Kinetic Theory of Temperature and Pressure AP Physics 2 FRQ – Exam Style Questions etc.
Kinetic Theory of Temperature and Pressure AP Physics 2 FRQ
Unit 9: Thermodynamics
Weightage : 15–18%
Exam Style Practice Questions, Kinetic Theory of Temperature and Pressure AP Physics 2 FRQ
Question
Students observe that a graphite rod gets hot when there is an electric potential difference \(\Delta V\) applied across it that causes an electric current I in it. The graphite rod is placed in an apparatus that consists of a clear plastic container with a lid, as shown above. The lid is equipped with electrical connectors and an opening that can be sealed around an inserted sensor. The graphite rod is connected to the electrical connectors by wires and sealed inside the container so that all the energy emitted by the rod goes into heating the air in the container. The teacher tells the students that in this situation the change in the internal energy of the air is equal to (5 2)\(Nk_{B}\Delta T\), where N is the number of molecules and T is the temperature, and the air can be treated as an ideal gas.
(a) Derive an expression for the temperature change of the air as a function of time t as a result of the electrical energy dissipated by the rod and delivered to the air in the container. Express your answer in terms of I, \(\Delta V\), N, and physical constants, as appropriate. Assume that the temperature of the graphite rod remains constant while the air is being heated.
The students are asked to design an experiment using the apparatus shown to investigate this heating. The students have an ammeter, a voltmeter, a fixed DC power supply, a stopwatch, an electronic temperature sensor, and a pressure sensor. Assume that the electrical connectors and connecting wires have negligible resistance.
(b) Outline an experimental procedure that can be used to gather data to determine how the air temperature in the container depends on the electrical energy delivered to the rod. Indicate the measurements to be taken and how the measurements will be used to obtain the data needed. On the diagram on the previous page, show how the container will be connected to instruments to take the necessary measurements.
(c) On the axes below, sketch the line or curve you predict will represent a plot of the temperature of the air in the container as a function of electrical energy delivered to the rod.
(d) i. On the axes below, sketch a line or curve you predict will represent a plot of the pressure of the air in the container as a function of time. Clearly label the sketch as \(R_{1}\).
Explain why your graph has this shape.
ii. The rod is now replaced by a second graphite rod that has twice the length but the same radius. The potential difference across the new rod is the same as that across the original rod. On the axes in part (d)(i), sketch a line or curve you predict will represent a plot of the pressure of the air in the container as a function of time for the second rod. Clearly label the sketch as \(R_{2}\). Compare this graph to the graph from part (d)(i) and explain why it is the same or different.
(e) Another group of students performing this experiment notices a gap in the seal of the container opening and thinks that some gas has leaked out of the container. If this is true, how would this group’s graph of air temperature as a function of electrical energy compare to the graph you drew in part (c)?
Answer/Explanation
Ans:
(a) \(\Delta E=I\Delta Vt\)
\(I\Delta Vt=5/2 Nk_{B}\Delta T\)
\(\Delta T=2\Delta VIt/5Nk_{B}\)
(b) For a valid description of the setup and procedure, including a diagram
For measuring current and potential difference, with symbols defined as needed
For measuring the temperature change of the air, with a symbol defined as needed
For a description of how the measurements will be used to calculate the energy
(c) 
(d) i) 
ii) 
(e) The new graph will have a different slope, and with no indication that the shape of the new graph would be different.
Question
Th e figure above shows a clear plastic container with a movable piston that contains a fixed amount of gas. A group of students is asked to determine whether the gas is ideal. Th e students design and conduct an experiment. They measure the three quantities recorded in the data table below.
(a) Describe an experimental procedure the group of students could have used to obtain these data. Include all the equipment needed and a labeled diagram of the setup. Clearly indicate what measurements would be taken and how each of the manipulated variables would be varied. Include enough detail so that someone else could carry out the procedure.
(b) Select a set of data points from the table and plot those points on the axes below to create a graph to determine whether the gas exhibits properties of an ideal gas. Fill in blank columns in the table for any quantities you graph other than the given data. Label the axes and indicate the scale for each. Draw a best-fit line or curve through your data points.
(c) Indicate whether the gas exhibits properties of an ideal gas, and explain what characteristic of your graph provides the evidence.
(d) Th e students repeat their experiment with an identical container that contains half as much gas. They take data for the same values of volume and temperature as in the table. Would the new data result in a different conclusion about whether the gas is ideal? Justify your answer in terms of interactions between the molecules of the gas and the container walls.
Answer/Explanation
Ans:(a)
For clearly indicating which variables are manipulated and which are controlled
For clearly describing an experimental setup
For an experimental setup that allows the manipulation and control of the indicated variables
For an experimental setup that allows multiple measurements of P, V, and T
Example:
Hold temperature constant while volume is manipulated and pressure is measured.Then change the temperature and hold it constant again while volume is manipulated and pressure is measured, etc. To do this: Measure the cross-sectional area of the piston in units of meters cubed.Put the container in an insulated bath that can be filled with water at one of the three
different temperatures. Fill the bath with water. Allow some time for the gas in the container to equilibrate in temperature with the surrounding water bath. Measure the temperature of the bath. Measure six fixed heights, in units of meters, along the side of the container. Slowly add weights to the top of the piston so that the piston can be depressed to each height without changing the temperature. Multiply the height by the cross-sectional area of the piston to get the volume.
Divide the weight in newtons by the cross-sectional area of the piston to get the pressure and add atmospheric pressure and the pressure from the piston. Repeat the process twice using water at each of the other two temperatures.
(b)
For plotting appropriate quantities to examine ideal behavior
For labeling and scaling the axes
For plotting the data reasonably correctly
For plotting a set of data with one variable controlled and drawing a reasonable best-t line
OR
plotting more than one data set and more than one reasonable best-t line
Example:
Plot P as a function of for a single value of T. Draw a best-t line through the data.
(c)
For a correct conclusion with a reasonable attempt at an explanation
For a correct explanation relating to characteristics exhibited in the ideal gas law
Example:
\(PV=nRT\), so a graph of P versus for an ideal gas should be linear if n and T are held constant. In the graphed set of data, n and T are held constant and the graph is linear, so there is evidence that the gas is ideal.
(d) For indicating that the conclusion would be the same, since the amount of gas does not aect the relationship between the state variables
For indicating that the rate of collision with the walls will be lower, so the pressure would be lower
Example:
No, the conclusion would be the same. Reducing the amount of gas by half would result in there being half the rate of collisions with the container walls and half as much weight needed to compress the gas to the same volume at the same temperature. The graph of P as a function of would still be linear, but with half the slope.