Thermodynamics: An Engineering Approach

Yunus A. Çengel, Michael A. Boles

Chapter 12 Thermodynamic Property Relations - all with Video Answers

Educators

+ 3 more educators

Chapter Questions

00:53

Problem 1

What is the difference between partial differentials and ordinary differentials?

Farnaz Mohseni

Numerade Educator

00:47

Problem 2

Consider a function $z(x, y)$ and its partial derivative $(\partial z / \partial y)_{x^{*}}$ Under what conditions is this partial derivative equal to the total derivative $d z / d y ?$

Farnaz Mohseni

Numerade Educator

00:37

Problem 3

Consider a function $z(x, y)$ and its partial derivative $(\partial z / \partial y)_{x^{*}}$ If this partial derivative is equal to zero for all values of $x,$ what does it indicate?

Farnaz Mohseni

Numerade Educator

00:58

Consider the function $z(x, y),$ its partial derivatives $(\partial z / \partial x)_{y}$ and $(\partial z / \partial y)_{x},$ and the total derivative $d z / d x$
(a) How do the magnitudes $(\partial x)_{y}$ and $d x$ compare?
(b) How do the magnitudes $(\partial z)_{y}$ and $d z$ compare?
(c) Is there any relation among $d z$ ( $\partial z_{x},$ and $(\partial z)_{y} ?$

Farnaz Mohseni

Numerade Educator

06:35

Problem 5

Consider air at $350 \mathrm{K}$ and $0.75 \mathrm{m}^{3} / \mathrm{kg} .$ 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.

Farnaz Mohseni

Numerade Educator

03:59

Problem 6

Repeat Problem $12-5$ for helium.

Farnaz Mohseni

Numerade Educator

03:43

Problem 7

Nitrogen gas at $400 \mathrm{K}$ and 300 kPa behaves as an ideal gas. Estimate the $c_{p}$ and $c_{v}$ 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 $\mathrm{A}-2 b$.

Farnaz Mohseni

Numerade Educator

05:00

Problem 8

Nitrogen gas at $800 \mathrm{R}$ and 50 psia behaves as an ideal gas. Estimate the $c_{p}$ and $c_{v}$ of the nitrogen at this state, using enthalpy and internal energy data from Table $\mathrm{A}-18 \mathrm{E},$ and compare them to the values listed in Table

Farnaz Mohseni

Numerade Educator

04:59

Problem 9

Consider an ideal gas at $300 \mathrm{K}$ and $100 \mathrm{kPa} .$ As a result of some disturbance, the conditions of the gas change to $305 \mathrm{K}$ and $96 \mathrm{kPa}$. Estimate the change in the specific.

Farnaz Mohseni

Numerade Educator

06:08

Problem 10

Using the equation of state $P(v-a)=R T,$ verify $(a)$ the cyclic relation and ( $b$ ) the reciprocity relation at constant $v$

Farnaz Mohseni

Numerade Educator

02:08

Problem 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.

Sheh Lit Chang

University of Washington

05:06

Problem 12

Verify the validity of the last Maxwell relation (Eq. $12-19$ ) for refrigerant- 134 a at $50^{\circ} \mathrm{C}$ and $0.7 \mathrm{MPa}$.

Keshav Singh

Numerade Educator

01:00

Problem 13

Reconsider Prob. $12-12 .$ Using EES (or other)
software, verify the validity of the last Max well relation for refrigerant-134a at the specified state.

Mayukh Banik

Numerade Educator

04:16

Problem 14

Verify the validity of the last Maxwell relation (Eq. $12-19$ ) for steam at $600^{\circ} \mathrm{F}$ and 275 psia.

Sheh Lit Chang

University of Washington

01:22

Problem 15

Using the Maxwell relations, determine a relation for $(\partial s / \partial P)_{T}$ for a gas whose equation of state is $P(v-b)=$
$R T .

Sheh Lit Chang

University of Washington

01:02

Problem 16

$12-17 \quad$ Using the Maxwell relations and the ideal-gas equation of state, determine a relation for $(\partial s / \partial v)_{T}$ for an ideal gas. Answer: $R / v$.

