the sublimation pressure of water at -30°C and compare
to the value in Table A–8.
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12–32C
Can the variation of specific heat cp with pressure at a given
temperature be determined from a knowledge of Pv-T data alone?
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12–33
Show that the enthalpy of an ideal gas is a function of temperature
only and that for an incompressible substance it also depends on
pressure.
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12–34
Derive expressions for (a) du,( b) dh, and (c) ds for a gas that obeys
the van der Waals equation of state for an isothermal process.
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12–35
Derive expressions for (a) du,( b) dh, and (c) ds for a gas whose
equation of state is P(v - a) = RT for an isothermal process.
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12–36 Derive expressions for (du/dP)T and (dh/dv)T in terms of P, v, and T only.
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12–37
Derive an expression for the specific-heat difference cp = cv for (a)
an ideal gas, (b) a van der Waals gas, and (c) an incompressible
substance.
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12–38 Estimate the specific-heat difference cp = cv for liquid water at 15 MPa and 80°C.
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12–39E Estimate the specific-heat difference cp = cv for liquid water at 1000 psia and 150°F.
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12–40
Derive a relation for the volume expansivity b and the isothermal
compressibility a (a) for an ideal gas and (b) for a gas whose equation
of state is P(v - a) = RT.
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12–41 Estimate the volume expansivity b and the isothermal compressibility a of refrigerant-134a at 200 kPa and 30°C.
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12–42C What does the Joule-Thomson coefficient represent?
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12–43C Describe the inversion line and the maximum inversion temperature.
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12–44C
The pressure of a fluid always decreases during an adiabatic throttling
process. Is this also the case for the temperature?
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12–45C Does the Joule-Thomson coefficient of a substance change with temperature at a fixed pressure?
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12–46C Will the temperature of helium change if it is throttled adiabatically from 300 K and 600 kPa to 150 kPa?
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12–47
Consider a gas whose equation of state is P(v - a) = RT, where a is a
positive constant. Is it possible to cool this gas by throttling?
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12–48
Derive a relation for the Joule-Thomson coefficient and the inversion
temperature for a gas whose equation of state is (P + a/v2)v = RT.
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12–49 Estimate the Joule-Thomson coefficient of steam at (a) 3 MPa and 300°C and (b) 6 MPa and 500°C.
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12–50E
Estimate the Joule-Thomson coefficient of nitrogen at (a) 200 psia and
500 R and (b) 2000 psia and 400 R. Use nitrogen properties from EES or
other source.
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12–51E
Reconsider Prob. 12–50E. Using EES (or other) software, plot the
Joule-Thomson coefficient for nitrogen over the pressure range 100 to
1500 psia at the enthalpy values 100,175,and 225 Btu/lbm. Discuss the
results.
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12–52 Estimate the Joule-Thomson coefficient of refrigerant-134a at 0.7 MPa and 50°C.
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12–53
Steam is throttled slightly from 1 MPa and 300°C. Will the temperature
of the steam increase, decrease, or remain the same during this process?
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12–54C What is the enthalpy departure?
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12–55C
On the generalized enthalpy departure chart, the normalized enthalpy
departure values seem to approach zero as the reduced pressure PR
approaches zero. How do you explain this behavior?
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12–56C Why is the generalized enthalpy departure chart prepared by using PR and TR as the parameters instead of P and T?
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12–57
Determine the enthalpy of nitrogen, in kJ/kg, at 175 K and 8 MPa using
(a) data from the ideal-gas nitrogen table and (b) the generalized
enthalpy departure chart. Compare your results to the actual value of
125.5 kJ/kg.
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12–58E
Determine the enthalpy of nitrogen, in Btu/lbm, at 400 R and 2000 psia
using (a) data from the ideal-gas nitrogen table and (b) the generalized
enthalpy chart. Compare your results to the actual value of 177.8
Btu/lbm.
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12–59
What is the error involved in the (a) enthalpy and (b) internal energy
of CO2 at 350 K and 10 MPa if it is assumed to be an ideal gas? Answers:
(a) 50%, (b) 49%
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12–60
Determine the enthalpy change and the entropy change of nitrogen per
unit mole as it undergoes a change of state from 225 K and 6 MPa to 320 K
and 12 MPa, (a) by assuming ideal-gas behavior and (b) by accounting
for the deviation from ideal-gas behavior through the use of generalized
charts.
