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5.6. Expand the following equations in the form Pu= RT (1+BP+CP²+...), and determine the second virial coefficient B in each case: (a) (P+)(u-b)=RT (b) (PRT) (u-b)=RT (van der Waals equation of state). (Dieterici equation of state),

5.6. Expand the following equations in the form Pu= RT (1+BP+CP²+...), and determine the second virial coefficient B in each case: (a) (P+)(u-b)=RT (b) (PRT) (u-b)=RT (van der Waals equation of state). (Dieterici equation of state),

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Expand the following equations in the form Pv = RT(1 + BP + CP^2+ ...), and determine the second virial coefficient B in Dieterici equation of state 

CHAPTER 5: Ideal Gas 135 5.6. Expand the following equations in the form Pv = RT(1+ BP + CP² + ...), and determine the second virial coefficient B in each case: (a) (P+)(v- b) = RT (van der Waals equation of state). (b) (PelRT")(v – b) = RT (Dieterici equation of state). (e) (P+)v – b) =- RT (Berthelot equation of state). a (d) v4cT - b) = RT (Clausius equation of state). +, B' C' (e) Pu = RT(1+++) (another type of virial expansion). (e) Pv = 5.7. An ideal gas is contained in a cylinder equipped with a frictionless, nonleaking piston of area A. When the pressure is atmospheric Po, the piston face is a distance / from the closed end. The gas is compressed by moving the piston a distance x. Calculate the spring constant 7/x of the gas: (a) Under isothermal conditions. (b) Under adiabatic conditions (c) In what respect is a gas cushion superior to a steel spring? (d) Using Eq. (4.14), show that Cp - Cy = nR for the ideal gas. 5.8. The temperature of an ideal gas in a tube of very small, constant cross-sectional area varies linearly from one end (x= 0) to the other end (x = L) according to the equation TL – To T = To + - If the volume of the tube is V and the pressure P is uniform throughout the tube, show that the equation of state for n moles of gas is given by TL - To "In(Tz/To)" PV = nR- Show that, when TL = To = T, the equation of state reduces to the obvious one, PV = nRT. 5.9. Prove that the work done by an ideal gas with constant heat capacities during a quasi-static adiabatic expansion is equal to: (a) W = -Cy(T, – T). P;Vj – P,V, (b) W = 1-1 136 PART I: Fundamental Concepts (7-1)/ P,V (c) W = 7-1 5.10. (a) Show that the heat transferred during an infinitesimal quasi-static process of an ideal gas can be written -vaP + Pav . paV. Applying this equation to an adiabatic process, show that PV = const. (b) An ideal gas of volume 0.05 ft and pressure 120 lb/in? undergoes a quasi-static adiabatic expansion until the pressure drops to 15 lb/in?. Assuming y to remain constant at the value 1.4, calculate the final volume. Calculate the work. 5.11. (a) Derive the following formula for a quasi-static adiabatic process for the ideal gas, assuming 7 to be constant TV- = const. (b) At about 0.1 ms after detonation of a 20-kiloton nuclear fission bomb, the "fireball" consists of a sphere of gas with a radius of about 40 ft and a uniform temperature of 300,000 K. Making rough assumptions, estimate the radius at a temperature of 3000 K. 5.12. (a) Derive the following formula for a quasi-static adiabatic process for the ideal gas, assuming y to be constant: T = const. P-5 (b) Helium (7 =) at 300 K and latm pressure is compressed quasi-statically and adiabatically to a pressure of 5 atm. Assuming that the helium behaves like the ideal gas, calculate the final temperature. 5.13. A horizontal, insulated cylinder contains a frictionless nonconducting piston. On each side of the piston is 54 liters of an inert monatomic ideal gas at 1 atm and 273 K. Heat is slowly supplied to the gas on the left side until the piston has com- pressed the gas on the right side to 7.59 atm. (a) How much work is done on the gas on the right side? (b) What is the final temperature of the gas on the right side? (c) What is the final temperature of the gas on the left side? (d) How much heat was added to the gas on the left side? 5.14. An evacuated bottle with nonconducting walls is connected through a valve to a large supply of gas, where the pressure is Po and the temperature is To. The valve is opened elightly and helium flows into the bottle unti1 the pressure inside the botle is P.