Sheh Lit Chang

University of Washington

00:45

Problem 17

$12-17 \quad$ Using the Maxwell relations and the ideal-gas equation of state, determine a relation for $(\partial s / \partial v)_{T}$ for an ideal gas.

Sheh Lit Chang

University of Washington

06:23

Problem 18

Prove that $$\left(\frac{\partial P}{\partial T}\right)_{s}=\frac{k}{k-1}\left(\frac{\partial P}{\partial T}\right).$$

Sheh Lit Chang

University of Washington

00:55

Problem 19

What is the value of the Clapeyron equation in thermodynamics?

Sheh Lit Chang

University of Washington

00:26

Problem 20

Does the Clapeyron equation involve any approximations, or is it exact?

Sheh Lit Chang

University of Washington

04:21

Problem 21

Using the Clapeyron equation, estimate the enthalpy of vaporization of refrigerant- 134 a at $40^{\circ} \mathrm{C},$ and compare it to the tabulated value.

Sheh Lit Chang

University of Washington

02:35

Problem 22

Reconsider Prob. $12-21 .$ Using EES (or other) software, plot the enthalpy of vaporization of refrigerant-134a as a function of temperature over the temperature range -20 to $80^{\circ} \mathrm{C}$ by using the Clapeyron equation and the refrigerant-134a data in EES. Discuss your results.

Naman Kumar

Numerade Educator

04:47

Problem 23

Using the Clapeyron equation, estimate the enthalpy of vaporization of steam at $300 \mathrm{kPa}$, and compare it to the tabulated value

Sheh Lit Chang

University of Washington

08:06

Problem 24

Determine the $h_{f g}$ of refrigerant-134a at $10^{\circ} \mathrm{F}$ on the basis of ( $a$ ) the Clapeyron equation and $(b)$ the Clapeyron-Clausius equation. Compare your results to the tabulated $h_{f g}$ value.

Sheh Lit Chang

University of Washington

05:28

Problem 25

$0.5-16 \mathrm{m}$ of a saturated vapor is converted to saturated liquid by being cooled in a weighted piston-cylinder device maintained at 50 psia. During the phase conversion, the system volume decreases by $1.5 \mathrm{ft}^{3} ; 250$ Btu of heat are removed; and the temperature remains fixed at $15^{\circ} \mathrm{F}$. Estimate the boiling point temperature of this substance when its pressure is 60 psia.

Sheh Lit Chang

University of Washington

04:57

Problem 26

Estimate the saturation pressure $P_{\mathrm{su}}$ of the substance in Prob. $12-25 \mathrm{E}$ when its temperature is $20^{\circ} \mathrm{F}$.

Sheh Lit Chang

University of Washington

02:41

Problem 27

Estimate the $s_{f g}$ of the substance in Problem $12-25 \mathrm{E}$ .

Sheh Lit Chang

University of Washington

05:31

Problem 28

A table of properties for methyl chloride lists the saturation pressure as 116.7 psia at $100^{\circ} \mathrm{F}$. At $100^{\circ} \mathrm{F}$, this table also lists $h_{f_{R}}=154.85 \mathrm{Btu} / \mathrm{lbm},$ and $v_{f g}=0.86332 \mathrm{ft}^{3} / \mathrm{bm}$.
Estimate the saturation pressure $P_{\text {sat }}$ of methyl chloride at $90^{\circ} \mathrm{F}$ and $110^{\circ} \mathrm{F}$

Sheh Lit Chang

University of Washington

04:41

Problem 29

Using the Clapeyron-Clausius equation and the triplepoint data of water, estimate the sublimation pressure of water at $-30^{\circ} \mathrm{C}$ and compare to the yalue in Table $\mathrm{A}-8$.

Sheh Lit Chang

University of Washington

03:00

Problem 30

Show that $$c_{p, g}-c_{p f}=T\left(\frac{\partial\left(h_{f g} / T\right)}{\partial T}\right)_{p}+v_{f s}\left(\frac{\partial P}{\partial T}\right)_{\mathrm{sat}}$$

Sheh Lit Chang

University of Washington

01:20

Problem 31

Can the variation of specific heat $c_{p}$ with pressure at a given temperature be determined from a knowledge of $P-v-T$ data alone?

