Determine the change in the entropy of helium, in kJ/kg,

Problem 1P What is the difference between the classical and the statistical approaches to thermodynamics?

Problem 3P Consider a function z(x, y) and its partial derivative (?z/?y)x. If this partial derivative is equal to zero for all values of x, what does it indicate?

Problem 104P 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(l + V/b)

Problem 2P 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?

12-5 Consider air at \(350 \mathrm{~K} \text { and } 0.75 \mathrm{~m}^{3} / \mathrm{kg}\). Using Eq. 12-3, determine the change in pressure corresponding to an increase of 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. Equation Transcription: Text Transcription: 350 K and 0.75 m3/kg

Problem 4P Consider the function z(x, y), its partial derivatives (?z/?x)yand (?z/?x)x and the total derivative dz/dx. (a) How do the magnitudes (?x)yand dx compare? ________________ (b) How do the magnitudes (?z)yand dz compare? ________________ (c) Is there any relation among dz, (?z)x, and (?z)y?

Repeat Problem 12–5 for helium.

Problem 8P Nitrogen gas at 800 R and 50 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.

Problem 7P Nitrogen gas at 400 K and 300 kPa behaves as an ideal gas. Estimate the cp and cvof 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.

Problem 11P Derive a relation for the slope of the V = constant lines on a T-Pdiagram for a gas that obeys the van der Waals equation of state.

Consider an ideal gas at \(300 \mathrm{~K} \text { and } 100 \mathrm{kPa}\). As a result of some disturbance, the conditions of the gas change to \(305 \mathrm{~K} \text { and } 96 \mathrm{kPa}\). Estimate the change in the specific volume of the gas using (a) Eq. 12–3 and (b) the ideal-gas relation at each state. Equation Transcription: Text Transcription: 300 K and 100 kPa 305 K and 96 kPa

Problem 10P Using the equation of state P(V? a) = RT, verify (a) the cyclic relation and (b) the reciprocity relation at constant V.

Problem 12P Verify the validity of the last Maxwell relation (Eq. 12–19) for refrigerant-134a at 50°C and 0.7 MPa.

Problem 15P Using the Maxwell relations, determine a relation for (?s/?P)T for a gas whose equation of state is P(V? b) = RT.

Verify the validity of the last Maxwell relation (Eq. 12–19) for steam at \(600^{\circ} \mathrm{F} \text { and } \mathrm{psia}\) Equation Transcription: Text Transcription: 600°F and psia

Problem 16P Using the Maxwell relations, determine a relation for (?s/?v)T for a gas whose equation of state is (P? a/V2) (V? b) = RT.

12-18 Prove that \(\left(\frac{\partial P}{\partial T}\right)_{S}=\frac{k}{k-1}\left(\frac{\partial P}{\partial T}\right)_{V}\) Equation Transcription: Text Transcription: (\frac{\partial P \partial T)_S=\frac{k k-1 (\frac{\partial P \partial T)_V

Problem 17P Using the Maxwell relations and the ideal-gas equation of state, determine a relation for (?s/?v)T for an ideal gas.

Problem 19P What is the value of the Clapeyron equation in thermodynamics?

Problem 21P Using the Clapeyron equation, estimate the enthalpy of vaporization of refrigerant-134a at 40°C, and compare it to the tabulated value.

Problem 23P Using the Clapeyron equation, estimate the enthalpy of vaporization of steam at 300 kPa, and compare it to the tabulated value.

Problem 20P Does the Clapeyron equation involve any approximations, or is it exact?

12-25E \(0.5-1 b m\) of a saturated vapor is converted to a saturated liquid by being cooled in a weighted piston-cylinder device maintained at 50 psia. During the phase conversion, the system volume decreases by \(1.5 f t^{3} ; 250 B t u\) 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. Equation Transcription: Text Transcription: 0.5-lbm 1.5 ft3;250 Btu 15°F

12-24E Determine the \(h_{f g}\) of refrigerant-134a at \(10^{\circ} F\) ( 5 ) on the basis of the Clapeyron equation and (b) the Clapeyron-Clausius equation. Compare your results to the tabulated \(h_{f g}\) value. Equation Transcription: Text Transcription: hfg 10°F hfg

Problem 27P Estimate the sfgof the substance in Problem 12–27E atl5°F.

