# Biological Physics/Boyles Law

This section explores Boyle’s Law and the relation between the pressure and volume of an ideal gas. In general, Boyle’s Law states that as the volume decreases, the pressure of an ideal gas increases. The opposite is also true: as the volume of an ideal gas increases, the pressure decreases as a result. In other words, these two properties are inversely related. For simplicity, we will only consider gases that are at a constant temperature.

Figure 1. Weights were added to a cup steadied on top of a syringe. Using LoggerPro and a Vernier pressure sensor, the pressure (mg/A) and volume can be measured to see the dependencies for Boyle's Law. One can also measure temperature to verify the process is isothermal.

Let us consider the isothermal process in which an ideal gas remains at a constant temperature. The formula PV = NkBT describes the state of a gas at some point. This can be rearranged so that pressure equals: ${\displaystyle P={\frac {Nk_{B}T}{V}}}$, where N is the number of particles, kB is the Boltzmann constant, T is temperature in Kelvin, and V is volume in m3. To illustrate this relationship, Figure 1 shows the inverse relationship of pressure related to volume.

Figure 2. Pressure vs. volume obtained using the method described here.

To calculate the work done on an ideal gas, the formula ${\displaystyle W=-P\Delta V}$ must be considered. Recall that pressure and volume are inversely related, so if pressure changes then volume will also undergo a change. Since both variables are changing, we must use the integral to consider this change: ${\displaystyle W=-PdV=-\int _{V_{i}}^{V_{f}}{\frac {Nk_{B}T}{V}}dV}$. If we then take out the constants, this formula can be rewritten as ${\displaystyle W=-Nk_{B}T\int _{V_{i}}^{V_{f}}{\frac {1}{V}}dV}$. Evaluating the integral gives us the following equation: ${\displaystyle W=-Nk_{B}T{\Bigg (}ln(V_{f})-ln(V_{i}){\Bigg )}}$, which can also be written as: ${\displaystyle W=Nk_{B}Tln{\Bigg (}{\frac {V_{i}}{V_{f}}}{\Bigg )}}$. This is the work done for an ideal gas with a constant temperature and a change in volume.

To illustrate Boyle's Law, you can do a simple experiment involving a syringe, weights, and a pressure meter. Figure 1 and 3 shows the setup used for this demonstration. First, measure the pressure when there are no weights placed on top of the syringe. This should be atmospheric pressure, which is around 101,325 Pascals. To measure the change in pressure and volume, begin by placing weights on top of the syringe until the plunger moves to the next milliliter mark. At this point, measure the mass, pressure, volume, and distance from one milliliter marking to the next. Continue with this procedure until the plunger reaches the 1 milliliter marking or until the pressure sensor reaches its maximum reading. Once you have collected all the data, graph the pressure versus volume. You can also calculate the work done by using the derived formula above and compare it to the formula ${\displaystyle W=mgx}$ where m is the mass in kg of the weights, g is the gravitational acceleration (9.8 m/s2, and x is the distance the plunger moved in meters.

Figure 3 displays a diagram of the experimental setup where the plunger has gone down one milliliter (from 5 mL to 4 mL) due to the force of the 400 g pushing on the top of the plunger. The syringe is hooked up to a pressure meter which displays the pressure readout in kPa on the computer screen using LoggerPro.

Figure 3. Another schematic of a Boyles law experiment.