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)
Students are watching a science program about the North Pole. The narrator says that cold air sinking near the North Pole causes high air pressure. Based on the narrator’s statement, a student makes the following claim:
“Since cold air near the North Pole is at high pressure, temperature and pressure must be inversely related.”
(a) Do you agree or disagree with the student’s claim about the relationship between pressure and temperature? Justify your answer.
After hearing the student’s hypothesis, you want to design an experiment to investigate the relationship between temperature and pressure for a fixed amount of gas. The following equipment is available.
-A cylinder with a movable piston, shown above on the left
-A cylinder with a fixed lid, shown above on the right
Note: The two cylinders have gaskets through which measurement instruments can be inserted without gas escaping.
-A pressure sensor -A source of mixed ice and water
-A basin that is large enough to hold -A meterstick
either cylinder with a lot of extra room -A thermometer
-A source of hot water -A stopwatch
(b) Put a check in the blank next to each of the items above that you would need for your investigation. Outline the experimental procedure you would use to gather the necessary data. Make sure the outline contains sufficient detail so that another student could follow your procedure.
The table below shows data from a different experiment in which the volume, temperature, and pressure of a sample of gas are varied.
(c) What subset of the experimental trials would be most useful in creating a graph to determine the relationship between temperature and pressure for a fixed amount of gas? Explain why the trials you selected are most useful.
(d) Plot the subset of data chosen in part (c) on the axes below. Be sure to label the axes appropriately.
Draw a curve or line that best represents the relationship between the variables.
(e) What can be concluded from your curve or line about the relationship between temperature and pressure?
▶️Answer/Explanation
Ans:
I do not agree with the students claim. Using the ideal gas law Pv = nRT, P and T are directly related, not inversely.
___A cylinder with a movable piston, shown above on the left
√__A cylinder with a fixed lid, shown above on the right
Note: The two cylinders have gaskets through which measurement instruments can be inserted without gas escaping.
√__A pressure sensor √__A source of mixed ice and water
√__A basin that is large enough to hold ___A meterstick
either cylinder with a lot of extra room √__A thermometer
√__A source of hot water √__A stopwatch
(b)
Fill the basin with the hot water, place a thermometer and a pressure senser in the fixed cylinder. Place the cylinder in the basin of hot water every minute record the temperature and pressure of the gas in the cylinder. Repeat for a cold water both, with the gas originally beginning at room temperature for both cases.
(c)
Trials 2, 5, 8, 11, and 14 are useful. The volume is constant for these values, so the only unknown variables being tested are temperature and pressure, also, there are more data points, which can reduce uncertainly.
Temperature (0C)
(e)
The relationship between temperature and pressure is linear, thus they are directly related.