12-26E Estimate the saturation pressure \(P_{s a t}\) of the substance in Prob. 12-25E when its temperature is \(20^{\circ} \mathrm{F}\) Equation Transcription: Text Transcription: Psat 20°F

Problem 28P A table of properties for methyl chloride lists the saturation pressure as 116.7 psia at 100°F. At 100°F, this table also lists hfg = 154.85 Btu/lbm, and Vfg = 0.86332 ft3/lbm. Estimate the saturation pressure Psat of methyl chloride at 90°F and 110°F.

12-30 Show that \(c_{p, g}-c_{p, f}=T\left(\frac{\partial\left(h_{f g} / T\right)}{\partial T}\right)_{P}+v_{f g}\left(\frac{\partial P}{\partial T}\right)_{s a t}\) Equation Transcription: Text Transcription: c_p, g-c_p, f=T\left(\frac{\partial(h_f g / T)}{\partial T)_P+v_f g(\frac{\partial P}{\partial T)_s a t

Problem 31P Can the variation of specific heat cp with pressure at a given temperature be determined from a knowledge of P-V-Tdata alone?

Problem 32P Estimate the volume expansivity ? and the isothermal compressibility ? of refrigerant-134a at 200 kPa and 30°C.

Using the Clapeyron-Clausius equation and the triplepoint data of water, estimate the sublimation pressure of water at \(230^{\circ} \mathrm{C}\) and compare to the value in Table A–8. Equation Transcription: Text Transcription: 230°C

Problem 33P Estimate the specific heat difference cp – cv for liquid water at 15 MPa and 80°C.

Problem 34P Determine the change in the internal energy of air, in kJ/kg, as it undergoes a change of state from 100 kPa 20°C to 600 kPa and 300°C using the equation of state P(V ? a) = RTwhere a= 1 m3/kg, and compare the result to the value obtained by using the ideal gas equation of state.

Problem 35P Determine the change in the enthalpy of air, in kJ/kg, as it undergoes a change of state from 100 kPa and 34°C to 800 kPa and 420°C using the equation of state P(V – a) = RT where a = 0.01 m3/kg, and compare the result to the value obtained by using the ideal gas equation of state.

Problem 40P Derive expressions for (a) ?u, (b) ?h, and (c) ?sfor a gas whose equation of state is P(V? a) = RTfor an isothermal process.

Problem 38P Determine the change in the enthalpy of helium, in kJ/kg, as it undergoes a change of state from 150 kPa and 20°C to 750 kPa and 380°C using the equation of state P(V – a) = RT where a = 0.01 m3/kg, and compare the result to the value obtained by using the ideal gas equation of state.

Problem 37P Determine the change in the internal energy of/helium, in kJ/kg, as it undergoes a change of state from 100 kPa and 20°C to 600 kPa and 300°C using the equation of state P(V ? a) = RTwhere a= 0.01 m3/kg, and compare the result to the value obtained by using the ideal gas equation of state.

Problem 39P Determine the change in the entropy of helium, in kJ/kg, as it undergoes a change of state from 100 kPa and 20°C to 600 kPa and 300°C using the equation of state P(V ? a) = RTwhere a =0.01 m3/kg, and compare the result to the value obtained by using the ideal gas equation of state.

Problem 36P Determine the change in the entropy of air, in kJ/kg, as it undergoes a change of state from 100 kPa and 20°C to 600 kPa and 300°C using the equation of state P(V ? a) = RTwhere a= 0.01 m3/kg, and compare the result to the value obtained by using the ideal gas equation of state.

Problem 42P 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.