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12–61
Determine the enthalpy change and the entropy change of CO2 per unit
mass as it undergoes a change of state from 250 K and 7 MPa to 280 K and
12 MPa, (a) by assuming ideal-gas behavior and (b) by accounting for
the deviation from ideal-gas behavior.
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12–62
Methane is compressed adiabatically by a steady-flow compressor from 2
MPa and -10°C to 10 MPa and 110°C at a rate of 0.55 kg/s. Using the
generalized charts, determine the required power input to the
compressor.
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12–63
Propane is compressed isothermally by a piston– cylinder device from
100°C and 1 MPa to 4 MPa. Using the generalized charts, determine the
work done and the heat transfer per unit mass of propane.
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12–64
Reconsider Prob. 12–63. Using EES (or other) software, extend the
problem to compare the solutions based on the ideal-gas assumption,
generalized chart data, and real fluid data. Also extend the solution to
methane.
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12–65E
Propane is compressed isothermally by a piston– cylinder device from
200°F and 200 psia to 800 psia. Using the generalized charts, determine
the work done and the heat transfer per unit mass of the propane.
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12–66 Determine the exergy destruction associated with the process described in Prob. 12–63. Assume T0 = 30°C.
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12–67
Carbon dioxide enters an adiabatic nozzle at 8 MPa and 450 K with a low
velocity and leaves at 2 MPa and 350 K. Using the generalized enthalpy
departure chart, determine the exit velocity of the carbon dioxide.
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12–68
Reconsider Prob. 12–67. Using EES (or other) software, compare the exit
velocity to the nozzle assuming ideal-gas behavior, the generalized
chart data, and EES data for carbon dioxide.
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12–69
A 0.08-m3 well-insulated rigid tank contains oxygen at 220 K and 10
MPa. A paddle wheel placed in the tank is turned on, and the temperature
of the oxygen rises to 250 K. Using the generalized charts, determine
(a) the final pressure in the tank and (b) the paddle-wheel work done
during this process.
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12–70
Carbon dioxide is contained in a constant-volume tank and is heated
from 100°C and 1 MPa to 8 MPa. Determine the heat transfer and entropy
change per unit mass of the carbon dioxide using (a) the ideal-gas
assumption, (b) the generalized charts, and (c) real fluid data from EES
or other sources.
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12–71
For B>= 0, prove that at every point of a singlephase region of an
h-s diagram, the slope of a constantpressure (P = constant) line is
greater than the slope of a constant-temperature (T = constant) line,
but less than the slope of a constant-volume (v = constant) line.
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12–72 Using the cyclic relation and the first Maxwell relation, derive the other three Maxwell relations.
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12–73
Starting with the relation dh = T ds + v dP, show that the slope of a
constant-pressure line on an h-s diagram (a) is constant in the
saturation region and (b) increases with temperature in the superheated
region.
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12–74
Derive relations for (a) du,( b) dh, and (c) ds of a gas that obeys the
equation of state (P + a/v2)v = RT for an isothermal process.
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12–75 Show that

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12–76
Estimate the cp of nitrogen at 300 kPa and 400 K, using (a) the
relation in the above problem and (b) its definition. Compare your
results to the value listed in Table A–2b.
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12–77
Steam is throttled from 4.5 MPa and 300°C to 2.5 MPa. Estimate the
temperature change of the steam during this process and the average
Joule-Thomson coefficient.
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12–78
A rigid tank contains 1.2 m3 of argon at -100°C and 1 MPa. Heat is now
transferred to argon until the temperature in the tank rises to 0°C.
Using the generalized charts, determine (a) the mass of the argon in the
tank, (b) the final pressure, and (c) the heat transfer. Answers: (a)
35.1 kg, (b) 1531 kPa, (c) 1251 kJ
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12–79
Argon gas enters a turbine at 7 MPa and 600 K with a velocity of 100
m/s and leaves at 1 MPa and 280 K with a velocity of 150 m/s at a rate
of 5 kg/s. Heat is being lost to the surroundings at 25°C at a rate of
60 kW. Using the generalized charts, determine (a) the power output of
the turbine and (b) the exergy destruction associated with the process.
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12–80
Reconsider Prob. 12–79. Using EES (or other) software, solve the
problem assuming steam is the working fluid by using the generalized
chart method and EES data for steam. Plot the power output and the
exergy destruction rate for these two calculation methods against the
turbine exit pressure as it varies over the range 0.1 to 1 MPa when the
turbine exit temperature is 455 K.
