Name: gas laws behind the ideal equation
Uploaded: 2023-04-30T12:30:10.000Z
Duration: 1 min 17 s
Channel: JoVE

K12 Standards

Tennessee Academic Standards

High School

Chemistry

Matter and Its Interactions

CHEM1.PS1.6

Ideal Gas Equation

Переменными состояния исследуемого газа являются давление, объем, температура и количество моль.
Закон Авогадро гласит, что объем газа пропорционален количеству его молей при постоянной температуре и давлении.
Аналогично, закон Бойля гласит, что давление обратно пропорционально объему газа для данного числа молей и при постоянной температуре.
Кроме того, закон Чарльза дает соотношение между объемом и температурой газа при постоянном давлении и заданным числом молей.
Точно так же закон Гей-Люссака гласит, что давление газа пропорционально его температуре для данного числа молей и при постоянном объеме.
Сочетание этих четырех законов дает уравнение идеального газа. Константа пропорциональности — это универсальная газовая постоянная со значением 8,314 джоулей на моль кельвина в единицах СИ. Он не зависит от типа исследуемого газа.
Уравнение идеального газа описывает любой газ при более высоких температурах и низких давлениях.

<ol>
	<li>Янг, Х.Д. и Фридман, Р.А. (2012). Университетская физика с современной физикой. Сан-Франциско, Калифорния: Пирсон; раздел 18.1; стр. 591–593.</li>
	<li>OpenStax. (2019). Университетская физика Том 2. [Веб-версия]. Получено из <a href="https://openstax.org/details/books/university-physics-volume-2">https://openstax.org/details/books/university-physics-volume-2</a>; раздел 2.1; стр. 68–73.</li>
</ol>

Уравнение идеального газа

Уравнение идеального газа - это уравнение состояния, которое связывает переменные состояния давление, объем, температуру и количество молей гипотетического газа. Это уравнение сочетает в себе четыре эмпирических закона: закон Бойля, закон Шарля, закон Авогадро и закон Гая-Люссака. Когда сочетаются пропорциональности вышеперечисленных четырех эмпирических законов, получается единая константа пропорциональности, известная как универсальная газовая постоянная.

<img alt="Equation1" src="/files/ftp_upload/13667/13667_Equation_1.svg" style="text-align: center; height: 12px;" />

Эта газовая постоянная является одной и той же для всех реальных газов при определенных условиях. Значение универсальной газовой постоянной равно

<img alt="Equation2" src="/files/ftp_upload/13667/13667_Equation_2.svg" style="text-align: center; height: 10px;" />

Но, когда уравнение идеального газа используется в химических расчетах, значение универсальной газовой постоянной равно

<img alt="Equation3" src="/files/ftp_upload/13667/13667_Equation_3.svg" style="text-align: center; height: 12px;" />

Уравнение идеального газа описывает природу любого реального газа, когда его плотность достаточно низка или температура достаточно высока, чтобы он находился далеко от конденсации.

В реальных условиях, уравнение идеального газа применяется к образцу газа с постоянным числом молекул; например, газ может находиться в герметичном контейнере. Если количество молей постоянно, то отношение произведения давления на объем к температуре газа является константой. В таких случаях, отношение произведения давления на объем к температурам в двух разных состояниях газа можно приравнять. Здесь температура должна быть выражена в кельвинах, а давление должно быть абсолютным, то есть суммой избыточного давления и атмосферного давления.

Уравнения идеального газа также можно выразить через постоянную Больцмана

<img alt="Equation4" src="/files/ftp_upload/13667/13667_Equation_4.svg" style="text-align: center; height: 12px;" />

Здесь kB - постоянная Больцмана, и NA - число Авогадро. Уравнение идеального газа может быть выведено с использованием кинетической теории газов. Уравнение идеального газа тесно связано с энергией. Единицы на обеих сторонах уравнения - джоули.

Уравнение идеального газа — это уравнение состояния, которое связывает переменные состояния давление, объем, температуру и количество молей гипотетического ...

Education

JoVE Core

Physics

Физика

Unit 1

Раздел 1

The Kinetic Theory of Gases

Кинетическая теория газов

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

gas laws behind the ideal equation

The ideal gas equation links pressure, volume, temperature, and amount of gas. It describes a hypothetical gas and is built from four empirical gas laws: Boyle’s law, Charles’s law, Avogadro’s law, and Gay-Lussac’s law.

When these proportionalities are combined, they give one constant called the universal gas constant. This gas constant has the same value for all real gases under certain conditions. In chemical calculations, that constant is used in the ideal gas equation.

The equation works best when a gas has low density or a high temperature, so it is far from liquefaction. In many real-world cases, the gas sample has a constant number of molecules. A sealed container is one example. If the number of moles stays constant, then the ratio of pressure multiplied by volume to temperature stays constant as well.