Problem 41P Derive expressions for (a) ?u, (b) ?h, and (c) ?s for a gas that obeys the van der Waals equation of state for an isothermal process.

12-43 Show that \(c_{p}-c_{v}=T\left(\frac{\partial P}{\partial T}\right)_{v}\left(\frac{\partial v}{\partial T}\right)_{P}\) Equation Transcription: Text Transcription: c_p-c_v=T (\frac{\partial P\partial T\right)_v\left(\frac{\partial v \partial T)_P

12-44 Temperature may alternatively be defined as \(T=\left(\frac{\partial u}{\partial s}\right)_{v}\) Equation Transcription: Text Transcription: T=(\frac{\partial u \partial s)_v

12-46 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 and are empirical constants. Equation Transcription: Text Transcription: P=\frac{R T v-b-\frac{a v(v+b) T^1 / 2

Problem 45P Derive a relation for the volume expansivity ? and the isothermal compressibility ? (a) for an ideal gas and (b) for a gas whose equation of state is P(v – a) = RT.

12-47 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 and are empirical constants. Equation Transcription: Text Transcription: P=\frac{R T v-b-\frac{a v^2 T

12-49 Demonstrate that \(k=\frac{c_{p}}{c_{v}}=-\frac{v \alpha}{(\partial v / \partial P)_{s}}) Equation Transcription: Text Transcription: k=\frac{c_p c_v=-\frac{v \alpha(\partial v / \partial P)_s

The Helmholtz function of a substance has the form \(a=-R T \ln \frac{\nu}{\nu_{0}}-c T_{0}\left(1-\frac{T}{T_{0}}+\frac{T}{T_{0}} \ln \frac{T}{T_{0}}\right)\) 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. Equation Transcription: _ Text Transcription: a = _RT ln frac{v}{v_0} - cT_0 (1 - frac{T}{T_0} + frac{T}{T_0} ln frac{T}{T_0}) T_0 v_0 P, h, s, c_v c_p

Problem 48P Show that ? = ?(?P/?T)V.

Problem 51P 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.

Problem 52P What does the Joule-Thomson coefficient represent?

Problem 53P Describe the inversion line and the maximum inversion temperature.

Problem 54P The pressure of a fluid always decreases during an adiabatic throttling process. Is this also the case for the temperature?

Problem 56P Will the temperature of helium change if it is throttled adiabatically from 300 K and 600 kPa to 150 kPa?

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

Problem 55P Does the Joule-Thomson coefficient of a substance change with temperature at a fixed pressure?

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.

Problem 59P Steam is throttled slightly from 2 MPa and 500°C. Will the temperature of the steam increase, decrease, or remain the same during this process?

Problem 61P Estimate the Joule-Thomson-coefficient of refriger-ant-134a at 40 psia and 60°F.

Problem 60P Estimate the Joule-Thomson coefficient of steam at (a) 3 MPa and 300°C and (b) 6 MPa and 500°C.

12-62 Demonstrate that the Joule-Thomson coefficient is given by \(\mu=\frac{T^{2}}{c_{p}}\left[\frac{\partial(v / T)}{\partial T}\right]_{P}\) Equation Transcription: Text Transcription: \mu=\frac{T^{2 c_p[\frac{\partial(v / T)\partial T]_P

Problem 64P 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.

Problem 65P On the generalized enthalpy departure chart, the normalized enthalpy departure values seem to approach zero as the reduced pressure PRapproaches zero. How do you explain this behavior?

Problem 63P 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?

Problem 66P Why is the generalized enthalpy departure chart prepared by using PRand TRas the parameters instead of P and T?

Problem 68P 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.

Problem 69P 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.

Problem 71P Water vapor at 1000 kPa and 600°C is expanded to 500 kPa and 400°C. Calculate the change in the specific entropy and enthalpy of this water vapor using the departure charts and the property tables.

Problem 67P 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.

