F. Gases and Atmospheric Chemistry · Gases and Atmospheric ChemistryInvestigating Gas Laws: Planning, Data and Error
F2.2 A1.1 A1.5 A1.6 A1.8 A1.10 A1.13About 70 minutes
Learning goals- I can identify independent, dependent and controlled variables in a gas law investigation.
- I can write a testable hypothesis with a reason.
- I can analyse pressure, volume and temperature data with tables and graphs.
- I can extrapolate data to estimate absolute zero and calculate percentage error.
- I can describe specific sources of error and suggest improvements.
Words to know
independent variable
The variable that is deliberately changed in an experiment.
dependent variable
The variable that is measured to see how it responds.
controlled variable
A variable kept constant so it does not affect the results.
hypothesis
A testable prediction with a reason, often in the form "If... then... because...".
percentage error
|experimental − accepted| ÷ |accepted| × 100 %.
line of best fit
A straight line (or curve) drawn to show the overall trend of plotted data.
From laws to evidence
So far you've been told the gas laws. Scientists didn't get them that way: Boyle, Charles and Gay-Lussac measured and graphed data until patterns appeared. In this lesson you'll do what they did: plan a fair investigation, analyse real-looking data, find the relationship from a graph, and evaluate the sources of error. These are the inquiry skills in strand A of the course, and you'll use them in every unit.
Planning a fair test
Every gas investigation has four variables: P, V, T and n. To study the relationship between two, you must hold the other two constant.
Before you start, write the variables in your lab notebook (see the table below). A hypothesis should be testable and give a reason, for example: If the volume of the trapped air is halved at constant temperature, then its pressure will double, because the particles will hit the walls twice as often.
| Investigation | Independent variable | Dependent variable | Controlled variables |
|---|
| Pressure–volume (Boyle) | volume of trapped air | pressure | temperature, amount of air (no leaks) |
| Pressure–temperature (Gay-Lussac) | temperature of water bath | pressure | volume (rigid flask), amount of air |
| Volume–temperature (Charles) | temperature | volume | pressure, amount of air |
Investigation 1: Pressure and volume
In a school lab, a plastic syringe is connected to an electronic gas pressure sensor. The plunger starts at 20.0 mL. A student pushes it in 2.5 mL at a time, waits a few seconds for the reading to settle, and records the pressure. The plunger is moved slowly so the gas stays at room temperature.
| V (mL) | P (kPa) | P × V (kPa·mL) | 1/V (mL⁻¹) |
|---|
| 20.0 | 101.5 | 2030 | 0.0500 |
| 17.5 | 116.0 | 2030 | 0.0571 |
| 15.0 | 135.2 | 2028 | 0.0667 |
| 12.5 | 162.1 | 2026 | 0.0800 |
| 10.0 | 203.4 | 2034 | 0.1000 |
| 7.5 | 270.0 | 2025 | 0.1333 |
Worked exampleExample 1: Analysing the P–V data
Use the table above to decide whether the data support Boyle's law, and find the constant k in P = k/V.
- Calculate P × V for each row: 2030, 2030, 2028, 2026, 2034, 2025 kPa·mL.
- The values differ by at most 9 out of about 2030, less than 0.5 %. They are constant within experimental error.
- A graph of P versus 1/V would be a straight line through the origin, with slope k.
- Average of P × V: (2030 + 2030 + 2028 + 2026 + 2034 + 2025) ÷ 6 = 2029 kPa·mL.
The data support Boyle's law: P = 2029/V (P in kPa, V in mL), or about 2.03 × 10³ kPa·mL.
Investigation 2: Pressure and temperature
A sealed flask of air is connected to a pressure sensor. The flask is held in water baths at different temperatures. At each temperature the student waits until the reading stops changing (so the air inside has reached the water's temperature) and records the pressure.
| Temperature (°C) | Pressure (kPa) |
|---|
| 0.5 | 96.4 |
| 21.0 | 103.6 |
| 40.2 | 110.5 |
| 61.5 | 117.9 |
| 80.3 | 124.2 |
Worked exampleExample 2: Finding absolute zero from data
Use the first and last data points of Investigation 2 to find the equation of the line, the temperature at which P would be zero, and the percentage error compared with the accepted value, −273 °C.
