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.
Question
Students use a sample of gas to investigate the behavior of the pressure P of the gas at constant temperature T as the volume V changes. The gas is in a cylinder with a movable piston and volume markings. Pressure and temperature probes can be inserted into the cylinder. A hot water bath and a cold water bath are also available.
(A) Describe a procedure that would allow the students to obtain data for the pressure P of the gas at constant temperature T as volume changes.
(B) One student suggests that the temperature probe is not needed. Is the student correct? Briefly explain your answer.
(C) Describe a method of analyzing the pressure and volume data that could be used to determine whether the gas is ideal. Explicitly indicate the results of the analysis that would indicate an ideal gas.
The students are now given a sample of ideal gas in a similar container with a piston. They investigate the behavior of the temperature T of the gas at known constant pressure P as the volume V changes. Their graph of the data, including a best-fit line, is shown below.
(D) Describe a method for using the graph to determine the number of moles of gas in the container.
(E) From the graph, determine the students’ experimental value for absolute zero temperature on the Celsius scale. Describe the method you used.
▶️Answer/Explanation
Ans:
(A) One point for using one of the baths to regulate the temperature. One point for using the piston to change the volume of the gas. One point for waiting for the temperature to reach equilibrium before measuring pressure One point for explicitly taking more than two measurements.
(B) One point for indicating that the student is incorrect with an acceptable explanation that addresses the need to know the gas temperature.
Example of acceptable explanation (claim, evidence, and reasoning):
• The student is not correct (claim). As the pressure and volume change the temperature also changes (evidence). The gas temperature would need to be measured to verify that it has reached equilibrium with the water bath (reasoning).
(C) One point for indicating the information from the analysis that would indicate an ideal gas.
• Example 1: Graph pressure as a function of volume. If the gas is ideal the best fit to the data will be linear.
• Example 2: For each pressure-volume data pair, multiply the pressure and the volume. If the values are reasonably the same, the gas is ideal.
(D) One point for using the slope of the graph. One point for indicating that the slope equals \(\frac{nR}{P}\) . One point for noting that the fundamental constant R and the value at which P is held constant are known.
(E) One point for indicating a value near 300° C (but not so precise as 273° C since the graph cannot be read that precisely) Describe the method you used. One point for describing the extrapolation of the line to zero volume to determine absolute zero.
Question: (12 points-suggested time 25 minutes)
A mole of ideal gas is enclosed in a cylinder with a movable piston having a cross-sectional area of 1 x 10-2 m2. The gas is taken through a thermodynamic process, as shown in the figure.
(A) Calculate the temperature of the gas at state A, and describe the microscopic property of the gas that is related to the temperature.
(B) Calculate the force of the gas on the piston at state A, and explain how the atoms of the gas exert this force on the piston.
(C) Predict qualitatively the change in the internal energy of the gas as it is taken from state B to state C. Justify your prediction.
(D) Is heat transferred to or from the gas as it is taken from state B to state C? Justify your answer.
(E) Discuss any entropy changes in the gas as it is taken from state B to state C. Justify your answer.
(F) Calculate the change in the total kinetic energy of the gas atoms as the gas is taken from state C to state A.
(G) On the axis provided, sketch and label the distribution of the speeds of the atoms in the gas for states A and B. Make sure that the two sketches are proportionally accurate.
▶️Answer/Explanation
Ans:
Note that all the numbers in this problem are rounded to two significant digits because that is the accuracy of the data from the graph.
Part (A)
1 point-For the correct value of temperature with supporting equation and work: PV = nRT, T = 480 K.
1 point-For an explanation that the temperature of the gas is directly related to the average kinetic energy of the gas molecules.
Part (B)
1 point-For the correct force with supporting equation and work: F = PA = 2,000 N.
1 point-For an explanation of the mechanism that produces gas force on the piston.
For example: The gas molecules collide with the piston in a momentum collision that imparts a tiny force on the piston. The sum of all the individual molecular collision forces is the net force on the piston.
Part (C)
1 point-For indicating that the temperature of the gas is decreasing and explaining why this occurs.
For example: Since the PV value of the gas decreases, the temperature of the gas decreases in this process as indicated by the ideal gas law.
1 point-For indicating that the internal energy of the gas will decrease and explaining why this occurs.
For example: ΔU = nRΔT and the temperature is decreasing. Therefore, the internal energy of the gas also must decrease.
Part (D)
1 point-For indicating that the work in this process is positive because the process is moving to the left on the graph and that the thermal energy of the gas is decreasing because the temperature is decreasing.
1 point-For using the first law of thermodynamics to determine that heat is being removed from the gas.
For example: work is positive and the gas internal energy is decreasing. Therefore, using the first law of thermodynamics, ΔU = Q + W, we can see that heat must be leaving the gas during this process.
Part (E)
1 point-For indicating that the entropy of the gas is decreasing because thermal energy is being removed in this process. This reduces the spread of
the speed distribution of the gas, thus reducing disorder.
Part (F)
1 point-For the correct answer with supporting work:
Part (G)
Based on the PV values, the temperature at point A is higher than that at point B. Thus, the peak for A must be at a higher speed than for B.
1 point-For both curves showing a roughly bell shape, and curve A having a higher average speed than curve B.
The area under the graphs must be equal because the number of molecules remains the same. This means the peak for A must be lower than that for B.
1 point-For curve A having a lower maximum than curve B, and both curves having roughly the same area beneath them.