St. Petersburg State Polytechnic University
Institute of Applied Mathematics and Mechanics
Department of Theoretical Mechanics
MOLECULE OF CARBON DIOXIDE
course project
Direction of bachelor's training: 010800 Mechanics and mathematical modeling
Group 23604/1
Project Manager:
Admitted to the defense:
St. Petersburg
Chapter 1 Molecular Dynamics 3
1.2 Pair potentials 5
1.2.1 Morse potential. 5
1.2.2 Lennard-Jones potential. 6
1.2.3 Comparison of Morse and Lennard-Jones potentials 7
1.2.4 Graphs of comparison of potentials and forces. 7
1.2.5 Conclusion 9
1.2 Molecule carbon dioxide 9
Chapter 2 Writing Program 10
2.1 Program requirements 10
2.2 Program code. eleven
2.2.1 Variables. eleven
2.2.2 Particle creation function 12
2.2.3 Physics function 14
2.2.4 Power 18 function
2.3 Selection of optimal parameters 19
Results of work 20
Reference List 21
Introduction and problem statement
Modeling molecules, even the simplest ones, is a difficult task. To model them, it is necessary to use many-particle potentials, but their programming is also a very difficult task. The question arises as to whether it is possible to find an easier way to model the simplest molecules.
Paired potentials are well suited for modeling, because they have a simple form and are easy to program. But how can they be applied to molecular modeling? My work is dedicated to solving this problem.
Therefore, the task set before my project can be formulated as follows - to model a carbon dioxide molecule using a pair potential (2D model) and consider its simplest molecular dynamics.
Chapter 1 Molecular Dynamics
Classical method molecular dynamics
The method of molecular dynamics (MD method) is a method in which the temporal evolution of a system of interacting atoms or particles is tracked by integrating their equations of motion
Basic provisions:
- To describe the movement of atoms or particles, classical mechanics. The law of motion of particles is found using analytical mechanics. The forces of interatomic interaction can be represented in the form of classical potential forces (as the potential energy gradient of the system). Exact knowledge of the trajectories of the particles of the system over long periods of time is not necessary to obtain results of a macroscopic (thermodynamic) nature. The sets of configurations obtained in the course of calculations by the molecular dynamics method are distributed in accordance with some statistical distribution function, for example, corresponding to the microcanonical distribution.
The molecular dynamics method is applicable if the De Broglie wavelength of an atom (or particle) is much smaller than the interatomic distance.
Also, classical molecular dynamics is not applicable to modeling systems consisting of light atoms, such as helium or hydrogen. In addition, at low temperatures, quantum effects become decisive, and to consider such systems, it is necessary to use quantums - chemical methods. It is necessary that the times at which the behavior of the system is considered be longer than the relaxation time of the studied physical quantities.
The method of molecular dynamics, originally developed in theoretical physics, has become widespread in chemistry and, since the 1970s, in biochemistry and biophysics. It plays an important role in determining the structure of a protein and refining its properties if the interaction between objects can be described by a force field.
1.2 Pair potentials
In my work, I used two potentials: Lennard-Jones and Morse. They will be discussed below.
1.2.1 Morse potential.
- D is the bond energy, a is the bond length, b is a parameter characterizing the width of the potential well.
The potential has one dimensionless parameter ba. At ba=6, the Morse and Lennard-Jones interactions are close. As ba increases, the width of the potential well for the Morse interaction decreases, and the interaction becomes more rigid and brittle.
A decrease in ba leads to opposite changes - the potential well expands, the rigidity decreases.
The force corresponding to the Morse potential is calculated by the formula:
Or in vector form:
1.2.2 Lennard-Jones potential.
Paired power potential of interaction. Defined by the formula:
- r is the distance between particles, D is the bond energy, a is the bond length.
The potential is a special case of the Mie potential and has no dimensionless parameters.
The interaction force corresponding to the Lennard-Jones potential is calculated by the formula
For the Lennard-Jones potential, the bond stiffness, the critical bond length, and the bond strength, respectively, are
The vector force of interaction is determined by the formula
This expression contains only even powers of the interatomic distance r, which makes it possible not to use the root extraction operation in numerical calculations by the particle dynamics method.
1.2.3 Comparison of Morse and Lennard-Jones potentials
To determine the potential, consider each from a functional point of view.
Both potentials have two terms, one is responsible for the attraction, and the other for the attraction.
The Morse potential contains a negative exponent, one of the fastest decreasing functions. Let me remind you that the exponent has the form for the term responsible for repulsion, and for the term responsible for attraction.
Advantages:
The Lennard Jones potential, in turn, contains power function kind
Where n = 6 for the term responsible for attraction, and n = 12 for the term responsible for repulsion.