- Slope m = ΔP/Δt = (124.2 − 96.4) ÷ (80.3 − 0.5) = 27.8 ÷ 79.8 = 0.348 kPa/°C
- Intercept: P = mt + b → 96.4 = 0.348(0.5) + b → b = 96.2 kPa
- Set P = 0: 0 = 0.348t + 96.2 → t = −96.2 ÷ 0.348 = −276 °C
- Percentage error = |experimental − accepted| ÷ |accepted| × 100 % = |−276 − (−273)| ÷ 273 × 100 % = 1.1 %
Absolute zero ≈ −276 °C, a 1.1 % error. (A line of best fit through all five points gives almost the same result.)
Evaluating sources of error
A good conclusion doesn't just say "human error". It names specific, realistic causes and says which way each would push the result.
- Leaks at the syringe tip or the flask stopper: gas escapes, so pressures are lower than expected at high pressure (or high temperature).
- Temperature not equilibrated: if readings are taken too soon, the gas hasn't reached the bath temperature. In Investigation 2 this makes hot readings too low.
- "Dead" volume: the tubing between the syringe and the sensor contains extra gas not read on the syringe scale, so the true volume is a little larger than recorded.
- Compressing too fast: the gas warms up as it's squeezed, raising the pressure.
- Instrument precision: a syringe scale may only be readable to ±0.2 mL, so record measurements with the correct number of significant figures.
Improving the investigation: check for leaks first, wait for readings to stabilize, measure the tubing volume and add it, repeat each trial and average.
Hands-on activityBoyle's law with a syringe and books
You need: a plastic syringe of 20 mL or larger with a cap for the tip (from a pharmacy, no needle; or seal the tip with a blob of hot glue that a grown-up applies and lets cool), 4 to 6 identical hardcover books, a ruler, graph paper.
- Pull the plunger out to the largest mark, then cap the tip so air is trapped.
- Hold the syringe upright with the tip down on a table (a grown-up can steady it, or wedge it in a mug). Balance one book on the plunger. Gently tap the side to reduce friction. Record the volume.
- Repeat with 2, 3, 4... books, carefully stacked. Then remove them one at a time and record again.
- Analyse: the pressure on the gas is atmospheric pressure plus the pressure from the books, so P is not proportional to the number of books. Instead, plot 1/V on the y-axis against the number of books on the x-axis.
What to look for: the graph of 1/V versus number of books should be a fairly straight line that does not pass through the origin. Its y-intercept represents the atmosphere's push with zero books. The pattern agrees with Boyle's law (1/V ∝ P). Friction in the plunger is the biggest source of error: that's why the readings while removing books differ from the readings while adding them. Average them.
Just for funRobert Boyle wrote a book called The Sceptical Chymist (1661), arguing that chemists should trust experiments over old theories. In other words: "Show me the data." Science teachers have been quoting the idea ever since, often while holding a red pen.
Your turn
1In an investigation of how temperature affects the pressure of a gas, identify the independent, dependent and controlled variables.
Independent: temperature. Dependent: pressure. Controlled: volume of the container and amount of gas.
2Write a testable hypothesis for an investigation of Charles's law.
Example: If the Kelvin temperature of a gas at constant pressure is increased, then its volume will increase proportionally, because faster particles push the container walls outward until the pressure inside matches the pressure outside.
3A student's P × V values are 2010, 2008, 1995, 1960 and 1902 kPa·mL as the volume gets smaller. Suggest the most likely cause.
A leak: as the pressure rises, gas escapes, so the product P × V decreases steadily.
4Why should the student push the syringe plunger slowly in a Boyle's law experiment?
Compressing a gas quickly warms it. Boyle's law requires constant temperature, so slow compression gives the gas time to stay at room temperature.
5A student's experimental value for absolute zero is −265 °C. Calculate the percentage error.
|−265 − (−273)| ÷ 273 × 100 % = 2.9 %
6Why does a graph of P versus 1/V test Boyle's law better than a graph of P versus V?
A straight line through the origin is easy to recognize and to fit. A curve could be many shapes; a straight line on a P versus 1/V graph confirms the inverse proportion exactly.
More practice: warm-up
1What is an independent variable?
The variable the experimenter deliberately changes.
2What is a controlled variable?
A variable kept constant so that it does not affect the result.
3Write the formula for percentage error.
|experimental − accepted| ÷ |accepted| × 100 %
4What does it mean to extrapolate a graph?
