# Gas Laws: Boyle's Law

## Concepts: Boyle's Experiment

Some of the earliest quantitative scientific measurements were performed on gases. One early study was conducted by Robert Boyle in 1662.

Robert Boyle employed a U-shaped piece of glass tubing that was sealed on one end. A gas (air) was trapped in the sealed end of the tube and varying amounts of mercury were added to the U-shaped tube to vary the pressure of the system. Boyle systematically varied the pressure and measured the volume of the gas. These measurements were performed using a fixed amount of gas and a constant temperature. In this way Boyle was able to examine the pressure-volume relationship without complications from other factors such as changes in temperature or amount of gas.

The manometer that was employed to measure pressure can also be used to perform Boyle's experiment. A small amount of air is trapped in the sealed end of the manometer. The scale that is used to read the height of a column of mercury can also be used to read the height of the column of air.

The pressure of the gas will be varied in exactly the same manner used by Robert Boyle: mercury is added or removed from the open end of the manometer. After changing the pressure (by changing the amount of mercury in the manometer), the pressure and volume are recorded.

## Data Analysis

Once the volume-pressure data has been obtained, the next challenge is to determine the mathematical relationship between the two properties. Although an enormous number of relationships are possible, one simple possibility is that the volume might be directly related to the pressure raised to some power:

*V* = *b P ^{ a}*

The exponent *a* is expected to be independent of the mass of gas and temperature; the goal is to determine the value of *a* from the "experimental" data. The constant *b* is expected to vary with the mass of gas and the temperature. At this point, the value of *b* is not of interest.

A simple way to determine the value of *a* is to prepare a plot of ln *V* vs ln *P*. If the proposed relationship is valid (and it might not be valid), this plot should yield a straight line of slope *a*. Thus the linearity of the plot serves as a test of our original hypothesis (that the volume-pressure relation may be described by the equation shown above).

ln *V* = ln *b* + *a* ln *P*

## Experiment: Part 1

### Objective

- Measure the atmospheric pressure.

The basic details of the experiment have been described above. The open tube of the manometer is exposed to the atmosphere. Consequently the atmospheric pressure (*P _{atm}*) contributes to the pressure on the gas.

The first step in the experiment is thus to measure the atmospheric pressure using the manometer containing no trapped air. Record the value value of *P _{atm}*. This value is needed in the next part of the experiment.

## Experiment: Part 2

### Objective

- Determine how the volume of a gas changes with the pressure for a fixed amount of gas and temperature.

A sample of air is now trapped in the closed end of the manometer. (The trapped air has artificially been given a light green color.)

Carefully measure the heights of the columns of mercury on each side of the manometer and the column of trapped air. The scale is in units of millimeters (mm). The effective top of the left side of the manometer is **1041 mm** (which is above the calibrated scale). The tube has an inside diameter of **4.000 cm**. The volume (in mL) of the trapped air is the height of the column of air (in cm) multiplied by the cross-sectional area of the tube (area = **ℼ** *r*^{ 2} = ¼ **ℼ** *d*^{ 2}).

Use this data to calculate the volume of the trapped air and the pressure of the air. You will need to use the value for *P _{atm}* measured in Part 1 of this experiment.

Change the amount of mercury in the manometer and once again measure the volume and pressure. Continue this process until data is obtained for at least five different pressures.

Notice that sometimes the column of mercury on the left is higher than that on the right and sometimes the reverse is true.

If the mercury is higher on the left than the right, then the atmospheric pressure equals pressure of the gas plus the pressure exerted by the column of mercury.

*P _{Hg}* +

*P*=

*P*

_{atm}If the mercury is higher on the right than the left, then the pressure of the gas equals the atmospheric pressure plus the pressuer exerted by the column of mercury. Take this effect into account in calculating the pressure.

*P* = *P _{Hg}* +

*P*

_{atm}For each pair of volume-pressure values for the trapped air, enter the data in the table. The point will automatically be plotted on the graph.

Carefully examine the plots of *V* vs *P* and ln *V* vs ln *P* and determine the value of *a*. Bear in mind *a* is expected to be an integer. However, experimental error, however, will result in a non-integer value for the slope.

Once *a*, write a simple mathematical equation for Boyle's Law.

**Reminders**

The effective top of the left side of the manometer is at a scale reading of **1041 mm**.

The tube has an inside diameter of **4.000 cm**

*BoylesLaw.html version 3.0*

*© 2000, 2014, 2023 David N. Blauch*