Advantages:
- no extraction required square root, since the powers are even when programmed Smoother rise and fall compared to Morse potential
1.2.4 Graphs of comparison of potentials and forces.
1.2.5 Conclusion
From these graphs, 1 conclusion can be drawn - the Morse potential is more flexible, therefore it is more suitable for my needs, because it is necessary to describe the interactions between three particles, and this will require 3 types of potential:
For the interaction between oxygen and carbon (it is the same for each oxygen in the molecule) For the interaction between oxygens in the carbon dioxide molecule (let's call it stabilizing) For the interaction between particles from different molecules
Therefore, in the future I will use only the Morse potential, and I will omit the name.
1.2 Carbon dioxide molecule
Carbon dioxide (carbon dioxide) is an odorless and colorless gas. The carbon dioxide molecule has a linear structure and covalent polar bonds, although the molecule itself is not polar. Dipole moment = 0.
DEFINITION
Carbon dioxide(carbon monoxide (IV), carbon dioxide, carbon dioxide) under normal conditions is a colorless gas, heavier than air, thermally stable, and when compressed and cooled, it easily turns into a liquid and solid (“dry ice”) state.
It is poorly soluble in water, partially reacting with it.
The main carbon dioxide constants are given in the table below.
Table 1. Physical properties and density of carbon dioxide.
Carbon dioxide plays an important role in biological (photosynthesis), natural (greenhouse effect) and geochemical (dissolution in the oceans and formation of carbonates) processes. In large quantities, it enters the environment as a result of burning fossil fuels, rotting waste, etc.
Chemical composition and structure of the carbon dioxide molecule
The chemical composition of a carbon dioxide molecule is expressed by the empirical formula CO 2 . The carbon dioxide molecule (Fig. 1) is linear, which corresponds to the minimum repulsion of binding electron pairs, the length of the C=H bond is 0.116 nm, and its average energy is 806 kJ/mol. Within the framework of the method of valence bonds, two σ -C-O connections formed sp-hybridized orbital of the carbon atom and 2p z - orbitals of oxygen atoms. The 2p x and 2p y orbitals of the carbon atom that do not participate in sp hybridization overlap with similar orbitals of oxygen atoms. In this case, two π-orbitals are formed, located in mutually perpendicular planes.
Rice. 1. The structure of the carbon dioxide molecule.
Due to the symmetrical arrangement of oxygen atoms, the CO 2 molecule is non-polar, therefore the dioxide is slightly soluble in water (one volume of CO 2 in one volume of H 2 O at 1 atm and 15 o C). The non-polarity of the molecule leads to weak intermolecular interactions and a low temperature of the triple point: t = -57.2 o C and P = 5.2 atm.
Brief description of the chemical properties and density of carbon dioxide
Chemically, carbon dioxide is inert, which is due to the high energy of the O=C=O bonds. With strong reducing agents at high temperatures, carbon dioxide exhibits oxidizing properties. With coal, it is reduced to carbon monoxide CO:
C + CO 2 \u003d 2CO (t \u003d 1000 o C).
Magnesium, ignited in air, continues to burn in an atmosphere of carbon dioxide:
CO 2 + 2Mg \u003d 2MgO + C.
Carbon monoxide (IV) partially reacts with water:
CO 2 (l) + H 2 O \u003d CO 2 × H 2 O (l) ↔ H 2 CO 3 (l).
Shows acidic properties:
CO 2 + NaOH dilute = NaHCO 2 ;
CO 2 + 2NaOH conc \u003d Na 2 CO 3 + H 2 O;
CO 2 + Ba(OH) 2 = BaCO 3 ↓ + H 2 O;
CO 2 + BaCO 3 (s) + H 2 O \u003d Ba (HCO 3) 2 (l).
When heated to a temperature above 2000 o C, carbon dioxide decomposes:
2CO 2 \u003d 2CO + O 2.