Sheh Lit Chang

University of Washington

04:37

Problem 32

Estimate the volume expansivity $\beta$ and the isothermal compressibility $\alpha$ of refrigerant- 134 a at $200 \mathrm{kPa}$ and $30^{\circ} \mathrm{C}$.

Sheh Lit Chang

University of Washington

03:37

Problem 33

Estimate the specific heat difference $c_{p}-c_{v}$ for liquid water at $15 \mathrm{MPa}$ and $80^{\circ} \mathrm{C}$.

Sheh Lit Chang

University of Washington

02:44

Problem 34

Determine the change in the internal energy of air, in $\mathrm{kJ} / \mathrm{kg},$ as it undergoes a change of state from $100 \mathrm{kPa} 20^{\circ} \mathrm{C}$
to $600 \mathrm{kPa}$ and $300^{\circ} \mathrm{C}$ using the equation of state $P(v-a)=$ $R T$ where $a=1 \mathrm{m}^{3} / \mathrm{kg},$ and compare the result to the value obtained by using the ideal gas equation of state.

Sheh Lit Chang

University of Washington

03:32

Problem 35

Determine the change in the enthalpy of air, in $\mathrm{kJ} /$ $\mathrm{kg},$ as it undergoes a change of state from $100 \mathrm{kPa}$ and $34^{\circ} \mathrm{C}$ to $800 \mathrm{kPa}$ and $420^{\circ} \mathrm{C}$ using the equation of state $P(v-a)=$ $R T$ where $a=0.01 \mathrm{m}^{3} / \mathrm{kg},$ and compare the result to the value obtained by using the ideal gas equation of state.

Sheh Lit Chang

University of Washington

02:38

Problem 36

Determine the change in the entropy of air, in $\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K},$ as it undergoes a change of state from $100 \mathrm{kPa}$ and $20^{\circ} \mathrm{C}$ to $600 \mathrm{kPa}$ and $300^{\circ} \mathrm{C}$ using the equation of state $P(v-a)$ $=R T$ where $a=0.01 \mathrm{~m}^{3} / \mathrm{~kg},$ and compare the result to the value obtained by using the ideal gas equation of state.

Sheh Lit Chang

University of Washington

02:09

Problem 37

Determine the change in the internal energy of helium, in $\mathrm{kJ} / \mathrm{kg},$ as it undergoes a change of state from $100 \mathrm{kPa}$ and $20^{\circ} \mathrm{C}$ to $600 \mathrm{kPa}$ and $300^{\circ} \mathrm{C}$ using the equation of state $P(v-a)=R T$ where $a=0.01 \mathrm{m}^{3} / \mathrm{kg},$ and compare the result to the value obtained by using the ideal gas equation of state.

Sheh Lit Chang

University of Washington

03:30

Problem 38

Determine the change in the enthalpy of helium, in $\mathrm{kJ} / \mathrm{kg},$ as it undergoes a change of state from $150 \mathrm{kPa}$ and $20^{\circ} \mathrm{C}$ to $750 \mathrm{kPa}$ and $380^{\circ} \mathrm{C}$ using the equation of state $P(v-a)=R T$ where $a=0.01 \mathrm{m}^{3} / \mathrm{kg},$ and compare the result to the value obtained by using the ideal gas equation of state.

Sheh Lit Chang

University of Washington

02:25

Problem 39

Determine the change in the entropy of helium, in $\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K},$ as it undergoes a change of state from $100 \mathrm{kPa}$ and $20^{\circ} \mathrm{C}$ to $600 \mathrm{kPa}$ and $300^{\circ} \mathrm{C}$ using the equation of state $P(v-a)=R T$ where $a=0.01 \mathrm{m}^{3} / \mathrm{kg},$ and compare the result to the value obtained by using the ideal gas equation of state.