12-72 Methane is compressed adiabatically by a steady-flow compressor from \(0.8 \mathrm{MP} \text { a and }-10^{\circ} \mathrm{C} \text { to } 6 \mathrm{MP} \text { a and } 175^{\circ} \mathrm{C}\) at a rate of . Using the generalized charts, determine the required power input to the compressor. Answer: Equation Transcription: Text Transcription: 0.8 MPa and -10°C to 6MPa and 175°C

Problem 70P Saturated water vapor at 400°F is expanded while its pressure is kept constant until its temperature is 800°F. Calculate the change in the specific enthalpy and entropy using (a) the departure charts, and (b) the property tables.

Problem 73P 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.

12-75E Oxygen is adiabatically and reversibly expanded in a nozzle from 200 psia and \(600^{\circ} F\) 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}) Equation Transcription: Text Transcription: 600°F 1738ft/s, 1740ft/s

12-76 Aropane is compressed isothermally by a piston- (k) cylinder device from \(100^{\circ} \mathrm{C}\) and \(1 M P a t o 4 M P a\). Using the generalized charts, determine the work done and the heat transfer per unit mass of propane. Equation Transcription: Text Transcription: 100°C 1MPa to 4MPa

Problem 79P A 0.05-m3 well-insulated rigid tank contains oxygen at 175 K and 6 MPa. A paddle wheel placed in the tank is turned on, and the temperature of the oxygen rises to 225 K. Using the generalized charts, determine (a) the final pressure in the tank, and (b) the paddle-wheel work done during this process.

Problem 78P Determine the exergy destruction associated with the process described in Prob. 12–76. Assume T0 = 25°C. Problem 12–76 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.

Problem 80P Derive relations for (a) ?u, (b) ?h, and (c) ?s of a gas that obeys the equation of state (P + a/V2)V = RT for an isothermal process.

Problem 81P Starting with the relation dh = T ds + V dP, show that the slope of a constant-pressure line on an h-sdiagram (a) is constant in the saturation region, and (b) increases with temperature in the superheated region.

12-82 Show that \(c_{v}=-T\left(\frac{\partial v}{\partial T}\right)_{s}\left(\frac{\partial P}{\partial T}\right)_{V} \text { and } c_{p}=T\left(\frac{\partial P}{\partial T}\right)_{s}\left(\frac{\partial v}{\partial T}\right)_{P}\) Equation Transcription: Text Transcription: c_v=-T(\frac{\partial v \partial T)_s(\frac{\partial P \partial T)_V and c_p=T(\frac{\partial P\partial T)_s(\frac{\partial v\partial T)_P

12-83 Temperature and pressure may be defined as \(T=\left(\frac{\partial u}{\partial s}\right)_{V} \text { and } P=-\left(\frac{\partial u}{\partial v}\right)_{s}\) Equation Transcription: Text Transcription: T=(\frac{\partial u\partial s)_V and P=-(\frac{\partial u \partial v)_s

Problem 85P Starting with ?JT = (l/cp) [T(?V/?T)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-Pplane is given by the equation (?Z/?T)P = 0.

Repeat Prob. 12–86 for an isobaric process.

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 is the isentropic expansion exponent defined as \(k=\frac{v}{P}\left(\frac{\partial P}{\partial V}\right)_{s}\) Also, show that the isentropic expansion exponent reduces to the specific heat ratio \(c_{p} / c_{v}\) Equation Transcription: Text Transcription: s=s(P,v) PVk= k=vP(PV)s cp/cv

12-84 For ideal gases, the development of the constantpressure specific heat yields \(\left(\frac{\partial h}{\partial P}\right)_{T}=0\) Equation Transcription: Text Transcription: (\frac{\partial h\partial P)_T=0\

12-86 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 the change in specific volume can be expressed in terms of the volume expansivity \(\beta \) and isothermal compressibility \(\alpha\) as \(\frac{d v}{v}=\beta d T=\alpha d P\) Equation Transcription: Text Transcription: v=v(P, T) \beta \alpha \frac{d v v=\beta d T=\alpha d P

Problem 89P Estimate the cpof 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.