Examples of problem solving
EXAMPLE 1
| Exercise | During the combustion of 0.77 g of organic matter, consisting of carbon, hydrogen and oxygen, 2.4 g of carbon dioxide and 0.7 g of water were formed. The vapor density of the substance in terms of oxygen is 1.34. Determine the molecular formula of the substance. |
| Solution |
m(C) = n(C)×M(C) = n(CO 2)×M(C) = ×M(C); m(C)=×12=0.65 g; m (H) \u003d 2 × 0.7 / 18 × 1 \u003d 0.08 g. m(O) \u003d m (C x H y O z) - m (C) - m (H) \u003d 0.77 - 0.65 - 0.08 \u003d 0.04 g. x:y:z = m(C)/Ar(C) : m(H)/Ar(H) : m(O)/Ar(O); x:y:z = 0.65/12:0.08/1: 0.04/16; x:y:z = 0.054: 0.08: 0.0025 = 22:32:1. Means the simplest formula compounds C 22 H 32 O, and its molar mass is 46 g / mol. The value of the molar mass of an organic substance can be determined using its oxygen density: M substance = M(O 2) × D(O 2) ; M substance \u003d 32 × 1.34 \u003d 43 g / mol. M substance / M (C 22 H 32 O) \u003d 43 / 312 \u003d 0.13. So all the coefficients in the formula must be multiplied by 0.13. So the molecular formula of the substance will look like C 3 H 4 O. |
| Answer | Molecular formula of the substance C 3 H 4 O |
EXAMPLE 2
| Exercise | When burning organic matter weighing 10.5 g, 16.8 liters of carbon dioxide (N.O.) and 13.5 g of water were obtained. The vapor density of the substance in air is 2.9. Derive the molecular formula of the substance. |
| Solution | Let's make a scheme of the combustion reaction organic compound denoting the number of carbon, hydrogen and oxygen atoms as "x", "y" and "z", respectively: C x H y O z + O z →CO 2 + H 2 O. Let us determine the masses of the elements that make up this substance. Relative atomic mass values taken from Periodic table DI. Mendeleev, rounded up to integers: Ar(C) = 12 a.m.u., Ar(H) = 1 a.m.u., Ar(O) = 16 a.m.u. m(C) = n(C)×M(C) = n(CO 2)×M(C) = ×M(C); m(H) = n(H)×M(H) = 2×n(H 2 O)×M(H) = ×M(H); Calculate the molar masses of carbon dioxide and water. As is known, the molar mass of a molecule is equal to the sum of the relative atomic masses of the atoms that make up the molecule (M = Mr): M(CO 2) \u003d Ar (C) + 2 × Ar (O) \u003d 12+ 2 × 16 \u003d 12 + 32 \u003d 44 g / mol; M(H 2 O) \u003d 2 × Ar (H) + Ar (O) \u003d 2 × 1 + 16 \u003d 2 + 16 \u003d 18 g / mol. m(C) = ×12 = 9 g; m(H) \u003d 2 × 13.5 / 18 × 1 \u003d 1.5 g. m(O) \u003d m (C x H y O z) - m (C) - m (H) \u003d 10.5 - 9 - 1.5 \u003d 0 g. Let's define chemical formula connections: x:y = m(C)/Ar(C) : m(H)/Ar(H); x:y = 9/12: 1.5/1; x:y = 0.75: 1.5 = 1: 2. This means that the simplest formula of the compound is CH 2, and its molar mass is 14 g / mol. The value of the molar mass of an organic substance can be determined using its density in air: Msubstance = M(air) × D(air) ; M substance \u003d 29 × 2.9 \u003d 84 g / mol. To find the true formula of an organic compound, we find the ratio of the obtained molar masses: M substance / M (CH 2) \u003d 84 / 14 \u003d 6. This means that the indices of carbon and hydrogen atoms should be 6 times higher, i.e. the formula of the substance will look like C 6 H 12. |
| Answer | Molecular formula of the substance C 6 H 12 |
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Carbon dioxide, carbon monoxide, carbon dioxide are all names for the same substance we know as carbon dioxide. So what are the properties of this gas, and what are its applications?
Carbon dioxide and its physical properties
Carbon dioxide is made up of carbon and oxygen. The formula for carbon dioxide is CO₂. In nature, it is formed by burning or rotting. organic matter. In the air and mineral springs, the gas content is also quite high. in addition, humans and animals also release carbon dioxide when they exhale.
Rice. 1. Molecule of carbon dioxide.
Carbon dioxide is a completely colorless gas and cannot be seen. It also has no odor. However, with its high concentration, a person may develop hypercapnia, that is, suffocation. Lack of carbon dioxide can also cause health problems. As a result of a lack of this gas, the reverse state of suffocation can develop - hypocapnia.
If carbon dioxide is placed in conditions of low temperature, then at -72 degrees it crystallizes and becomes like snow. Therefore, carbon dioxide in the solid state is called "dry snow".
Rice. 2. Dry snow is carbon dioxide.
Carbon dioxide is 1.5 times denser than air. Its density is 1.98 kg/m³ chemical bond in a carbon dioxide molecule, covalent polar. It is polar because oxygen has a higher electronegativity value.
An important concept in the study of substances is the molecular and molar mass. Molar mass carbon dioxide is 44. This number is formed from the sum of the relative atomic masses of the atoms that make up the molecule. The values of relative atomic masses are taken from the table of D.I. Mendeleev and rounded up to whole numbers. Accordingly, the molar mass of CO₂ = 12+2*16.