That means the ratio can be set equal for two different states of the same gas. The temperature must be written in kelvin. The pressure must be absolute pressure, which is gauge pressure plus atmospheric pressure.

The ideal gas equation can also be written with Boltzmann’s constant, kB, and Avogadro’s number, NA. It can be derived from the kinetic theory of gases. The equation is also closely related to energy, and the units on both sides are joules.

Gas Laws Behind the Ideal Equation

equation of state

Gas Behavior in the Equation of State

The equation of state describes how a gas’s pressure, volume, temperature, and number of moles are related. These macroscopic properties help define the condition, or state, of the matter in a thermodynamics system. The relationship between these quantities can be written as a simple mathematical expression. In some cases, it is too complex to express in a formula. Even then, the link between the quantities still exists. When an equation of state cannot be written mathematically, scientists...

ideal gas equation

The ideal gas equation links pressure, volume, temperature, and amount of gas. It describes a hypothetical gas and is built from four empirical gas laws: Boyle’s law, Charles’s law, Avogadro’s law, and Gay-Lussac’s law. When these proportionalities are combined, they give one constant called the universal gas constant. This gas constant has the same value for all real gases under certain conditions. In chemical calculations, that constant is used in the ideal gas equation. The equation works...

van der waals equation

How Real Gases Deviate from Ideal Behavior

Real gases often deviate from the ideal gas law, especially at low temperatures and high pressures. The van der Waals equation improves the ideal gas model by adding two corrections that account for these real-world effects. One correction accounts for attractive forces between molecules. These forces become stronger at higher densities and lower the measured pressure, so the equation adds a pressure term based on the square of the molar density and a positive constant a. The second...

pv-diagrams

Liquid-Gas Changes on pV Curves

pV curves show how gas pressure changes with volume. A pV diagram is a graph of pressure versus volume for the gas being studied. It helps describe how the substance behaves. When the gas acts like an ideal gas, the ideal gas equation gives the link between pressure and volume. A common pV curve is an isotherm, which is a line drawn at fixed temperature and fixed number of molecules. For an ideal gas, pressure times volume stays constant. That means volume goes down as pressure goes up. The...

kinetic theory of an ideal gas

Avogadro's Number and Gas Particle Motion

Avogadro's number and gas particle motion help explain how gases behave at the particle level. A mole is the amount of a substance that contains as many molecules as there are atoms in exactly 12 grams of carbon-12. This idea came from Amedeo Avogadro, who proposed that equal volumes of gas at the same pressure and temperature contain equal numbers of molecules, no matter what the gas is. Avogadro's number is the number of molecules in one mole. Its value is 6.02 × 10^23 mol−1. This number...

molecular kinetic energy

Gas Pressure and Molecular Motion

Gas pressure comes from the motion of molecules in a container. The word gas came from a Flemish word meaning “chaos,” and J. B. van Helmont first used it to describe vapors. In a gas, molecules move in a continuous and random way. When a molecule hits a container wall, its velocity changes. The part of the velocity parallel to the wall stays the same. The part perpendicular to the wall reverses direction, but its size does not change. If the molecule’s velocity changes in the x-direction, its...

distribution of molecular speeds

Molecular Speed Curves and Vapor Pressure

Molecular speed curves show how the speeds of gas particles are spread out at a given temperature. For a gas with many molecules, this spread is predictable even though each molecule moves randomly in direction and speed. The pattern is called the Maxwell-Boltzmann distribution. Liquids have a similar speed distribution, but it is not exactly the same as in gases. This difference helps explain boiling and vapor pressure. A molecule must reach at least a certain speed, v a, to escape from the...

maxwell-boltzmann distribution: problem solving

maxwell-boltzmann-distribution-problem-solving

Maxwell-Boltzmann Gas Speed Ratio

The Maxwell-Boltzmann distribution describes how molecular speeds are spread in a gas. Even though individual molecules move in random directions, the whole gas has a predictable speed pattern. This distribution is written as f(v), where v is molecular speed. The distribution function is used to find the expected number of particles with speeds between v1 and v2. Because N is dimensionless, f(v) has units of seconds per meter. The relationship can also be written in differential form for speed...

phase diagram

Water Phase Changes on a Pressure Chart

Water phase changes can be mapped on a pressure and temperature chart called a phase diagram. These plots show which phase a substance is in at different conditions. They also reveal where phase transitions happen, because the boundary lines mark the temperatures and pressures where two phases coexist. The water phase diagram shows how pressure affects boiling. As pressure rises, the boiling temperature increases gradually until it reaches 374 °C at 218 atm. A pressure cooker works this way.