Problem 90P Steam is throttled from 2.5 MPa and 400°C to 1.2 MPa. Estimate the temperature change of the steam during this process and the average Joule-Thomson coefficient.

Problem 91P The volume expansivity ? values of copper at 300 K and 500 K are 49.2 × 10–6 K–1 and 54.2 × 10–6 K–1, respectively, and ? 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.

Argon gas enters a turbine at 1000 psia and with a velocity of \(300 \mathrm{ft} / \mathrm{s}\) and leaves at 150 psia and with a velocity of \(450 \mathrm{ft} / \mathrm{s}\) at a rate of \(12 \mathrm{lbm} / \mathrm{s}\) Heat is being lost to the surroundings at \(75^{\circ} \mathrm{F}\) at a rate of \(80 \mathrm{Btu} / \mathrm{s}\). Using the generalized charts, determine the power output of the turbine and the exergy destruction associated with the process. Equation Transcription: Text Transcription: 300 ft/s 450ft/s 12lbm/s 75°F 80Btu/s

An adiabatic \(0.2-m^{3}\) storage tank that is initially evacuated is connected to a supply line that carries nitrogen at and . 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 . Determine the final temperature in the tank (a) treating nitrogen as an ideal gas, and using generalized charts. Compare your results to the actual value of . Equation Transcription: Text Transcription: 0.2-m^3

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. Equation Transcription: Text Transcription: 100°F

Problem 96P 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.

Problem 98P Methane at 50 psia and 100°F is compressed in a steady-flow device to 500 psia and 1100°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.

Problem 99P Determine the second-law efficiency of the compression process described in Prob. 12-103E. Take T0 = 77°F.

Problem 100P 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.

Problem 101P Consider the liquid–vapor saturation curve of a pure substance on the P-Tdiagram. 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 Vfgat that temperature. ________________ (e) inversely proportional to the entropy change sfgat that temperature.

Problem 103P Based on data from the refrigerant-134a tables, the Joule-Thompson coefficient of refrigerant-134a at 0.8 MPa and ifJ0°C is approximately (a) 0 ________________ (b) ?5°C/MPa ________________ (c) 11°C/MPa ________________ (d) 8°C/MPa ________________ (e) 26°C/MPa

Problem 102P Based on the generalized charts, the error involved in the enthalpy of C02 at 300 K and 5 MPa if it is assumed to be an ideal gas is (a) 0% ________________ (b) 9% ________________ (c) 16% ________________ (d) 22% ________________ (e) 27%

What is the difference between partial differentials and ordinary differentials?

Consider a function and its partial derivative . Under what conditions is this partial derivative equal to the total derivative ?

Consider a function z(x, y) and its partial derivative (z/y)x. If this partial derivative is equal to zero for all values of x, what does it indicate?

Consider the function , its partial derivatives and , and the total derivative . (a) How do the magnitudes and compare? (b) How do the magnitudes and compare? (c) Is there any relation among , , and

Consider air at and . Using Eq. 123, 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.

Consider air at and . 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. Repeat problem 12-5 for helium.

Nitrogen gas at and behaves as an ideal gas. Estimate the and 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.

Nitrogen gas at and behaves as an ideal gas. Estimate the and of the nitrogen at this state, using enthalpy and internal energy data from Table A-18E, and comparing them to the values listed in Table A-2Eb.

Consider an ideal gas at and . As a result of some disturbance, the conditions of the gas change to and . Estimate the change in the specific volume of the gas using (a) Eq. 12-3 and (b) the ideal-gas relation at each state.

Using the equation of state , verify (a) the cyclic relation and (b) the reciprocity relation at constant .

Derive a relation for the slope of the = constant lines on a T-P diagram for a gas that obeys the van der Waals equation of state.

Verify the validity of the last Maxwell relation (Eq. 1219) for refrigerant-134a at and

Reconsider Prob. 1212. Using EES (or other) software, verify the validity of the last Maxwell relation for refrigerant-134a at the specified state.