To calculate the mass fractions of elements in carbon dioxide, you must follow the formula for calculating the mass fractions of each chemical element in substance.
n is the number of atoms or molecules.
A r- relative atomic mass chemical element.
Mr is the relative molecular weight of the substance.
Calculate the relative molecular weight carbon dioxide.
Mr(CO₂) = 14 + 16 * 2 = 44 w(C) = 1 * 12 / 44 = 0.27 or 27% Since carbon dioxide contains two oxygen atoms, n = 2 w(O) = 2 * 16 / 44 = 0.73 or 73%
Answer: w(C) = 0.27 or 27%; w(O) = 0.73 or 73%
Chemical and biological properties of carbon dioxide
Carbon dioxide has acid properties, since it is an acidic oxide, and when dissolved in water forms carbonic acid:
CO₂+H₂O=H₂CO₃
It reacts with alkalis, resulting in the formation of carbonates and bicarbonates. This gas is non-flammable. It only burns some active metals such as magnesium.
When heated, carbon dioxide breaks down into carbon monoxide and oxygen:
2CO₃=2CO+O₃.
Like other acidic oxides, this gas easily reacts with other oxides:
СaO+Co₃=CaCO₃.
Carbon dioxide is a constituent of all organic substances. The circulation of this gas in nature is carried out with the help of producers, consumers and decomposers. In the process of life, a person produces about 1 kg of carbon dioxide per day. When we inhale, we get oxygen, but at this moment carbon dioxide is formed in the alveoli. At this point, an exchange occurs: oxygen enters the blood, and carbon dioxide goes out.
Carbon dioxide is produced during the production of alcohol. Also, this gas is a by-product in the production of nitrogen, oxygen and argon. The use of carbon dioxide is necessary in Food Industry, where carbon dioxide acts as a preservative, and carbon dioxide in the form of a liquid is contained in fire extinguishers.
Rice. 3. Fire extinguisher.
What have we learned?
Carbon dioxide is a substance that under normal conditions is colorless and odorless. In addition to its common name, carbon dioxide, it is also called carbon monoxide or carbon dioxide.
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But if molecules from the same atoms differ so much, what a variety there must be among molecules from different atoms! Let's look again in the air - maybe we will find such molecules there too? Of course we will!
Do you know what molecules you exhale into the air? (Of course, not only you - all people and all animals.) Molecules of your old friend - carbon dioxide! Bubbles of carbon dioxide pleasantly tingle your tongue when you drink sparkling water or lemonade. Pieces of dry ice that are put in ice cream boxes are also made of such molecules; dry ice is solid carbon dioxide.
In a carbon dioxide molecule, two oxygen atoms are attached from opposite sides to one carbon atom. "Carbon" means "the one who gives birth to coal." But carbon gives birth to more than just coal. When you draw with a simple pencil, small flakes of graphite remain on the paper - they also consist of carbon atoms. Diamond and ordinary soot are “made” of them. Again the same atoms - and completely dissimilar substances!
When carbon atoms combine not only with each other, but also with “foreign” atoms, then so many different substances are born that it is difficult to count them! Especially many substances are born when carbon atoms combine with atoms of the lightest gas in the world - hydrogen. All these substances are called by a common name - hydrocarbons, but each hydrocarbon has its own name.
The simplest of the hydrocarbons is spoken of in the verses you know: “But we have gas in our apartment - this is it!” The name of the gas that burns in the kitchen is methane. The methane molecule has one carbon atom and four hydrogen atoms. In the flame of a kitchen burner, methane molecules are destroyed, a carbon atom combines with two oxygen atoms, and you get the already familiar carbon dioxide molecule. Hydrogen atoms also combine with oxygen atoms, and as a result, molecules of the most important and necessary substance in the world are obtained!
Molecules of this substance are also in the air - there are a lot of them there. By the way, to some extent you are also involved in this, because you exhale these molecules into the air along with carbon dioxide molecules. What is this substance? If you didn’t guess, breathe on the cold glass, and here it is in front of you - water!
Interesting:
The molecule is so tiny that if we lined up one hundred million water molecules one after another, then this whole line would easily fit between two adjacent rulers in your notebook. But scientists still managed to find out what a water molecule looks like. Here is her portrait. True, it looks like the head of a bear cub Winnie the Pooh! Look how you pricked up your ears! Of course, these are not ears, but two hydrogen atoms attached to the “head” - the oxygen atom. But jokes are jokes, but really - do these “ears on top” have anything to do with the extraordinary properties of water?
