Any fact and any object in the universe obeys precise laws that mankind has been able to explain over the centuries, thanks to the research of scientists, physicists, and mathematicians from the past and present. Let’s examine the secrets of physics hidden in the creation of espresso.

**Pressure**

Pressure is the force exerted per unit of area. The formula for its calculation is:

where:

*P*= pressure, expressed in pascals (Pa)*F*= force, expressed in newton (N)*S*= surface, expressed in square meters (m²)

The inverse formulas for determining strength and surface are simply obtained from this equation. The pascal is a unit of measurement for pressure, which is equivalent to the pressure exerted by 1 N on the surface of 1 m^{2}. It is a very small unit of measurement; hence, some more practical multiples are employed for convenience:

- 1 kilopascal (1 kPa) = 10
^{3}Pa - 1 megapascal (1 MPa) = 10
^{6}Pa - 1 gigapascal (1 GPa) = 10
^{9}Pa

There are alternative units of measurement for pressure control, such as the bar and the atmosphere.

**The Moka coffee maker**

Let’s talk about the famous Moka coffee maker, a real steam machine used by millions of people today for the preparation of their espresso coffee. Its design follows many rules of physics and its project was obviously patented a long time ago. The Moka is made up of three main parts:

- The base, also known as the boiler, which also includes a safety valve
- The metal filter
- The collector, made up of a central chimney inside

The Moka’s functioning is both simple and unique. Figure 1 shows the various phases that occur one after the other during the water-boiling process. The boiler is half-filled with water. The other half is, of course, occupied by air. When you turn on the stove, the air heats up and increases its pressure, trying to expand. In other words, it exploits the pressure exerted by the steam (after boiling the water) to trace and mark the path of the drink. This expansion forces the water lower, where it finds its only way out through the filter tube and climbs toward the coffee powder. As a result, more and more steam forms, which rapidly saturates the space and, having no space left, begins to force the water down, pushing it to flow through the only way out. Finally, the liquid condenses in the collector, infused with the aroma of coffee. As can be seen, it is a game of thrusts and pressure that forcefully deviate the liquid’s route. Each physical attribute of the Moka’s components, such as the length of the chimney and the filter pipe, is properly calibrated. These characteristics have a decisive influence on the temperature of the drink and on the concentration of coffee, making them important for the drink’s flavor.

Here’s a detailed description of coffee making:

- Initially, the water heats up in a hermetically sealed space, where the water occupies most of the available space.
- The water in the Moka quickly reaches the boiling temperature.
- The additional heat increases the pressure of the steam inside.
- When the compressed air reaches the equilibrium point, it forces the boiling water via the single escape route, reaching the coffee powder stored in the filter.
- Finally, the hot water, now blended with the coffee powder, aroma, and perfume, reaches the collector as a delicious, readymade drink.

The taste of the final product depends on several factors, such as:

- Water temperature
- Type of mixture used
- Coffee beans
- Quantity and compression of the coffee powder
- Type of filter
- Time of preparation

As you can see, the variables are many, and it is probably not possible to find a single mathematical formula capable of expressing the taste of the drink.

**Water boiling**

The water-boiling temperature is not always constant but largely depends on external pressure. Figure 2 shows the graph of the boiling point as a function of external pressure. At the pressure of one atmosphere, the boiling temperature of water is equal to 100˚C. At an altitude of 2,000 meters above sea level, the boiling temperature of water is approximately 90˚C. The x-axis is in logarithmic scale. One of the laws governing this process is dictated by the “Vapor Pressure Model” formula:

In this specific case, the values of the coefficients *a*, *b*, and *c* are equivalent to:

*a*= 4.62239*b*= –0.023197*c*= 0.250226

The graph, and especially the formula, make it clear that if the pressure is high, the water begins to boil at much higher temperatures. Thus, a Moka can be an extremely dangerous device, and if it were to explode, it would have serious effects. Another interesting formula for calculating the boiling temperature, knowing the altitude above sea level, is the following:

Or, with the more generic rational model:

where:

*a*= 100*b*= –49615971E-03*c*= –48858384E-06*d*= –70131171E-12

These formulas also state that, at least at low altitudes, for each unit increase in altitude, the boiling temperature drops by 3.3-thousandths of a degree. In other words, water does not boil for a purely thermal reason, but it boils when its surface tension equals the pressure to which it is subjected (usually atmospheric pressure). At 3,000 meters above sea level, the atmospheric pressure is much lower than what can be calculated at sea level, so the water boils at a lower temperature. The table below shows the boiling temperatures of water at various altitudes. On Mount Everest (about 8,800 meters), the water boils at about 71˚C. As the height is increased, the water boils at body temperature (about 60,000 feet, or 18,000 meters). This altitude is known as the “Armstrong limit.”

Another alternative formula is the following:

where:

*Te(z)*= the absolute boiling temperature of water, expressed in kelvins (K)*z*= the altitude, expressed in meters (m)

To obtain the temperature in degrees Celsius, you have to subtract 273K from the absolute temperature of the result of the equation.

**Features of a perfect Espresso coffee**

Coffee preparation is obviously different from country to country. There are countries in the world that like to prepare coffee with a lot of water, while other countries prefer it very strong and concentrated. The following are the characteristics and conditions of a perfect Italian espresso coffee, one of the best in the world, referring to the quantity of a cup:

- The temperature of the water leaving the boiler must be between 86˚C and 90˚
- The temperature of the drink in the cup must be very hot, between 64˚C and 70˚
- The quantity of the drink in the cup (including the cream) should be about 25 mL.
- The amount of caffeine must be less than 100 mg.
- The necessary quantity of ground coffee should be about 7 g.
- The water pressure should reach 9 bar.
- Finally, the viscosity of the coffee, at a temperature of 45˚C, should be greater than 1.5 MPa.

These characteristics cannot be controlled when preparing coffee in a café or at home but appropriate laboratories have studied and defined them. Another fundamental aspect is the brewing speed of the coffee machine. An excellent parameter is 1 mL/s; therefore, dispensing the drink in a normal cup would take exactly 25 seconds (see Figure 3). If the coffee is delivered in 12 seconds, therefore too fast, it means that a part of the precious substance is left in the powder. As a result, the drink will be unbalanced and more bitter than usual. If, on the other hand, the extraction time is 40 seconds, the woody and bitter substances will also be extracted from the coffee powder.

**A coffee pot can become a bomb**

When the coffee powder is excessively compressed, it closes the small holes in the filter, obstructing the entire system. The effective radius of the capillaries, in fact, becomes extremely reduced. In this way, the air chamber becomes isolated and watertight. A Moka can become really dangerous. For example, suppose that the filter and the safety valve are completely clogged. A small amount of water is heated in a modest volume of space. The water turns into steam at a temperature of about 370˚C (due to the pressure). The steam pressure inside the Moka can be calculated using the following equation:

The pressure is approximately 100,000,000 Pa, which corresponds to about 1,000 atmospheres, and the energy is extremely high, equal to 50 kJ. An explosion would be terrible, much like a real bomb, with shrapnel flying at speeds exceeding 700 km/h.

**Conclusion**

As demonstrated in the article, even a nice cup of coffee is subject to the severe laws of physics and mathematics. This is the fate of everything in the universe. The matter still hides many secrets. When you take a sip of hot coffee, realize that it is the result of a lengthy history of innovations, discoveries, and ongoing technical advancements that has resulted in the drink that makes millions of people around the world happy.