Boyle's Law and the Ideal Gas Law: Laboratory and Theoretical and Practical Guide

Fundamental Principles of Pressure and Volume

• The relationship between pressure ($P$) and volume ($V$) is intuitive and observable in basic biological functions such as breathing.
• Human Respiration Example:
  • When your lungs compress, the volume inside the lungs decreases, which causes the pressure to increase. This elevated internal pressure pushes air out of the lungs.
  • When you inhale, you increase your lung capacity (chest volume). This expansion causes the internal pressure to drop, allowing external air to flow into the lungs.
  • This phenomenon is a direct application of Boyle's Law.

Boyle's Law and Mathematical Relationships

• Definition: Boyle's Law states that for a fixed amount of gas at a constant temperature, the pressure of the gas is inversely proportional to its volume.
• Mathematical Formula:   $P \times V = k$
  • In this equation, $k$ represents a constant.
• Inverse Proportionality:
  • If you increase the pressure, the volume must decrease to maintain the constant value of $k$.
  • Numerical Example: If you have arbitrary units where $P = 1$ and $V = 1$, then $k = 1 \times 1 = 1$. If you double the pressure ($P = 2$), the volume must decrease to half ($V = 0.5$) to ensure the product remains $1$ ($2 \times 0.5 = 1$).

The Ideal Gas Law

• Boyle's Law is a specific component of the broader Ideal Gas Law.
• Formula:   $PV = nRT$
  • $P$: Pressure.
  • $V$: Volume.
  • $n$: Number of moles.
  • $R$: Ideal gas constant.
  • $T$: Temperature.
• Temperature Requirements: In the Ideal Gas Law, temperature must always be measured in Kelvin ($K$) units, never in degrees Celsius ($^{\circ}C$) or Fahrenheit ($^{\circ}F$).
• Ideal Gas Constant ($R$):
  • The value of $R$ varies depending on the units used for pressure and volume.
  • When using atmospheres ($atm$), liters ($L$), moles ($mol$), and Kelvin ($K$), the value of $R$ is approximately $0.082$.
  • Explicitly: $R = 0.082\,L\,atm\,mol^{-1}\,K^{-1}$.
  • More significant figures can be used for precise calculations if necessary.
• Units used for the Ideal Gas Law:
  • Pressure: Atmospheres ($atm$) or Millimeters of Mercury ($mmHg$).
  • Volume: Liters ($L$) or Milliliters ($mL$).
  • The units of $R$ are designed specifically to cancel out the units of pressure, volume, moles, and temperature in the equation.

Standard Pressure Units and Conversions

• Atmosphere ($atm$): Originally defined by the pressure on Earth at sea level on a standard day.
• Millimeters of Mercury ($mmHg$): A more convenient unit developed later, based on the height of a mercury column supported by atmospheric pressure.
• Conversion Factor:   $1\,atm = 760\,mmHg$
• Physical Measurement: In a barometer, air pressure pushes down on mercury, forcing it to rise in a column. This column reaches a height of $760\,mmHg$, which is equal to $76\,cmHg$ (about three-quarters of a 39-inch meter stick).
• Kilopascals ($kPa$): The LabQuest device used in the lab defaults to kilopascals ($kPa$). A sample laboratory pressure reading provided was $102.58\,kPa$.

Laboratory Experiment: Boyle’s Law with LabQuest

• Equipment:
  • LabQuest device (requires a power supply and a long press of the top button to power on).
  • Pressure sensor (a black sensor found in a white box).
  • Syringe (analog device used to vary volume).
• Sig Fig Precision for Syringe:
  • The syringe has divisions of $1\,mL$ between major markings (e.g., between $5\,mL$ and $10\,mL$).
  • As an analog device, you must record one digit beyond the smallest division.
  • Therefore, you read the syringe to the nearest tenth of a milliliter ($0.1\,mL$).
  • Example: If set on the line for 10, record as $10.0\,mL$ (three significant figures).
• Procedure:
  1. Set the syringe plunger to exactly $10.0\,mL$ before attaching it to the pressure sensor.
  2. Attach the syringe by turning it in a clockwise motion.
  3. Take approximately 11 to 12 readings between $5\,mL$ and $17\,mL$.
  4. Compressing below $5\,mL$ is difficult due to the gas fighting back; do not force it or break the syringe.
  5. Pulling beyond $20\,mL$ may pull the plunger out of the syringe.

Data Recording and Excel Calculations

• Students must prepare a lab notebook with a data table containing five specific columns (discard the sample values $5.8$ and $0.17$ in the handout):
  1. Volume ($V$): Recorded from the syringe with proper sig figs (2 sig figs for values under $10$, 3 sig figs for values $10$ and over).
  2. Reciprocal Volume ($1/V$): Calculated by dividing 1 by the volume. The sig figs of the result should match the volume (2 or 3).
  3. Pressure ($P$): Read directly from the LabQuest instrument (typically 4-5 sig figs).
  4. Constant Ratio ($P/V$): Calculated as pressure divided by volume. Units: $kPa\,mL^{-1}$.
  5. Constant Product ($P \times V$): Calculated by multiplying pressure by volume. Units: $kPa \cdot mL$.

Analysis and Graphical Representation

• Required Graphs in Excel:
  1. Graph 1: Pressure ($P$) vs. Volume ($V$):
  • This graph will yield a curve known as a hyperbolic or rational function.
  • It is a geometric function seen in nature, similar to the bend of telephone lines.
  • The curve never touches the axes.
  1. Graph 2: Pressure ($P$) vs. Reciprocal Volume ($1/V$):
  • This graph will yield a straight line.
  • Students must find the linear regression equation: $y = mx + b$.
• Variables in Linear Regression:
  • $y$: Pressure ($P$).
  • $x$: Reciprocal volume ($1/V$).
  • $m$: The slope of the line (must include units).
  • $b$: The y-intercept (must include units corresponding to the y-axis). It should be value very close to zero.
• Graphing Standards: Every graph must include a title, axis labels with correct units, and the indicated mathematical function (curve or linear).

Questions & Discussion

• Homework Assignment: Due Sunday night, two days after the lab if performed on Friday or Saturday. The assignment includes completing the data table, constructing two Excel graphs, and answering post-lab questions.
• Excluded Content: Question 1 regarding the equation $y = ax^{b}$ should be ignored and marked with an "x".
• Corrected Typo: The manual refers blindly to "y is equal to m b plus b"; this is a typo and must be corrected to the standard linear equation $y = mx + b$.
• Attendance Assignment / Secret Password:
  • The password is: $PV = k$
  • Alternatively, if typing math symbols is difficult: "Pressure Times Volume is a Constant".