Sheh Lit Chang

University of Washington

04:20

Problem 40

Derive expressions for $(a) \Delta u,(b) \Delta h,$ and $(c) \Delta s$ for a gas whose equation of state is $P(v-a)=R T$ for an isothermal process.

Sheh Lit Chang

University of Washington

07:39

Problem 41

Derive expressions for $(a) \Delta u,(b) \Delta h,$ and $(c) \Delta s$ for a gas that obeys the van der Waals equation of state for an isothermal process.

Sheh Lit Chang

University of Washington

08:19

Problem 42

Derive an expression for the specific heat difference $c_{p}-c_{v}$ for $(a)$ an ideal gas, $(b)$ a van der Waals gas, and
$(c)$ an in compressible substance.

Sheh Lit Chang

University of Washington

03:23

Problem 43

$\text { Show that } c_{p}-c_{v}=T\left(\frac{\partial P}{\partial T}\right)_{v}\left(\frac{\partial v}{\partial T}\right)_{P}$.

Sheh Lit Chang

University of Washington

06:08

Problem 44

Temperature may alternatively be defined as $$T=\left(\frac{\partial u}{\partial s}\right)$$.Prove that this definition reduces the net entropy change of two constant-volume systems filled with simple compressible substances to zero as the two systems approach thermal equilibrium.

Sheh Lit Chang

University of Washington

03:08

Problem 45

Derive a relation for the volume expansivity $\beta$ and the isothermal compressibility $\alpha(a)$ for an ideal gas and
(b) for a gas whose equation of state is $P(v-a)=R T$

Sheh Lit Chang

University of Washington

06:49

Problem 46

Derive an expression for the isothermal compressibility of a substance whose equation of state is $$P=\frac{R T}{v-b}-\frac{a}{v(v+b) T^{1 / 2}}$$.where $a$ and $b$ are empirical constants.

Sheh Lit Chang

University of Washington

03:56

Problem 47

Derive an expression for the volume expansivity of a substance whose equation of state is $$P=\frac{R T}{v-b}-\frac{a}{v^{2} T}$$ where $a$ and $b$ are empirical constants.

Sheh Lit Chang

University of Washington

02:39

Problem 48

Show that $$\beta=\alpha(\partial P / \partial T)_{v}.$$

Sheh Lit Chang

University of Washington

04:21

Problem 49

Demonstrate that $$k=\frac{c_{p}}{c_{v}}=-\frac{v \alpha}{(\partial v / \partial P)_{s}}$$

Sheh Lit Chang

University of Washington

06:13

Problem 50

The Helmholtz function of a substance has the form $$a=-R T \ln \frac{v}{v_{0}}-c T_{0}\left(1-\frac{T}{T_{0}}+\frac{T}{T_{0}} \ln \frac{T}{T_{0}}\right)$$ where $T_{0}$ and $v_{0}$ are the temperature and specific volume at a reference state. Show how to obtain $P, h, s, c_{v}$ and $c_{p}$ from this expression.

Sheh Lit Chang

University of Washington

04:48

Problem 51

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.

Sheh Lit Chang

University of Washington

00:48

Problem 52

What does the Joule-Thomson coefficient represent?

Sheh Lit Chang

University of Washington

01:09

Problem 53

Describe the inversion line and the maximum inversion temperature.

Sheh Lit Chang

University of Washington

00:46

Problem 54

The pressure of a fluid always decreases during an adiabatic throttling process. Is this also the case for the temperature?

Sheh Lit Chang

University of Washington

00:19

Problem 55

Does the Joule-Thomson coefficient of a substance shange with temperature at a fixed pressure?

Sheh Lit Chang

University of Washington

View

Problem 56

Will the temperature of helium change if it is throttled adiabatically from $300 \mathrm{K}$ and $600 \mathrm{kPa}$ to $150 \mathrm{kPa} ?$

Ankur S

Numerade Educator

02:01

Problem 57

Estimate the Joule-Thomson coefficient of nitrogen at $(a) 120$ psia and $350 \mathrm{R},$ and (b) 1200 psia and 700 R. Use nitrogen properties from EES or other source.