Transcribed Image Text:CHAPTER 5: Ideal Gas 135 5.6. Expand the following equations in the form Pv = RT(1+ BP + CP² + ...), and determine the second virial coefficient B in each case: (a) (P+)(v- b) = RT (van der Waals equation of state). (b) (PelRT")(v – b) = RT (Dieterici equation of state). (e) (P+)v – b) =- RT (Berthelot equation of state). a (d) v4cT - b) = RT (Clausius equation of state). +, B' C' (e) Pu = RT(1+++) (another type of virial expansion). (e) Pv = 5.7. An ideal gas is contained in a cylinder equipped with a frictionless, nonleaking piston of area A. When the pressure is atmospheric Po, the piston face is a distance / from the closed end. The gas is compressed by moving the piston a distance x. Calculate the spring constant 7/x of the gas: (a) Under isothermal conditions. (b) Under adiabatic conditions (c) In what respect is a gas cushion superior to a steel spring? (d) Using Eq. (4.14), show that Cp - Cy = nR for the ideal gas. 5.8. The temperature of an ideal gas in a tube of very small, constant cross-sectional area varies linearly from one end (x= 0) to the other end (x = L) according to the equation TL – To T = To + - If the volume of the tube is V and the pressure P is uniform throughout the tube, show that the equation of state for n moles of gas is given by TL - To "In(Tz/To)" PV = nR- Show that, when TL = To = T, the equation of state reduces to the obvious one, PV = nRT. 5.9. Prove that the work done by an ideal gas with constant heat capacities during a quasi-static adiabatic expansion is equal to: (a) W = -Cy(T, – T). P;Vj – P,V, (b) W = 1-1 136 PART I: Fundamental Concepts (7-1)/ P,V (c) W = 7-1 5.10. (a) Show that the heat transferred during an infinitesimal quasi-static process of an ideal gas can be written -vaP + Pav . paV. Applying this equation to an adiabatic process, show that PV = const. (b) An ideal gas of volume 0.05 ft and pressure 120 lb/in? undergoes a quasi-static adiabatic expansion until the pressure drops to 15 lb/in?. Assuming y to remain constant at the value 1.4, calculate the final volume. Calculate the work. 5.11. (a) Derive the following formula for a quasi-static adiabatic process for the ideal gas, assuming 7 to be constant TV- = const. (b) At about 0.1 ms after detonation of a 20-kiloton nuclear fission bomb, the "fireball" consists of a sphere of gas with a radius of about 40 ft and a uniform temperature of 300,000 K. Making rough assumptions, estimate the radius at a temperature of 3000 K. 5.12. (a) Derive the following formula for a quasi-static adiabatic process for the ideal gas, assuming y to be constant: T = const. P-5 (b) Helium (7 =) at 300 K and latm pressure is compressed quasi-statically and adiabatically to a pressure of 5 atm. Assuming that the helium behaves like the ideal gas, calculate the final temperature. 5.13. A horizontal, insulated cylinder contains a frictionless nonconducting piston. On each side of the piston is 54 liters of an inert monatomic ideal gas at 1 atm and 273 K. Heat is slowly supplied to the gas on the left side until the piston has com- pressed the gas on the right side to 7.59 atm. (a) How much work is done on the gas on the right side? (b) What is the final temperature of the gas on the right side? (c) What is the final temperature of the gas on the left side? (d) How much heat was added to the gas on the left side? 5.14. An evacuated bottle with nonconducting walls is connected through a valve to a large supply of gas, where the pressure is Po and the temperature is To. The valve is opened elightly and helium flows into the bottle unti1 the pressure inside the botle is P.
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