The Ideal Gas Law AP Physics 2 FRQ – Exam Style Questions etc.
The Ideal Gas Law AP Physics 2 FRQ
Unit 9: Thermodynamics
Weightage : 15–18%
Exam Style Practice Questions, The Ideal Gas Law AP Physics 2 FRQ
Question
An ideal gas expands from points A to C along three possible paths.
(a) Is it expected that final temperature at point C be path dependent? Justify your answer qualitatively with no calculations.
(b) Discuss and compare the flow of thermal energy along the three paths ( AC , ABC , ADC ). Indicate the direction of heat for each pathway (into or out of the gas) and the relative amount of thermal energy involved. Justify your answer qualitatively with no calculations.
(c) Calculate the work done along path:
i. ABC
ii. AC
iii. ADC
(d) Is there a way to go from A to C with no thermal energy exchanged with the environment? If so, describe and sketch the path on a P – V diagram.
(e) Is there a way to go from A to C with no temperature change? If so, describe and sketch the path on a P – V diagram.
Answer/Explanation
Ans:
(a) No, the temperature is a state function of pressure and volume. So the temperature at point C is not path dependent.
(b) ΔU = Q + W
For an ideal gas, the change in internal energy ( ΔU) is fixed by the change in the temperature. So the difference in thermal energy exchange ( Q ) will be dictated by the amount of work done on each path (see part (c) for a calculation of work). Without doing a calculation, we can visually tell which path is doing more work by examining the areas under the pathways. The product PV is lower at point C than at point A . Therefore, we can conclude that the temperature is lower at point C (ideal gas law PV = nRT ). Since the temperature is decreasing, ΔU is negative while all the work done by the gas represents a loss of energy as well. The difference between the given pathway and an adiabatic expansion (see answer (d)) represents the amount of thermal energy exchanged with the environment ( Q ).
Path AC represents moderate work done by the gas, requiring some thermal energy to be added to the gas during the process but not as much as path ABC , which represents the most work done by the gas. Since pathway ABC is so far from adiabatic, it will require significant thermal energy to be added to the gas. Finally, path ADC is the least amount of work done and the only pathway to require thermal energy to be given to the environment from the gas.
(c) The work done is going to be equal to the area under each segment of the P – V graph. Also recall that 1 atm = 101 kPa = 101,000 N/\(m^{2}\) and 1 L = 0.001 \(m^{3}\):
Pa · \(m^{3}\) = J
i. Along path ABC , no work is done from B → C because there is no change in volume. Thus the area is just the area under AB = (8 atm)(6 L) = 4,848 J.
ii. Along path AC , the total area is equal to:
\(\frac{1}{2}(6atm)(6L)+(2atm)(6L)=3,030J\)
iii. Along path ADC , no work is done from A → D because there is no change in volume. Thus the area is just the area under DC :
(2 atm)(6 L) = 1,212 J
(d) Yes, if the expansion is adiabatic, you can go from A to C with no thermal energy exchanged. This would connect point A and point C with a steep hyperbola. In this case, the work done is solely responsible for the change in internal energy as no heat is involved.
(e) Yes, if the expansion is isothermal, you can go from A to C with no temperature change, In this case, thermal energy will have to be added to the system at the same rate as work is being done by the gas in order to effect no ΔU, This would connect point A and point C with a hyperbola but one less steep than in part (d).
Question
An air bubble is formed at the bottom of a swimming pool and then released. The air bubble ascends toward the surface of the pool.
(A) In a clear, coherent, paragraph-length response, describe any changes in the bubble size and describe the motion of the bubble as it ascends to the surface. Explain the factors that affect the size of the bubble and the bubble’s motion. Include a description of any forces acting on the bubble from the time it is at the bottom of the pool until the bubble is just below the surface of the pool.
(B) On the figure, draw a vector for each force acting on the bubble. Make sure all vectors are drawn in correct proportion to each other.
(C) The bubble does not collapse under the pressure of the water. Explain how the behavior of the gas atoms keeps the bubble from collapsing.
(D) The bubble begins at a depth of D below the surface of the water where the bubble has an initial volume of \(V_{D}\). The atmospheric pressure at the surface of the pool is \(\rho\). The density of the water in the pool is \(P_{S}\). Assume that the air temperature in the bubble remains constant as it rises to the surface. Derive an expression for the volume (\(V_{S}\)) of the bubble when it reaches the surface of the pool.
(E) In part (D) it was assumed that the temperature of the bubble remains constant. Now assume that the air temperature in the bubble can change but that the bubble rises so quickly to the surface that there is negligible thermal energy transfer between the bubble and the swimming pool water. Base your answers on this assumption.
I. Sketch the process on the PV diagram. Indicate on the axis the initial and final pressures and volumes.
II. How does the value \(P_{S}V_{S}\) compare to the value \(P_{D}V_{D}\)?
_Greater than \(P_{D}V_{D}\) _Equal to \(P_{D}V_{D}\) _Less than \(P_{D}V_{D}\)
Justify your answer.
Answer/Explanation
Ans:
Part (A)
The amount of air inside the bubble remains the same during ascent. The external pressure from the water on the bubble decreases as the
bubble rises. The bubble increases in volume as it moves upward. As the bubble volume increases, the buoyancy force increases, but the gravity force on the bubble remains constant. The bubble accelerates upward at an increasing rate as it ascends.
Part (B)
For buoyancy force upward and gravity force downward. Buoyancy must be drawn larger than gravity. If extra vectors are drawn, this point is not awarded.
Part (C)
The atoms inside the gas collide with and bounce off the water molecules. This causes a change in momentum of the gas atoms. This generates an equal and opposite force between the water and gas that keeps the bubble from collapsing.
Part (D)
For the correct relationship between the pressure at the surface of the water and at a depth D: \(P_{D}=P_{S}+\rho gD\)
For the correct application of the ideal gas law and the correct final expression:
\(P_{S}V_{S}=P_{D}V_{D}\)
\(P_{S}V_{S}=(P_{S}+\rho gD)V_{D}\)
\(V_{S}=\frac{(P_{S}+\rho gD)V_{D}}{P_{S}})
Part (E)
(i) 
(ii) The process is an adiabatic expansion. There is no heat transfer between the water and the bubble. Therefore: \(Q=0\), and \(\Delta U=W\). Since the work is negative as the bubble expands, the internal energy of the bubble must also decrease. Therefore, the temperature and the PV value must also decrease.