Jincy M Saji

Numerade Educator

01:40

Problem 58

Reconsider Prob. 12-57E. 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.

Jincy M Saji

Numerade Educator

01:09

Problem 59

Steam is throttled slightly from $2 \mathrm{MPa}$ and $500^{\circ} \mathrm{C}$ Will the temperature of the steam increase, decrease, or remain the same during this process?

Anand Jangid

Numerade Educator

11:33

Problem 60

Estimate the Joule-Thomson coefficient of steam at
$(a) 3$ MPa and $300^{\circ} \mathrm{C}$ and $(b) 6 \mathrm{MPa}$ and $500^{\circ} \mathrm{C}$

Sheh Lit Chang

University of Washington

03:34

Problem 61

Estimate the Joule-Thomson-coefficient of refrigerant-134a at 40 psia and $60^{\circ} \mathrm{F}$.

Sheh Lit Chang

University of Washington

02:15

Problem 62

Demonstrate that the Joule-Thomson coefficient is given by $$\mu=\frac{T^{2}}{c_{p}}\left[\frac{\partial(v / T)}{\partial T}\right]_{P}$$, $$\mu=\frac{T^{2}}{c_{p}}\left[\frac{\partial(v / T)}{\partial T}\right]_{p}$$

Sheh Lit Chang

University of Washington

02:06

Problem 63

Consider a gas whose equation of state is $P(v-a)=$
$R T,$ where $a$ is a positive constant. Is it possible to cool this gas by throttling?

Sheh Lit Chang

University of Washington

05:06

Problem 64

Derive a relation for the Joule-Thomson coefficient and the inversion temperature for a gas whose equation of state is $\left(P+a / v^{2}\right) v=R T$

Sheh Lit Chang

University of Washington

00:55

Problem 65

On the generalized enthalpy departure chart, the normalized enthalpy departure values seem to approach zero as the reduced pressure $P_{R}$ approaches zero. How do you explain this behavior?

Keshav Singh

Numerade Educator

01:11

Problem 66

Why is the generalized enthalpy departure chart prepared by using $P_{B}$ and $T_{B}$ as the parameters instead of $P$ and $T ?$

Keshav Singh

Numerade Educator

05:29

Problem 67

The Helmholtz function of a substance has the form $12-67$ Determine the enthalpy of nitrogen, in $\mathrm{kJ} / \mathrm{kg},$ at $175 \mathrm{K}$ and $8 \mathrm{MPa}$ using $(a)$ data from the ideal-gas nitrogen table and ( $b$ ) the generalized enthalpy departure chart. Compare your results to the actual value of $125.5 \mathrm{kJ} / \mathrm{kg}$.

Keshav Singh

Numerade Educator

05:55

Problem 68

Determine the enthalpy of nitrogen, in Btu/lbm, at $400 \mathrm{R}$ 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 $177.8 \mathrm{Btu} / \mathrm{lbm}$.

Keshav Singh

Numerade Educator

09:51

Problem 69

Determine the enthalpy change and the entropy change of $\mathrm{CO}_{2}$ per unit mass as it undergoes a change of state from $250 \mathrm{K}$ and $7 \mathrm{MPa}$ to $280 \mathrm{K}$ and $12 \mathrm{MPa},(a)$ by assuming ideal-gas behavior and ( $b$ ) by accounting for the deviation from ideal-gas behavior.

Keshav Singh

Numerade Educator

14:30

Problem 70

Saturated water vapor at $400^{\circ} \mathrm{F}$ is expanded while its pressure is kept constant until its temperature is $800^{\circ} \mathrm{F}$. Calculate the change in the specific enthalpy and entropy using ( $a$ ) the departure charts, and ( $b$ ) the property tables.

Keshav Singh

Numerade Educator

12:43

Problem 71

Water vapor at $1000 \mathrm{kPa}$ and $600^{\circ} \mathrm{C}$ is expanded to 500 kPa and $400^{\circ} \mathrm{C}$. Calculate the change in the specific entropy and enthalpy of this water vapor using the departure charts and the property tables.

