Consider nmoles of ideal gas kept in a cylinder with a piston. Two heat reservoirs 1 and
2 with the temperatures T1 < T2 are available, and at any given moment of time the heat exchange is established with only one of the reservoirs. In the initial equilibrium state the external pressure is p1 , and heat contact is with reservoir 1. At the same time moment the external pressure is quickly changed to
p2 , whereas heat exchange switches to reservoir 2, and we wait for the system to equilibrate. Then the external pressure is quickly returned to its initial value p1 , whereas heat exchange switches back to reservoir 1, and we wait for the system to equilibrate. The system, obviously, returns to its initial state and, therefore we can call our entire process a cycle. a) Calculate the amount of work wthe system produce on the environment, the amount of heat amount of heat
q2 , transferred from reservoir 2 (heat source) to the system, and the q1 , transferred to reservoir 1 from the system; all for an entire cycle.
b) (2 pts) For given T1 , T2 , and p1 find the range of values for p2 for which the produced work is positive w > 0 .
c) In the case w> 0 our process can be considered as a cycle of a heat engine that produces useful work w > 0 , using the amount of heat q2 from the heat source, and dumping the amount of heat q1 to the reservoir (inevitable loss). Find the efficiency ? = w/q2 of our heat engine.
d) Based on the first and second laws of thermodynamics find the maximal value?max of the efficiency for any heat engine that uses two heat reservoirs at temperatures T1 < T2 . Is our engine more or less efficient? Explain why.
Consider nmoles of ideal gas kept in a cylinder with a piston. Two heat reservoirs 1 and
2 with the temperatures T1 < T2 are available, and at any given moment of time the heat exchange is established with only one of the reservoirs. In the initial equilibrium state the external pressure is p1 , and heat contact is with reservoir 1. At the same time moment the external pressure is quickly changed to
p2 , whereas heat exchange switches to reservoir 2, and we wait for the system to equilibrate. Then the external pressure is quickly returned to its initial value p1 , whereas heat exchange switches back to reservoir 1, and we wait for the system to equilibrate. The system, obviously, returns to its initial state and, therefore we can call our entire process a cycle.a) Calculate the amount of work wthe system produce on the environment, the amount of heat amount of heat
q2 , transferred from reservoir 2 (heat source) to the system, and the q1 , transferred to reservoir 1 from the system; all for an entire cycle.
b) (2 pts) For given T1 , T2 , and p1 find the range of values for p2 for which the produced work is positive w > 0 .
c) In the case w> 0 our process can be considered as a cycle of a heat engine that produces useful work w > 0 , using the amount of heat q2 from the heat source, and dumping the amount of heat q1 to the reservoir (inevitable loss). Find the efficiency ? = w/q2 of our heat engine.
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Related questions
| pV work - Exercise 1 Consider a container with a frictionless piston that contains a given amount of an ideal gas. If the external pressure is kept constant, the piston will move up or down in response to a change in the internal pressure. The piston will move up if Pint > Pext and vice versa. The piston will stop moving when Pint = Pext (the system is equilibrated).
The following problem will slowly guide you through a work calculation. Some questions may appear trivial, so trust your instincts and donât assume there are hidden tricks. The goal of such a guided exercise is to help you picture the situation so you understand how all relevant variables change or stay the same. This is the first in a sequence of related problems. Remember to use three significant figures for all numerical answers. The margin of error for each numerical answer is 5%. To avoid rounding errors use the unrounded intermediate values in your final calculations. Letâs assume the initial volume of the gas is 3.90 L , the initial temperature of the gas is 26.0 °C , and the system is in equilibrium with an external pressure of 1.2 bar (given by the sum of a 1 bar atmospheric pressure and a 0.2 bar pressure due to a brick that rests on top of the piston).
| Part A How many moles of gas are there in the container?
SubmitMy AnswersGive Up You now heat the gas slowly until the temperature reaches 51.2 °C You do it slowly so the piston can move if necessary until a new equilibrium is reached. Part B What happens with the piston while you heat the gas? What happens with the piston while you heat the gas?
When the final temperature is reached, you stop heating the gas. Part C What is the final pressure of the gas?
SubmitMy AnswersGive Up Part D What is the final volume of the gas?
SubmitMy AnswersGive Up Part E What happens with the piston after you finish heating the gas? Assume you do not need to worry about the gas cooling down again because the outside of the container is at a lower temperature. That is, you manage to keep the gas at a constant temperature that equals 51.2 °C What happens with the piston after you finish heating the gas? Assume you do not need to worry about the gas cooling down again because the outside of the container is at a lower temperature. That is, you manage to keep the gas at a constant temperature that equals 51.2
SubmitMy AnswersGive Up So far this was all at the level of your general chemistry class. Letâs now calculate the work involved in the process. Recall the sign conventions we use in physical chemistry, and always think from the point of view of the system (the gas). Part F What is the sign of w? What is the sign of w?
SubmitMy AnswersGive Up Part G What is the value of w? Be careful with units. How do you convert bar*L to J?
SubmitMy AnswersGive Up |