To extend it beyond the measured data to predict values.
5Name one way to reduce random error in an experiment.
Repeat trials and take the average (also: use more precise instruments).
More practice: on your own
1In Investigation 2, the student read the 80.3 °C pressure only 10 s after moving the flask into the hot bath. Which way would this error shift the extrapolated absolute zero? Explain.
The air would not yet be at 80.3 °C, so the pressure would be too low. The line would be less steep, so its x-intercept would move farther from the data, to a more negative temperature (further below −273 °C).
2Use the data point (21.0 °C, 103.6 kPa) from Investigation 2 to predict the pressure at 100.0 °C using Gay-Lussac's law.
P₂ = 103.6 × 373 ÷ 294 = 131 kPa
3A syringe scale has marks every 0.5 mL. How should a volume of exactly 12.5 mL be recorded, and why does this matter?
Record it as 12.5 mL (an estimate to about ±0.2 mL is reasonable). The number of significant figures shows the precision of the measurement and limits the precision of the calculated results.
4Two students measure the volume of a balloon at different temperatures. One uses a water bath, the other uses a hair dryer. Which method is better? Explain.
The water bath: it heats evenly and lets the temperature be measured accurately. A hair dryer heats unevenly and the balloon's temperature is unknown.
5Calculate the slope of the line for Investigation 2 using the points (21.0, 103.6) and (61.5, 117.9).
Slope = (117.9 − 103.6) ÷ (61.5 − 21.0) = 14.3 ÷ 40.5 = 0.353 kPa/°C
Challenge
1A Boyle's law syringe is connected by tubing that holds 1.5 mL of air not counted on the scale. The student records 10.0 mL at 200 kPa and 20.0 mL at 107 kPa. Show that correcting for the tubing makes the data fit Boyle's law.
Uncorrected: P × V = 200 × 10.0 = 2000 and 107 × 20.0 = 2140 kPa·mL, which differ by 7 %. Corrected volumes include the tubing: 11.5 mL and 21.5 mL. Then 200 × 11.5 = 2300 and 107 × 21.5 = 2300 kPa·mL (2300.5). The products now agree, so the dead volume explains the mismatch.
2Design an investigation to test Charles's law at home using a balloon. Include variables, procedure, data table and one expected source of error.
Independent: temperature (freezer, fridge, room, warm water bath). Dependent: balloon volume (from circumference, V = C³/6π²). Controlled: amount of air (same tied balloon), outside pressure. Procedure: let the balloon reach each temperature for 20 minutes, measure the circumference quickly with string, record temperature. Table: temperature (°C), temperature (K), circumference (cm), volume (cm³), V/T. Error: the balloon warms or cools while measuring; the rubber's elasticity also changes the inside pressure slightly.
3Explain why extrapolating a graph is riskier than interpolating, using the gas law investigations as an example.
Interpolating predicts between measured points, where the pattern is known to hold. Extrapolating assumes the pattern continues where nothing was measured. For a real gas, the straight line would not continue to −273 °C: the gas would condense first. The extrapolated absolute zero is only meaningful because the ideal-gas pattern points there.
Quiz
1. In a Boyle's law investigation, which variable must be controlled?
2. A student's data give P × V values that steadily decrease as the gas is compressed. The most likely cause is:
3. An experimental absolute zero of −280 °C has a percentage error of about:
4. To test Gay-Lussac's law, which equipment set-up is most suitable?
5. Which graph should be a straight line through the origin if a gas obeys Boyle's law?
6. Why wait for the pressure reading to stop changing after moving a flask into a new water bath?
Big idea (F2.2 with A1.1, A1.5, A1.6, A1.8, A1.10, A1.13): students plan and analyse gas-law investigations: they identify variables, write hypotheses, organize and graph data, calculate percentage error, and evaluate specific sources of error.
How to help: ask "What did you change? What did you measure? What did you keep the same?" for any experiment. When your teen writes "human error", ask for a specific cause and which direction it would push the result.
Common difficulties:
- Vague error analysis. Push for concrete, directional explanations (a leak lowers P × V).
- Graphing the wrong axes. The independent variable goes on the x-axis (except where we deliberately plot 1/V to linearize data).
- Forgetting atmospheric pressure in the book experiment. That's why 1/V versus number of books has a non-zero intercept.
Safety: a grown-up should apply any hot glue. Use only syringes without needles.