Verify the validity of the last Maxwell relation (Eq. 1219) for steam at and .

Using the Maxwell relations, determine a relation for for a gas whose equation of state is .

Using the Maxwell relations, determine a relation for for a gas whose equation of state is

Using the Maxwell relations and the ideal-gas equation of state, determine a relation for for an ideal gas.

Prove that

What is the value of the Clapeyron equation in thermodynamics?

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

Using the Clapeyron equation, estimate the enthalpy of vaporization of refrigerant-134a at , and compare it to the tabulated value.

Reconsider Prob. 1221. Using EES (or other) software, plot the enthalpy of vaporization of refrigerant-134a as a function of temperature over the temperature range 220 to 808C by using the Clapeyron equation and the refrigerant-134a data in EES. Discuss your results.

Using the Clapeyron equation, estimate the enthalpy of vaporization of steam at 300 kPa, and compare it to the tabulated value.

Determine the hfg of refrigerant-134a at 108F on the basis of (a) the Clapeyron equation and (b) the Clapeyron-Clausius equation. Compare your results to the tabulated hfg value.

0.5-lbm of a saturated vapor is converted to a 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 ft3 ; 250 Btu of heat are removed; and the temperature remains fixed at 158F. Estimate the boiling point temperature of this substance when its pressure is 60 psia.

Estimate the saturation pressure Psat of the substance in Prob. 1225E when its temperature is 208F.

Estimate the sfg of the substance in Problem 1225E at 158F.

A table of properties for methyl chloride lists the saturation pressure as 116.7 psia at 1008F. At 1008F, this table also lists hfg 5 154.85 Btu/lbm, and vfg 5 0.86332 ft3 /lbm. Estimate the saturation pressure Psat of methyl chloride at 908F and 1108F.

Using the Clapeyron-Clausius equation and the triplepoint data of water, estimate the sublimation pressure of water at 2308C and compare to the value in Table A8.

Show that cp,g 2 cp,f 5 Ta 0(hfg/T) 0T b P 1 vfga 0P 0T b sat

Can the variation of specific heat cp with pressure at a given temperature be determined from a knowledge of P-v-T data alone?

Estimate the volume expansivity b and the isothermal compressibility a of refrigerant-134a at 200 kPa and 308C.

Estimate the specific heat difference cp 2 cv for liquid water at 15 MPa and 808C.

Determine the change in the internal energy of air, in kJ/kg, as it undergoes a change of state from 100 kPa 208C to 600 kPa and 3008C using the equation of state P(v 2 a) 5 RT where a 5 1 m3 /kg, and compare the result to the value obtained by using the ideal gas equation of state.

Determine the change in the enthalpy of air, in kJ/ kg, as it undergoes a change of state from 100 kPa and 348C to 800 kPa and 4208C using the equation of state P(v 2 a) 5 RT where a 5 0.01 m3 /kg, and compare the result to the value obtained by using the ideal gas equation of state.

Determine the change in the entropy of air, in kJ/kg?K, as it undergoes a change of state from 100 kPa and 208C to 600 kPa and 3008C using the equation of state P(v 2 a) 5 RT where a 5 0.01 m3 /kg, and compare the result to the value obtained by using the ideal gas equation of state.

Determine the change in the internal energy of helium, in kJ/kg, as it undergoes a change of state from 100 kPa and 208C to 600 kPa and 3008C using the equation of state P(v 2 a) 5 RT where a 5 0.01 m3 /kg, and compare the result to the value obtained by using the ideal gas equation of state.

Determine the change in the enthalpy of helium, in kJ/kg, as it undergoes a change of state from 150 kPa and 208C to 750 kPa and 3808C using the equation of state P(v 2 a) 5 RT where a 5 0.01 m3 /kg, and compare the result to the value obtained by using the ideal gas equation of state.