Keshav Singh

Numerade Educator

06:28

Problem 72

Methane is compressed adiabatically by a steady-flow compressor from $0.8 \mathrm{MPa}$ and $-10^{\circ} \mathrm{C}$ to $6 \mathrm{MPa}$ and $175^{\circ} \mathrm{C}$ at a rate of $0.33 \mathrm{kg} / \mathrm{s}$. Using the generalized charts, determine the required power input to the compressor.

Keshav Singh

Numerade Educator

07:07

Problem 73

Carbon dioxide enters an adiabatic nozzle at $8 \mathrm{MPa}$ and $450 \mathrm{K}$ with a low velocity and leaves at $2 \mathrm{MPa}$ and $350 \mathrm{K}$ Using the generalized enthalpy departure chart, determine the exit velocity of the carbon dioxide.

Keshav Singh

Numerade Educator

02:12

Problem 74

Reconsider Prob. $12-73 .$ 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.

Jincy M Saji

Numerade Educator

13:19

Problem 75

Oxygen is adiabatically and reversibly expanded in a nozzle from 200 psia and $600^{\circ} \mathrm{F}$ to 70 psia. Determine the velocity at which the oxygen leaves the nozzle, assuming that it enters with negligible velocity, treating the oxygen as an ideal gas with temperature variable specific heats and using the departure charts. Answers: $1738 \mathrm{ft} / \mathrm{s}, 1740 \mathrm{ft} / \mathrm{s}$

Keshav Singh

Numerade Educator

11:23

Problem 76

Propane is compressed isothermally by a pistoncylinder device from $100^{\circ} \mathrm{C}$ and $1 \mathrm{MPa}$ to $4 \mathrm{MPa}$. Using the generalized charts, determine the work done and the heat transfer per unit mass of propane.

Keshav Singh

Numerade Educator

12:46

Problem 77

Reconsider Prob. $12-76 .$ 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.

Khoobchandra Agrawal

Numerade Educator

09:10

Problem 78

Determine the exergy destruction associated with the process described in Prob. $12-76 .$ Assume $T_{0}=25^{\circ} \mathrm{C}$

Mohammad Mehran

Numerade Educator

10:43

Problem 79

A $0.05-\mathrm{m}^{3}$ well-insulated rigid tank contains oxygen at $175 \mathrm{K}$ and 6 MPa. A paddle wheel placed in the tank is turned on, and the temperature of the oxygen rises to $225 \mathrm{K}$ Using the generalized charts, determine ( $a$ ) the final pressure in the tank, and ( $b$ ) the paddle-wheel work done during this process.

Keshav Singh

Numerade Educator

04:26

Problem 80

Derive relations for $(a) \Delta u,(b) \Delta h,$ and $(c) \Delta s$ of a gas that obeys the equation of state $\left(P+a / v^{2}\right) v=R T$ for an isothermal process.

Sheh Lit Chang

University of Washington

03:20

Problem 81

Starting with the relation $d h=T d s+v d P,$ 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.

Sheh Lit Chang

University of Washington

02:26

Problem 82

Show that $$c_{v}=-T\left(\frac{\partial v}{\partial T}\right)_{s}\left(\frac{\partial P}{\partial T}\right)_{v} \text { and } c_{p}=T\left(\frac{\partial P}{\partial T}\right)_{s}\left(\frac{\partial v}{\partial T}\right)$$

Sheh Lit Chang

University of Washington

01:30

Problem 83

Temperature and pressure may be defined as $$T=\left(\frac{\partial u}{\partial s}\right)_{v} \text { and } P=-\left(\frac{\partial u}{\partial v}\right).$$ Using these definitions, prove that for a simple compressible substance $$\left(\frac{\partial s}{\partial v}\right)_{u}=\frac{P}{T}.$$

Sheh Lit Chang

University of Washington

04:53

Problem 84

For ideal gases, the development of the constant pressure specific heat yields $$\left(\frac{\partial h}{\partial P}\right)_{T}=0$$ Prove this by using the definitions of pressure and temperature, $T=(\partial u / \partial s)_{v}$ and $P=-(\partial u / \partial v)_{s}$.

