phys124s11-hw11

phys124s11-hw11 - HW 11 Solutions Isobaric, Isochoric,...

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1 HW 11 Solutions Isobaric, Isochoric, Isothermal, and Adiabatic Processes Consider a system consisting of an ideal gas confined within a container, one wall of which is a movable piston. Energy can be added to the gas in the form of heat by applying a flame to the outside of the container. Conversely, energy can also be removed from the gas in the form of heat by immersing the container in ice water. Energy can be added to the system in the form of work by pushing the piston in, thereby compressing the gas. Conversely, if the gas pushes the piston out, thereby pushing some atmosphere aside, the internal energy of the gas is reduced by the amount of work done. The internal energy of an ideal gas is directly proportional to its absolute temperature T . An ideal gas also obeys the ideal gas law PV nRT = , so the absolute temperature T is directly proportional to the product of the absolute pressure p and the volume V . Here n denotes the amount of gas in moles, which is a constant because the gas is confined, and R is the universal gas constant. A pV diagram is a convenient way to track the pressure and volume of a system. Energy transfers by heat and/or work are associated with processes, which are lines or curves on the pV diagram taking the system from one state (i.e., one point on the diagram) to another. Work corresponds geometrically to the area under the curve on a pV diagram. If the volume increases (i.e., the system expands) the work will be classified as an energy output from the system. a) What is the sign of Δ U as the system of ideal gas goes from point A to point B on the graph? Recall that U is proportional to T. ( ) ( ) 0 AB PV PV T T U =⇒ = Δ = b) How are Q and W related during this isothermal expansion? 0 UQW QW Δ=− = ⇒ =
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2 W provides energy output (the gas does some work on the environment, ln BB AA VV B A V nRT W PdV dV nRT ⎛⎞ == = ⎜⎟ ⎝⎠ ∫∫ , while Q provides energy input (the gas must be heated to be able to expand at constant T ). They are equal in magnitude. c) How are Q and W related during the isochoric part of the overall path from state A to state D? Q provides energy output (the gas must be cooled in this isochoric process), while W equals zero. () ( ) 33 0, 0 22 D A V DD AD V DA A V PV WP d V V V Q n c T T n R V P P nR nR === = = = < d) How are Q and W related during the isobaric part of the overall path from state D to state B? 55 0, 0 B D V DB D PB D V W PdV P V V Q nc T T n R P V V nR nR > =− = −= > W provides energy output, while Q provides energy input; Q is larger. e) Another way to get from state A to state B is to follow an adiabatic path from state A to state E, in which no heat energy transfer is allowed, and then to follow an isochoric path from state E vertically to state B. Notice that during the adiabatic part of this path, from state A to state E, Q =0 by definition and internal energy is lost due to work since the system is expanding. Which of the following statements are true about the isochoric part of the overall path, from state E to state B?
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phys124s11-hw11 - HW 11 Solutions Isobaric, Isochoric,...

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