Determine the change in the entropy of helium, in kJ/kg?K, as it undergoes a change of state from 100 kPa and 208C to 600 kPa and 3008C using the equation of state P(v 2 a) 5 RT where a 5 0.01 m3 /kg, and compare the result to the value obtained by using the ideal gas equation of state

Derive expressions for (a) Du, (b) Dh, and (c) Ds for a gas whose equation of state is P(v 2 a) 5 RT for an isothermal process.

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.

Derive an expression for the specific heat difference cp 2 cv for (a) an ideal gas, (b) a van der Waals gas, and (c) an incompressible substance.

Show that cp 2 cv 5 Ta 0P 0T b v a 0v 0T b P

Temperature may alternatively be defined as T 5 a 0u 0s b v 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.

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 2 a) 5 RT.

Derive an expression for the isothermal compressibility of a substance whose equation of state is P 5 RT v 2 b 2 a v(v 1 b)T1/2 where a and b are empirical constants.

Derive an expression for the volume expansivity of a substance whose equation of state is P 5 RT v 2 b 2 a v 2 T where a and b are empirical constants

Show that b 5 a(0P/0T)v.

Demonstrate that k 5 cp cv 5 2 va (0v/0P)s

The Helmholtz function of a substance has the form a 5 2RT ln v v0 2 cT0a1 2 T T0 1 T T0 ln T T0 b where T0 and v0 are the temperature and specific volume at a reference state. Show how to obtain P, h, s, cv , and cp from this expression.

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.

What does the Joule-Thomson coefficient represent?

Describe the inversion line and the maximum inversion temperature

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

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

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

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

Reconsider Prob. 1257E. 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.

Steam is throttled slightly from 2 MPa and 5008C. Will the temperature of the steam increase, decrease, or remain the same during this process?

Estimate the Joule-Thomson coefficient of steam at (a) 3 MPa and 3008C and (b) 6 MPa and 5008C.

Estimate the Joule-Thomson-coefficient of refrigerant-134a at 40 psia and 608F.

Demonstrate that the Joule-Thomson coefficient is given by m 5 T 2 cp c 0(v/T) 0T d P

Consider a gas whose equation of state is P(v 2 a) 5 RT, where a is a positive constant. Is it possible to cool this gas by throttling?

Derive a relation for the Joule-Thomson coefficient and the inversion temperature for a gas whose equation of state is (P 1 a/v2 )v 5 RT.

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?

Why is the generalized enthalpy departure chart prepared by using PR and TR as the parameters instead of P and T?

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.

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.

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.

Saturated water vapor at 4008F is expanded while its pressure is kept constant until its temperature is 8008F. Calculate the change in the specific enthalpy and entropy using (a) the departure charts, and (b) the property tables.

Water vapor at 1000 kPa and 6008C is expanded to 500 kPa and 4008C. Calculate the change in the specific entropy and enthalpy of this water vapor using the departure charts and the property tables.

Methane is compressed adiabatically by a steady-flow compressor from 0.8 MPa and 2108C to 6 MPa and 1758C at a rate of 0.33 kg/s. Using the generalized charts, determine the required power input to the compressor.

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.

Reconsider Prob. 1273. 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.

Oxygen is adiabatically and reversibly expanded in a nozzle from 200 psia and 6008F 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.

Propane is compressed isothermally by a piston cylinder device from 1008C and 1 MPa to 4 MPa. Using the generalized charts, determine the work done and the heat transfer per unit mass of propane.

Reconsider Prob. 1276. 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.

Determine the exergy destruction associated with the process described in Prob. 1276. Assume T0 5 258C.

A 0.05-m3 well-insulated rigid tank contains oxygen at 175 K and 6 MPa. A paddle wheel placed in the tank is turned on, and the temperature of the oxygen rises to 225 K. Using the generalized charts, determine (a) the final pressure in the tank, and (b) the paddle-wheel work done during this process.

Derive relations for (a) Du, (b) Dh, and (c) Ds of a gas that obeys the equation of state (P 1 a/v2 )v 5 RT for an isothermal process.

