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course Phy 202
2.11 12 pm2.7.11 Physics
ndividual_experimnts/brief_bottle_experiment_2a.htm
individual_experiments/brief_bottle_experiment_2b.htm
individual_experiments/brief_bottle_experiment_2c.htm
individual_experiments/brief_bottle_experiment_2d.htm
`q001. The gas in a bottle is heated, forcing water first up a thin tube then out of the top of the tube. After the water reaches its maximum temperature, the top of the tube remains open to the atmosphere, the heat source is removed and the gas is allowed to cool until the water in the tube has nearly reached the level of the water in the bottle. A valve is closed, preventing any air from entering the system through the tube, and the system is allowed to continue cooling to room temperature.
Sketch a pressure vs. volume graph for this cycle.
What shape is made on your graph?
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The shape of the graph is that of a square/rectangle.
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`q002. Suppose that in the preceding we have a large supply of cool air at temperature T_c and a large supply of warm water at temperature T_h. The gas in the bottle begins at temperature T_0 = T_c and pressure P_0, with volume V_0,the tube's open end is at height y relative to the water in the bottle. We will raise the temperature of the gas in the bottle to temperature T_h by bathing the bottle in warm water, and we will allow it to cool while surrounded by cool air.
In terms of T_c, P_0, V_0 and y, as well as the density rho of water and the acceleration g of gravity (not all of which will necessarily appear in every expression), what will be the pressure P_1, volume V_1 and temperature T_1 when water first reaches the top of the tube?
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P1=P0+`rho*g*y
V1=V0
T1=T09(`rho*g*y+P0/P0)
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In terms of the same variables, what will be the pressure P_2, volume V_2 and temperature T_2 when the gas reaches its maximum temperature T_h?
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P2=P1=P0=`rho*g*y
V2=V0(Th/T0)(P0/`rho*g*y+P0)
T2=Th, so T2/T1=Th/T1=Th/(T0*(`rho*g*y+P0))=Th/T0(P0/`rho*g*y+P0)
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In terms of the same variables, what will be the pressure P_3, volume V_3 and temperature T_3 when the water has in the tube has receded to just above the level of the water in the bottle and the tube is closed?
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P3=P0
V3=V2= V0(Th/T0)(P0/`rho*g*y+P0)
T3= Th (P0/`rho*g*y+P0)
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In terms of the same variables, what will be the pressure P_4, volume V_4 and temperature T_4 when the system has cooled to its original temperature T_c?
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Same as the original:
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`q003. In the preceding:
How much thermal energy enters or leaves the system during each of the four separate phases of the cycle? Assume the gas to be diatomic.
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`dQ01=5/2nR(Tc(`rho*g*y+P0/P0)-Tc
`dQ12=5/2nR(Th-Tc(`rho*g*y+P0/P0))
`dQ23=5/2nR(th*(P0/P0+`rho*g*y)-Th)
`dQ34=7/2n(Tc-Th(P0/P0+`rho*g*y))
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How much thermal energy is taken from the warm water?
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`dPE/`dQin= (V0(Th/Tc)(P0/`rho*g*y)-V0)*`rho*g*y)/5/2Nr(Tc(`rho*g*y+P0/P0)-Tc) + 7/2nR(Th-Tc(`rho*g*y+P0/P0)
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How much thermal energy is added to the cool air?
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@& To reduce the pressure at constant volume we allow the gas to cool, which requires that we remove KE from its molecules.
By how much does the temperature change, and how much energy is therefore removed from the system?
Then to contract the gas at constant pressure we have to again reduce the temperature. By how much does the temperature change, and how much energy is involved?*@
How much potential energy is gained by the water which flows out of the top of the tube, assuming it is caught in a reservoir at that height?
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`dPE=(V0(Th/Tc)(P0/`rho*g*y)-V0)*`rho*g*y
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`q004. We are still referring to the parameters of the system in question 2.
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How high would it be possible to raise water in the tube by heating the system from temperature T_c to temperature T_h, assuming the tube to be sufficiently long?
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@& What is the highest pressure we could atain in the system if we didn't let it expand?
How high a water column could we maintain at that temperature?*@
Let's call this height y_max.
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If the height y is half as great at y_max, then what is expression for the PE change of the water?
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@& If you are able to get an expression for y_max, then is can be substituted into the `dPE expression for y to obtain a result.*@
If the height y is adjusted to be a little greater than half of y_max, will the PE change increase or decrease relative to your preceding answer? University physics students could answer this question by considering the derivative of an appropriate function.
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If y = r * y_max, what is the expression for the PE change of the water?
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`dPE=(V0(Th/Tc)(P0/`rho*g*(r*y))-V0)*`rho*g(r*y)
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@& That would be
`dPE=(V0(Th/Tc)(P0/`rho*g*(r*y_max))-V0)*`rho*g(r*y_max)
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For what value of r will the PE change of the water be maximized? (University Physics students only)
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For what value of r will the ratio of PE change to energy absorbed from the warm water be maximized? (University Physics students only)
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`q005. (University Physics students only). If after reaching State 2, instead of letting the water gradually recede as the gas cools, we open a vent in the bottle and simply let the pressurized air suddenly escape, the pressure and volume will change in such a way that P V^gamma = constant, where gamma = C_p / C_v (gamma = 7/2 for a diatomic gas, 5/2 for a monatomic gas).
In this case what will be the temperature T_3 and the volume V_3 when the pressure has reached the original pressure P_3 = P_0?
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How much work does the gas do in this expansion?
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If the gas is then permitted to cool at constant pressure until it has reached its original temperature T_c, how much thermal energy must be transferred from the gas to the surrounding air?
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`q006. If the gas is returned to its original pressure P_0 by slowly venting it, while keeping it at the same temperature, how much energy will need to be supplied during this phase? How much work will the expanding gas do?
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... how can we design a system to take advantage of the insights we have gained ... ?
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@& See my notes and, if you can make some progress with reasonable time and effort, go ahead and submit a revision. If you do, use @@@@ to indicate your insertions.
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