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On an adiabatic process of an ideal gas pressure, volume and temperature change such that p V γ is constant with γ = 5 / 3 for monatomic gas such as helium and γ = 7 / 5 for diatomic gas such as hydrogen at room temperature. Use numerical values to plot two isotherms of 1 mol of helium gas using ideal gas law and two adiabatic processes mediating between them. Use T 1 = 500 K , V 1 = 1 L , and T 2 = 300 K for your plot.


The figure is a plot of pressure, p, in atmospheres on the vertical axis as a function of volume, V, in liters on the horizontal axis. The horizontal, V, axis runs from 1.0 to 2.0. The vertical, p, axis runs from 0 to about 40. Two isotherms are shown. One isotherm is for T equal to 500 K, with the pressure starting at about 40 atmospheres when the volume is 1.0 Liter and decreasing with volume to about 25 atmospheres at 2.0 liters. The second isotherm is for T equal to 300 K, with the pressure starting at about 25 atmospheres when the volume is 1.0 Liter and decreasing with volume to a little over 10 atmospheres at 2.0 liters. A third plot, labeled “Adiabatic” starts with the 500 K isotherm, at 1.0 L and about 40 atmospheres, and ends with the 300 K isotherm, at 2.0 L and just over 10 atmospheres.

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Two moles of a monatomic ideal gas such as helium is compressed adiabatically and reversibly from a state (3 atm, 5 L) to a state with pressure 4 atm. (a) Find the volume and temperature of the final state. (b) Find the temperature of the initial state of the gas. (c) Find the work done by the gas in the process. (d) Find the change in internal energy of the gas in the process.

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Additional problems

Consider the process shown below. During steps AB and BC , 3600 J and 2400 J of heat, respectively, are added to the system. (a) Find the work done in each of the processes AB , BC , AD , and DC . (b) Find the internal energy change in processes AB and BC . (c) Find the internal energy difference between states C and A . (d) Find the total heat added in the ADC process. (e) From the information given, can you find the heat added in process AD ? Why or why not?

The figure is a plot of pressure, p, in atmospheres on the vertical axis as a function of volume, V, in liters on the horizontal axis. The horizontal volume scale runs from 0 to 7, and the vertical pressure scale runs from 0 to 5. Four points, A, B, C, and D, are labeled on the plot and their pressures and volumes are labeled on the axes. Point A is at volume 3 L, pressure 2 atmospheres. Point B is at volume 3 L, pressure 5 atmospheres. Point C is at volume 7 L, pressure 5 atmospheres. Point D is at volume 7 L, pressure 2 atmospheres. A path goes from A up to B and across to C. Another path goes from A across to D and then up to C.

a. W A B = 0 , W B C = 2026 J , W A D = 810.4 J , W D C = 0 ; b. Δ E A B = 3600 J , Δ E B C = 374 J; c. Δ E A C = 3974 J ; d. Q A D C = 4784 J; e. No, because heat was added for both parts AD and DC . There is not enough information to figure out how much is from each segment of the path.

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A car tire contains 0.0380 m 3 of air at a pressure of 2.20 × 10 5 Pa (about 32 psi). How much more internal energy does this gas have than the same volume has at zero gauge pressure (which is equivalent to normal atmospheric pressure)?

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A helium-filled toy balloon has a gauge pressure of 0.200 atm and a volume of 10.0 L. How much greater is the internal energy of the helium in the balloon than it would be at zero gauge pressure?

300 J

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Steam to drive an old-fashioned steam locomotive is supplied at a constant gauge pressure of 1.75 × 10 6 N/m 2 (about 250 psi) to a piston with a 0.200-m radius. (a) By calculating p Δ V , find the work done by the steam when the piston moves 0.800 m. Note that this is the net work output, since gauge pressure is used. (b) Now find the amount of work by calculating the force exerted times the distance traveled. Is the answer the same as in part (a)?

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A hand-driven tire pump has a piston with a 2.50-cm diameter and a maximum stroke of 30.0 cm. (a) How much work do you do in one stroke if the average gauge pressure is 2.4 × 10 5 N/m 2 (about 35 psi)? (b) What average force do you exert on the piston, neglecting friction and gravitational force?

a. 59.5 J; b. 170 N

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Calculate the net work output of a heat engine following path ABCDA as shown below.

The figure is a plot of pressure, p, in nuits of 10 to the 6 Newtons per meter squared on the vertical axis as a function of volume, V, in 10 to the minus 3 cubic meters on the horizontal axis. The horizontal volume scale runs from 0 to 4, and the vertical pressure scale runs from 0 to about 4. Four points, A, B, C, and D, are labeled on the plot and their pressures and volumes are labeled on the axes. Point A is at volume 1.0 times 10 to the -3 cubic meters, pressure 2.6 times 10 to the 6 Newtons per meter squared. Point B is at volume 4.0 times 10 to the -3 cubic meters, pressure 2.0 times 10 to the 6 Newtons per meter squared. Point C is at volume 4.0 times 10 to the -3 cubic meters, pressure 0.6 times 10 to the 6 Newtons per meter squared. Point D is at volume 1.0 times 10 to the -3 cubic meters, pressure 1.0 times 10 to the 6 Newtons per meter squared. A straight line connects A to B, another straight line B to C, another straight line C to D, and another straight line back to A.
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What is the net work output of a heat engine that follows path ABDA in the preceding problem with a straight line from B to D ? Why is the work output less than for path ABCDA ?

2.4 × 10 3 J

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Five moles of a monatomic ideal gas in a cylinder at 27 ° C is expanded isothermally from a volume of 5 L to 10 L. (a) What is the change in internal energy? (b) How much work was done on the gas in the process? (c) How much heat was transferred to the gas?

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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