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12–81E
Argon gas enters a turbine at 1000 psia and 1000 R with a velocity of
300 ft/s and leaves at 150 psia and 500 R with a velocity of 450 ft/s at
a rate of 12 lbm/s. Heat is being lost to the surroundings at 75°F at a
rate of 80 Btu/s. Using the generalized charts, determine (a) the power
output of the turbine and (b) the exergy destruction associated with
the process. Answers: (a) 922 hp, (b) 121.5 Btu/s
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12–82
An adiabatic 0.2-m3 storage tank that is initially evacuated is
connected to a supply line that carries nitrogen at 225 K and 10 MPa. A
valve is opened, and nitrogen flows into the tank from the supply line.
The valve is closed when the pressure in the tank reaches 10 MPa.
Determine the final temperature in the tank (a) treating nitrogen as an
ideal gas and (b) using generalized charts. Compare your results to the
actual value of 293 K.
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12–83
For a homogeneous (single-phase) simple pure substance, the pressure
and temperature are independent properties, and any property can be
expressed as a function of these two properties. Taking v = v(P, T),
show that the change in specific volume can be expressed in terms of the
volume expansivity b and isothermal compressibility a as

Also, assuming constant average values for b and a, obtain a relation
for the ratio of the specific volumes v2/v1 as a homogeneous system
undergoes a process from state 1 to state 2.
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12–84 Repeat Prob. 12–83 for an isobaric process.
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12–85
The volume expansivity of water at 20°C is b = 0.207 x 10-6 K-1.
Treating this value as a constant, determine the change in volume of 1
m3 of water as it is heated from 10°C to 30°C at constant pressure.
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12–86
The volume expansivity b values of copper at 300 K and 500 K are 49.2 x
10-6 K-1 and 54.2 x 10-6 K-1, respectively, and b varies almost
linearly in this temperature range. Determine the percent change in the
volume of a copper block as it is heated from 300 K to 500 K at
atmospheric pressure.
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12–87
Starting with mJT = (1/cp) [T(dv/dT)p - v] and noting that Pv = ZRT,
where Z = Z(P, T) is the compressibility factor, show that the position
of the Joule-Thomson coefficient inversion curve on the T-P plane is
given by the equation (dZ/dT)P = 0.
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12–88
Consider an infinitesimal reversible adiabatic compression or expansion
process. By taking s = s(P, v) and using the Maxwell relations, show
that for this process Pvk = constant, where k is the isentropic
expansion exponent defined as

Also, show that the isentropic expansion exponent k reduces to the
specific heat ratio cp/cv for an ideal gas.
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12–89
Refrigerant-134a undergoes an isothermal process at 60°C from 3 to 0.1
MPa in a closed system. Determine the work done by the refrigerant-134a
by using the tabular (EES) data and the generalized charts, in kJ/kg.
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12–90
Methane is contained in a piston–cylinder device and is heated at
constant pressure of 4 MPa from 100 to 350°C. Determine the heat
transfer, work and entropy change per unit mass of the methane using (a)
the ideal-gas assumption, (b) the generalized charts, and (c) real
fluid data from EES or other sources.
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12–91
A substance whose Joule-Thomson coefficient is negative is throttled to
a lower pressure. During this process, (select the correct statement)
(a) the temperature of the substance will increase. (b) the temperature
of the substance will decrease. (c) the entropy of the substance will
remain constant. (d) the entropy of the substance will decrease. (e) the
enthalpy of the substance will decrease.
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12–92
Consider the liquid–vapor saturation curve of a pure substance on the
P-T diagram. The magnitude of the slope of the tangent line to this
curve at a temperature T (in Kelvin) is (a) proportional to the enthalpy
of vaporization hfg at that temperature. (b) proportional to the
temperature T. (c) proportional to the square of the temperature T. (d)
proportional to the volume change vfg at that temperature. (e) inversely
proportional to the entropy change sfg at that temperature.
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12–93
Based on the generalized charts, the error involved in the enthalpy of
CO2 at 350 K and 8 MPa if it is assumed to be an ideal gas is (a) 0 (b)
20% (c) 35% (d) 26% (e) 65%
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12–94
Based on data from the refrigerant-134a tables, the Joule-Thompson
coefficient of refrigerant-134a at 0.8 MPa and 100°C is approximately
(a) 0 ( b) -5°C/MPa (c) 11°C/MPa (d)8°C/MPa (e) 26°C/MPa
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12–95
For a gas whose equation of state is P(v - b) = RT, the specified heat
difference cp -cv is equal to (a) R (b) R - b (c) R + b (d) 0 (e) R(1 +
v/b)
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