In problem 18 at the end of Chapter \(19,\) you might have calculated the number of molecules of oxaloacetate in a typical mitochondrion. What about protons? A typical mitochondrion can be thought of as a cylinder \(1 \mu \mathrm{m}\) in diameter and \(2 \mu \mathrm{m}\) in length. If the \(\mathrm{pH}\) in the matrix is \(7.8,\) how many protons are contained in the mitochondrial matrix?

Calculating the volume of a mitochondrion is crucial for various cellular biology calculations, including determining the number of molecules or ions within this organelle. If we consider a mitochondrion as a simple cylindrical shape, we can use the formula for the volume of a cylinder: \( V = \pi r^2 h \) where \( r \) is the radius, and \( h \) is the height of the cylinder.

To find the volume of a typical mitochondrion, which is described as being \( 1 \mu m \) in diameter and \( 2 \mu m \) in length, we first calculate the radius by halving the diameter, resulting in \( r = 0.5 \mu m \) . Next, we apply the height \( h = 2 \mu m \) into the formula, doing all calculations in micrometers. However, to align with units commonly used in concentration calculations, we need to convert this volume from cubic micrometers to liters, since concentrations are typically given in moles per liter.

Through appropriate unit conversions, this step transcends from simple geometry to being an integral technique in quantitative cell biology studies.

Understanding pH and Proton Concentrations

The measure of acidity or alkalinity of a solution is expressed in terms of pH, which is the negative logarithm of the proton (\( H^+ \) ions) concentration. The lower the pH, the higher the concentration of protons. The relationship is given by the formula \( [H^+] = 10^{-\text{pH}} \).

For a pH of 7.8 inside the mitochondrial matrix, the proton concentration can be calculated using the aforementioned formula to find that the concentration is \( 10^{-7.8} \) moles of \( H^+ \) ions per liter. It's essential to appreciate how such a seemingly abstract number like pH can so directly indicate the presence of protons, which play vital roles in cellular processes such as energy production and homeostasis.

The Bridge from Moles to Molecules

Avogadro's number, \( 6.022 \times 10^{23} \), is a fundamental constant in chemistry that represents the number of units (atoms, molecules, ions, etc.) in one mole of substance. To translate the quantity of a substance from moles to actual countable units, we multiply the number of moles by Avogadro's number.

Applying this principle in our mitochondrial scenario, once we have calculated the number of moles of protons in the mitochondrial volume, we can find out the exact number of protons by multiplying it with Avogadro's number. This elucidates the tremendous scale on which biological interactions take place, honing the understanding that cellular structures, though minuscule, contain an astronomical number of particles, each contributing to the larger function of the cell and, ultimately, life itself.