Sheh Lit Chang

University of Washington

03:10

Problem 85

Starting with $\mu_{\mathrm{JT}}=\left(1 / c_{p}\right)\left[T(\partial v / \partial T)_{p}-v\right]$ and noting that $P v=Z R T,$ 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 $(\partial Z / \partial T)_{P}=0$.

Sheh Lit Chang

University of Washington

03:45

Problem 86

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 $\beta$ and isothermal compressibility $\alpha$ as \[\frac{d v}{v}=\beta d T=\alpha d P\] Also, assuming constant average values for $\beta$ and $\alpha,$ obtain a relation for the ratio of the specific volumes $v_{2} / v_{1}$ as a homogeneous system undergoes a process from state 1 to state 2.

Sheh Lit Chang

University of Washington

01:36

Problem 87

Repeat Prob. $12-86$ for an isobaric process.

Sheh Lit Chang

University of Washington

08:11

Problem 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 $P v^{k}=$ constant, where $k$ is the isentropic expansion exponent defined as $$k=\frac{V}{P}\left(\frac{\partial P}{\partial V}\right)$$ Also, show that the isentropic expansion exponent $k$ reduces to the specific heat ratio $c_{p} / c_{v}$ for an ideal gas.

Sheh Lit Chang

University of Washington

07:39

Problem 89

Estimate the $c_{p}$ of nitrogen at $300 \mathrm{kPa}$ and $400 \mathrm{K}$ using $(a)$ the relation in Prob. $12-88,$ and $(b)$ its definition. Compare your results to the value listed in Table $A-2 b$.

Sheh Lit Chang

University of Washington

03:37

Problem 90

Steam is throttled from 2.5 MPa and $400^{\circ} \mathrm{C}$ to 1.2 MPa. Estimate the temperature change of the steam during this process and the average Joule-Thomson coefficient.

Sheh Lit Chang

University of Washington

03:51

Problem 91

The volume expansivity $\beta$ values of copper at $300 \mathrm{K}$ and $500 \mathrm{K}$ are $49.2 \times 10^{-6} \mathrm{K}^{-1}$ and $54.2 \times 10^{-6} \mathrm{K}^{-1},$ respectively, and $\beta$ varies almost linearly in this temperature range. Determine the percent change in the volume of a copper block as it is heated from $300 \mathrm{K}$ to $500 \mathrm{K}$ at atmospheric pressure.

Sheh Lit Chang

University of Washington

04:53

Problem 92

Keshav Singh

Numerade Educator

12:35

Problem 93

An adiabatic $0.2-\mathrm{m}^{3}$ storage tank that is initially evacuated is connected to a supply line that carries nitrogen ane at $225 \mathrm{K}$ and $10 \mathrm{MPa}$. A valve is opened, and nitrogen flows depe 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 $293 \mathrm{K}$.

Keshav Singh

Numerade Educator

13:16

Problem 94

Methane is to be adiabatically and reversibly compressed from 50 psia and $100^{\circ} \mathrm{F}$ to 500 psia. Calculate the specific work required for this compression treating the methane as an ideal gas with variable specific heats and using the departure charts.

Keshav Singh

Numerade Educator

02:14

Problem 95

Refrigerant-134a undergoes an isothermal process at $40^{\circ} \mathrm{C}$ from 2 to $0.1 \mathrm{MPa}$ in a closed system. Determine the work done by the refrigerant-134a by using the tabular (EES) data and the generalized charts, in $\mathrm{kJ} / \mathrm{kg}$.

Dominador Tan

Numerade Educator

10:06

Problem 96

A rigid tank contains $1.2 \mathrm{m}^{3}$ of argon at $-100^{\circ} \mathrm{C}$ and 1 MPa. Heat is now transferred to argon until the temperature in the tank rises to $0^{\circ} \mathrm{C}$. Using the generalized charts,determine $(a)$ the mass of the argon in the tank, $(b)$ the final pressure, and (c) the heat transfer.

