Table of Contents

## Quantity sizes

### Volume

Volume is used to describe the **contents of a three-dimensional space**. Besides the SI unit m³, it is also common to state volumes in liters. The volume is calculated very differently depending on the type of body. In general, one could say that the volume is calculated from the base times the height of the body.

**Example: cylinder**. The base of a cylinder is a circle, which can be calculated with the following formula:

**A = π * r ^{2}**

From the formula for the volume (**V = A * h**) the following equation can be followed for a cylinder:

**V _{zylinder} = A * h = π * r^{2} * h**

V ⇒ Volume [m³]

### Mass

Mass is a property of matter, which can be divided into the following groups:

**Inertial mass**which provides resistance for the body during accelerated movement;**Gravitational mass**which states how heavy a body is.

Mass can be determined by measuring a body with a scale. In some cases, the mass can be calculated with the momentum conservation principle. However, there are numerous ways to calculate the mass that vary from case to case.

m ⇒ mass [kg]

### Particle count

The particle count (or, number of particles) corresponds to the **absolute sum of particles within a system**. It is directly proportional to the amount of substance. In macroscopically small systems, in which the particle count can no longer be correctly determined, one refers to Avogadro’s constant, with which the amount of substance can be calculated:

**N = n * N _{A}**

N ⇒ number of particles, no unit

NA ⇒ Avogadro’s constant [1/mole]

n ⇒ amount of substance [mole]

**Note:**One mole of a substance always contains N

_{A}= 6,02 * 10

^{23}particles of this substance.

### Amount of substance and molar mass

The **amount of substance** provides indirect information on the particle count of a system. It is stated in the unit ‘mole.’ The molar mass is required to establish the masses of a substance needed for an experiment. Information on the mass of an atom involved in a reaction can be found in the periodic table of the elements.

The amounts required during a reaction can be calculated with the following formula:

**m = n * M**

n ⇒ amount of substance [mole]

M ⇒ molar mass [g/mole]

**Example:** Magnesium oxide is the result of the reaction between magnesium and oxygen (the ‘burning of magnesium’). Balanced out, one receives the following reaction equation:

**2 Mg + O _{2} → 2 MgO**

According to the reaction equation, one requires two moles of magnesium and one mole of oxygen to receive two moles of magnesium oxide. The molar masses for the substances involved are:

Magnesium: 24.31 g/mol; oxygen: 15.9994 g/mol.

When you incorporate the values into the aforementioned equation, m = n * M, you receive the following dimensions:

Magnesium: m = 2 mole * 24.31 g/mol = 48.62 g.

Oxygen: m = 1 mole * 15.9994 g/mol = 15.9994 g.

Thus, for this reaction you require **48.62 g of magnesium** and **15.9994 g of oxygen**.

### Densities

With constant external conditions, the mass of a body depends on the volume. Both behave directly proportional to one another. This proportionality can be derived from the quotient of mass and volume, if it provides a constant value. At certain temperatures, and at a constant pressure, the quotient of mass and volume is characteristic of a specific substance. It is called density:

**ρ = m / V**

ρ ⇒ density (mass density) [kg/cm³]

**The density of solid, liquid and gaseous substances depends on the temperature. The density of gaseous bodies also depends on the pressure.**

### Particle density

The particle density is defined as the quotient of particle count and volume. The particle count indicates the density of the substance:

**C _{i} = N_{i}/V**

Ci ⇒ particle density [particles/cm³]

Ni ⇒ particle count of a specific substance

## Mass-based sizes

Mass-based sizes are sizes based on the mass of a substance.

### Specific volume

The specific volume is defined as the reciprocal of the density and depends on the volume of the unit of mass. It is described with the following equation:

**V = 1/p = V/m**

v ⇒ specific volume [m³/kg]

Among other things, it is used to create P-V diagrams in thermodynamics, which describe changes in volume and pressure in a system. A **P-V diagram** may look like this:

### Specific heat capacity

The increase in temperature of a body causes an increase in the kinetic energy of its smallest particles. Heating means an input of energy, and cooling means the extraction of energy. The heat that a body takes in is** proportional to the mass and temperature change** of the body. The constant of proportionality is called **specific heating capacity**. It is a material constant.

**Example:** The specific heating capacity for water is c_{H2O} = 4.19 J/(kg*K). This means that the energy of 4.19 kJ is necessary to increase one kilogram of water by one Kelvin.

The specific heating capacity, or the calculation of the heat, is achieved via the following formula:

**ΔQ = c * m * ΔT**

c ⇒ specific heating capacity [J/(kg*K]

ΔQ ⇒ heat output/intake

ΔT ⇒ increase in temperature/decrease in temperature

## Substance mixtures

In nature (including within the human body), matter is rarely encountered as a pure substance. Substances bind together and depending on the bond, they exhibit different features. Substance mixtures, such as cholesterol or calcium phosphate deposition, are **responsible for** the increase in flow resistance of the blood, and thus for **arteriosclerosis**.

Just as much as they are harmful, however, substance mixtures may also be very helpful. Such is the case when vital **iron is formed, stored and released as needed in the liver in the form of ferritin crystals**.

## Mole fraction

The mole fraction is the amount-of-substance fraction and is based on the **number of moles in a gas or liquid mixture**.

**Example:** A mole of air is based on 80 % nitrogen and 20 % oxygen. Therefore, the mole amount equates to 0.2 oxygen and 0.8 nitrogen.

