WORK NEEDED! a) A cylindrical container of fixed volume is initially in a cool r
ID: 2131307 • Letter: W
Question
WORK NEEDED!
a) A cylindrical container of fixed volume is initially in a cool room, and then is moved to a warmer room. Measurements show that the energy of the air in the cylinder increases by X Joules.) (i) How much heat is absorbed or emitted by the cylinder. Justify your answer by referring to the first law of thermodynamics. (ii) If the cylinder is equipped with a piston, and the air inside is allowed to expand freely, does its internal energy also increase by X Joules, fewer, or greater? Again, justify your answer.
In parts (b) and (c) consider an insulated cylinder with a piston attached to one end, arranged so that the volume of the air in the box can be changed from a minimum value Vmin to a maximum value Vmax. Initially the volume has an in-between value Vi, where Vmin < Vi < Vmax.
b) The volume is increased to Vf . If the pressure is constant during this process at the value p ?, what is the change of the energy of the air in the box?
c) Again, the volume is increase to Vf , but suppose that more careful obser- vations show that the pressure is not quite constant but changes according to
p=p 1?? V/ Vmax
where ? is a constant and is less than one (this insures that the pressure is positive).
What is the change in the energy of the air in the box?
Note: You will need to do an integral.
A cylindrical container of fixed volume is initially in a cool room, and then is moved to a warmer room. Measurements show that the energy of the air in the cylinder increases by X Joules.) (i) How much heat is absorbed or emitted by the cylinder. Justify your answer by referring to the first law of thermodynamics. (ii) If the cylinder is equipped with a piston, and the air inside is allowed to expand freely, does its internal energy also increase by X Joules, fewer, or greater? Again, justify your answer. In parts (b) and (c) consider an insulated cylinder with a piston attached to one end, arranged so that the volume of the air in the box can be changed from a minimum value Vmin to a maximum value Vmax. Initially the volume has an in-between value Vi, where VminExplanation / Answer
a) By first law of thermodynamics dQ = dU + dW. Since here dW = 0, So, dQ = dU. Also since volumne is fixed, dQ = n * C_v * (T2 - T1) = dU where T2 is the final temperature and T1 is the initial cool temperature.
If piston is present, then certain amount of work would be done by the gas and as a result, by first law of thermodynamics dU = dQ - dW i.e. internal energy would now decrease by an amount of work that is done by the gas
b) Change in energy of the box => dU = dQ - dW = n * C_v * (T2 - T1) - p * (Vf - Vi)
c) Here to calculate work you need to do an integral of pdV from volume Vi to Vf. The sign is not clear in the question, so I am assuming p = p1(1 - cV/V_max).
So, integral pdV = p1(V - cV^2/(2V_max) ). Substitute the limits and use previous equation to get answer
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