Starting with the relation dh 5 T ds 1 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.

Show that cv 5 2T a 0v 0T b s a 0P 0T b v and cp 5 T a 0P 0T b s a 0v 0T b P

Temperature and pressure may be defined as T 5 a 0u 0s b v and P 5 2a 0u 0v b s Using these definitions, prove that for a simple compressible substance

For ideal gases, the development of the constantpressure specific heat yields a 0h 0Pb T 5 0 Prove this by using the definitions of pressure and temperature, T 5 (u/s)v and P 5 2(u/v)s.

Starting with mJT 5 (1/cp)[T(v/T)p 2 v] and noting that Pv 5 ZRT, where Z 5 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 (Z/T)P 5 0.

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 5 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 dv v 5 b dT 5 a dP 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.

Repeat Prob. 1286 for an isobaric process.

Consider an infinitesimal reversible adiabatic compression or expansion process. By taking s 5 s(P, v) and using the Maxwell relations, show that for this process Pv k 5 constant, where k is the isentropic expansion exponent defined as k 5 v Pa 0P 0v b s Also, show that the isentropic expansion exponent k reduces to the specific heat ratio cp/cv for an ideal gas.

Estimate the cp of nitrogen at 300 kPa and 400 K, using (a) the relation in Prob. 1288, and (b) its definition. Compare your results to the value listed in Table A2b.

Steam is throttled from 2.5 MPa and 4008C to 1.2 MPa. Estimate the temperature change of the steam during this process and the average Joule-Thomson coefficient.

The volume expansivity b values of copper at 300 K and 500 K are 49.2 3 1026 K21 and 54.2 3 1026 K21 , 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.

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.

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 758F 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.

Methane is to be adiabatically and reversibly compressed from 50 psia and 1008F 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.

Refrigerant-134a undergoes an isothermal process at 408C from 2 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

A rigid tank contains 1.2 m3 of argon at 21008C and 1 MPa. Heat is now transferred to argon until the temperature in the tank rises to 08C. Using the generalized charts, determine (a) the mass of the argon in the tank, (b) the final pressure, and (c) the heat transfer.

Methane is contained in a pistoncylinder device and is heated at constant pressure of 5 MPa from 100 to 2508C. 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.

Methane at 50 psia and 1008F is compressed in a steady-flow device to 500 psia and 11008F. 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.

Determine the second-law efficiency of the compression process described in Prob. 1298E. Take T0 5 778F

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.

Consider the liquidvapor 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.

Based on the generalized charts, the error involved in the enthalpy of CO2 at 300 K and 5 MPa if it is assumed to be an ideal gas is (a) 0% (b) 9% (c) 16% (d) 22% (e) 27%

Based on data from the refrigerant-134a tables, the Joule-Thompson coefficient of refrigerant-134a at 0.8 MPa and 1008C is approximately (a) 0 (b) 258C/MPa (c) 118C/MPa (d) 88C/MPa (e) 268C/MPa

For a gas whose equation of state is P(v 2 b) 5 RT, the specified heat difference cp 2 cv is equal to (a) R (b) R 2 b (c) R 1 b (d) 0 (e) R(1 1 v/b)

Consider the function z 5 z(x, y). Write an essay on the physical interpretation of the ordinary derivative dz/dx and the partial derivative (z/x)y. Explain how these two derivatives are related to each other and when they become equivalent.

There have been several attempts to represent the thermodynamic relations geometrically, the best known of these being Koenigs thermodynamic square shown in the figure. There is a systematic way of obtaining the four Maxwell relations as well as the four relations for du, dh, dg, and da from this figure. By comparing these relations to Koenigs diagram, come up with the rules to obtain these eight thermodynamic relations from this diagram.

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 Bridgmans 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, cp, b, and a, which can be measured directly or indirectly with relative ease. Obtain a copy of Bridgmans table and explain, with examples, how it is used.