## Mass fraction

A mass fraction is the percent amount of a dissolved substance in relation to the entire mass of the solution.

## States of matter

The states of matter describe the **physical states of a substance** and depend on temperature and pressure. There are three different states:

**Solid**: A fixed alignment and bond between the atoms.**Liquid**: The atoms are mobile and unorganized.**Gas**: There is almost no bond between the atoms.

## Flows of liquids and gases

When measuring blood pressure, the dynamic pressure in the blood is measured. If this pressure is too high, it can be lowered via greater speed of flow. To do so, the volume flow must increase or the flow resistance must be lowered.

The expiratory air must pass through a narrowing in the vocal chords – the so-called glottis. Rheology is the science that describes how high the flow resistance or the volume flow must be to open or close the glottis, the relationships between those factors and what the Bernoulli equation states.

### Flow fields

The flow of ideal liquids and gases is illustrated with streamlines. The streamlines are drawn thicker the higher the flow speed is. There are two types of flow:

**Laminar flow**: Despite obstructions or constrictions, the lines do not cease but rather continue onward.**Turbulent flow**: There is turbulence in obstructions or constrictions.

### Flow resistance and Hagen-Poiseuille’s law

If a body is placed into a liquid, it undergoes a force that, in many cases, turns out to be proportional to the density of the fluid, to the square of the flow speed and to the cross-sectional area.

Newton’s Law states that **every force is counteracted by an equal force** (Newton’s action-reaction principle). This counteracting force is described as the resistance of a body. It is dependent on the viscosity, internal friction and obstructions in the flow.

The flow resistance is calculated with the Hagen-Poiseuille equation:

**R _{S} = 8π * η * Δl / A^{2}**

η ⇒ viscosity of the fluid [(n*s) / m²]

Δl ⇒ length of the flow / length of the pipe [m]

A ⇒ cross-section [m²]

### Volume flow rate

The volume flow rate states how much volume per unit of time is flowing through a cross-section. It is defined with the following equation:

**V = A / Δt * Δl**

V ⇒ volume flow rate [m³/s]

### Continuity equation

With laminar flow, the number of streamlines remains constant. If the cross-section of the pipe is smaller, the streamlines are condensed, which means the flow speed is increasing.

**The smaller the cross-sectional area, the greater the flow speed** as the volume flow rate remains constant.

**V _{1} = V_{2}**

The flow speeds thus behave in the opposite manner from the pipe cross-sections; they are indirectly proportional.

### Bernoulli’s equation

**When a liquid with a specific density flows horizontally through a pipe with a changing cross-section—and provided that friction is insignificant—then, the total pressure remains the same in all parts of the pipe:**

**p _{1} = 0,5ρ * v_{1}^{2} = p_{2} + 0,5ρ * v _{2}^{2}**

For pipes with a slant: The sum of static pressure, hydrostatic pressure and dynamic pressure is constant in every point on a streamline.

### Ohm’s law and real fluids

**Contrary to ideal fluids, real fluids undergo a loss in pressure through internal friction or viscosity**. Friction always causes a loss in kinetic energy, resulting in the fluid adhering to the walls of the pipe.

**That is why a fluid also flows more slowly on the edges**. In comparison, the flow speed in the middle of the flow is greater. The graphic course of such a fluid resembles a parabola:

The zenith of the parabolic course of the flow speed is in the center. Fluids that exhibit such a graphic course when flowing in cylindrical pipes are called **Newtonian fluids**. Ohm’s law applies to such fluids:

**V = Δρ / RS**

There exists a linear relationship between the volume flow rate and pressure difference.

### Series circuit of flow resistance

As described above, the pressure in real fluids decreases and thus sinks with the length of the pipe’; it is also dependent on the cross-section of the pipe. Consecutive resistance behaves like the resistance in an electrical system and is thus aggregated.

### Kirchhoff’s laws

Pipes through which fluids flow can be connected together in different ways:

**Consecutively**, i.e., in a row. The resistance is added together.**Branching**: At each branch node, the sum of the flows is constant or, in other words, the sum of the incoming liquid must be exactly as great as the sum of the outgoing liquid.

## Review Questions

The solutions can be found beneath the references.

**1. Approximately how great is the hydrostatic pressure of a one-meter-high water column?**

- 10
^{4}hPa - 10³ hPa
- 10² hPa
- 10 hPa
- 1 hPa

**2. A fluid flows through a capillary in accordance with Hagen-Poiseuille’s law. When will the current be twice as great?**

- When a capillary with twice the diameter is selected.
- When a capillary with four times the cross-section is selected.
- When a fluid with twice the dynamic viscosity is selected.
- When the pressure difference between the ends of the capillary is doubled.
- When a capillary with twice the length is used.

**3. Choose the corresponding unit for “pressure”!**

- kg * m / s²
- kg * m² / s²
- kg / ( m + s²)
- kg * m² / s³
- kg * m² / s

**4. Water is pumped through two pipes. The pipes have the following dimensions: Pipe 1: length l1 = 2m, radius r1 = 2cm; Pipe 2: length l2 = 1m, radius r2 = 1cm. The drop in pressure is the same in both ends, the flow is laminar. What is the ratio of the volume flow rates v1 and v2?**

- v1 : v2 = 1 : 1
- v1 : v2 = 2 : 1
- v1 : v2 = 4 : 1
- v1 : v2 = 8 : 1
- v1 .: v2 = 16 : 1