Keshav Singh

Numerade Educator

14:30

Problem 97

Methane is contained in a piston-cylinder device and is heated at constant pressure of 5 MPa from 100 to $250^{\circ} \mathrm{C}$. 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.

Keshav Singh

Numerade Educator

14:30

Problem 98

Methane at 50 psia and $100^{\circ} \mathrm{F}$ is compressed in a steady-flow device to 500 psia and $1100^{\circ} \mathrm{F}$. Calculate the change in the specific entropy of the methane and the specific work required for this compression ( $a$ ) treating the methane as an ideal gas with temperature variable specific heats, and (b) using the departure charts.

Keshav Singh

Numerade Educator

02:19

Problem 99

Determine the second-law efficiency of the compression process described in Prob. $12-98 \mathrm{E}$. Take $T_{0}=77^{\circ} \mathrm{F}$.

Keshav Singh

Numerade Educator

01:17

Problem 100

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.

Sheh Lit Chang

University of Washington

02:02

Problem 101

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 $h_{f g}$ at that temperature.
(b) proportional to the temperature $T$
$(c)$ proportional to the square of the temperature $T$
$(d)$ proportional to the volume change $v_{f g}$ at that temperature.
$(e)$ inversely proportional to the entropy change $s_{f g}$ at that temperature.

Sheh Lit Chang

University of Washington

01:17

Problem 102

Based on the generalized charts, the error involved in the enthalpy of $\mathrm{CO}_{2}$ at $300 \mathrm{K}$ and $5 \mathrm{MPa}$ if it is assumed to be an ideal gas is
$(a) 0 \%$
(b) $9 \%$
$(c) 16 \%$
$(d) 22 \%$
$(e) 27 \%$

Jincy M Saji

Numerade Educator

01:00

Problem 103

Based on data from the refrigerant- $134 a$ tables, the Joule-Thompson coefficient of refrigerant-134a at $0.8 \mathrm{MPa}$ and $100^{\circ} \mathrm{C}$ is approximately
$(a) 0$
$(b)-5^{\circ} \mathrm{C} / \mathrm{MPa}$
$(c) \quad 11^{\circ} \mathrm{C} / \mathrm{MPa}$
$(d) 8^{\circ} \mathrm{C} / \mathrm{MPa}$
$(e) 26^{\circ} \mathrm{C} / \mathrm{MPa}$

Jincy M Saji

Numerade Educator

01:53

Problem 104

$12-104 \quad$ For a gas whose equation of state is $P(v-b)=R T$ the specified heat difference $c_{p}-c_{v}$ is equal to
$(a) R$
(b) $R-b$
$(c) R+b$
$(d) 0$
$(e) R(1+v / b)$

Sheh Lit Chang

University of Washington

03:53

Problem 105

Consider the function $z=z(x, y) .$ Write an essay on the physical interpretation of the ordinary derivative $d z / d x$ and the partial derivative $(\partial z / \partial x)_{y} .$ Explain how these two derivatives are related to each other and when they become equivalent.

Paul Teng

Numerade Educator

01:18

Problem 106

There have been several attempts to represent the thermodynamic relations geometrically, the best known of these being Koenig's thermodynamic square shown in the figure. There is a systematic way of obtaining the four Maxwell relations as well as the four relations for $d u, d h$ $d g,$ and $d a$ from this figure. By comparing these relations to Koenig's diagram, come up with the rules to obtain these eight thermodynamic relations from this diagram.

Manik Pulyani

Numerade Educator

View

Problem 107

Several attempts have been made to express the partial derivatives of the most common thermodynamic properties in a compact and systematic manner in terms of measurable properties. The work of P. W. Bridgman is perhaps the most fruitful of all, and it resulted in the well-known Bridgman's table. The 28 entries in that table are sufficient to express the partial derivatives of the eight common properties $P, T, v, s, u, h, f,$ and $g$ in terms of the six properties $P, v, T$ $c_{p}, \beta,$ and $\alpha,$ which can be measured directly or indirectly with relative ease. Obtain a copy of Bridgman's table and explain, with examples, how it is used.

Eduard Sanchez

Numerade Educator