mean free path and mean free time

Gas collisions: path length and time

Gas collisions in a cylinder help explain the mean free path and mean free time. Gas molecules move in random motion. They collide with each other and change speed and direction. The mean free path is the average path length between collisions. The mean free path depends on how many molecules are packed into a volume. It varies inversely with density. When more molecules are present, collisions happen more often, so the mean free path becomes shorter. It also depends on molecule size. Larger...

heat capacity: problem-solving

Heat Capacity of Gases by Degrees of Freedom

Heat capacity of gases depends on the gas type and the motion of its molecules. It is the amount of heat energy needed to raise the temperature of a unit mass of gas by one degree Celsius. This thermodynamic property is important in science and industry. To solve heat capacity problems, first identify the gas. A gas's molecular structure affects its heat capacity because the molecules have different degrees of freedom. These values can be found from tables or from empirical equations. Next,...

dalton's law of partial pressure

Gas Mixtures and Their Partial Pressures

Gas mixtures have a total pressure that comes from the partial pressures of each gas. A partial pressure is the pressure one gas would make if it filled the whole container by itself. This idea helps explain how gases behave when they are mixed together. When two or more gases share a container, their molecules move randomly and collide with one another. If the gases are at the same temperature, they reach thermal equilibrium. Their molecules then have the same average kinetic energy. In that...

escape velocities of gases

Why Lighter Gases Escape Earth First

Earth’s escape velocity is the speed an object needs to leave gravity behind. Near the top of the atmosphere, about 100 km above Earth, that speed is 11.1 km/s. Gas molecules can escape only if their speeds become high enough to reach that limit. The chance of escape depends on gas temperature and molecular speed. According to kinetic theory, the average kinetic energy of gas molecules is proportional to temperature. As temperature rises, the rms speed, or root-mean-square speed, also rises.

key terms and concepts

Learn how Boyle’s, Charles’s, Avogadro’s, and Gay-Lussac’s laws combine into the ideal gas equation, and why kelvin and absolute pressure matter.

Gas Laws Behind the Ideal Equation | JoVE K12

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JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

Arizona Science Standards

Physical Science

HS+C.P1

HS+C.P1U1.3

Alabama Course of Study Science

9-12

Earth and Space Science

Earth's Systems

9-12.ESS.2.7.c

Florida Department of Education

SC.912.P.12

SC.912.P.12.10

Georgia Standards of Excellence

SC5.

gas-laws-pressure-volume-temperature-and-moles

Gas Laws: Pressure, Volume, Temperature, and Moles

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Formula Mass and Mole Concepts of Compounds

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Stoichiometry, Product Yield, and Limiting Reactants

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Understand and be prepared to use values specific to chemical processes: the mole, molar mass, molarity, and percent composition. EXPLANATION: During the early and mid-1800s, a tremendous amount of early research poured into building an understanding of the way that atoms combine. This work required analysis of the percent composition of different substances. Joseph Proust's work with tin oxides led him to publish his law of definite proportions noting that a given pure substance will have a constant ratio for the combination of elements making up the substance. Building on Proust's work, John Dalton published his law of multiple proportions and atomic theory noting that multiple patterns exist for the way that atoms combine, yet the combining atoms are invariable. Meanwhile Amedeo Avogadro hypothesizes that given equal pressures and temperatures, two equal volumes of gas will contain equal numbers of atoms. Acknowledging that the exact number of atoms was unknown in these gaseous volumes, the Latin word "mole" was used. "Mole" translates to "lump" recognizing the ambiguous quantity of gas. A number of attempts have been undertaken historically to quantify the mole. Robert Millikan's oil drop experiment revealed the change of a single electron, which could then be divided into Faraday's Constant for the charge of a mole of electrons. Dividing the charge of a mole of electrons by the charge of a single electron yields the number of electrons in a mole, $6.022\times10^{\wedge}23$. Students should be familiar with how to translate between the given quantities. Student's first use atomism/Dalton's atomic theory in third grade noting that matter is made up of particles too small to be seen. This the first time where students will undertake the process of counting these invisible particles. Electrolysis of water provides a tangible experience to demonstrate the constituent parts of water, though the definite proportions of water may not be easily seen due to the solubility of the gasses produced. The concept of relative mass can be explored by comparing small groups of like items. For example, relative masses of beans can be compared with pinto, white, and black beans. If bags are created with equal numbers of like beans, students can develop a concept of why the smallest bean will have a relative mass of 1, analogous to hydrogen.

CHEM1.PS1.1

Chemical Equations

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The Atomic Theory of Matter

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Demonstrate that atoms, and therefore mass, are conserved during a chemical reaction by balancing chemical reactions. EXPLANATION: Standard 7.PS1.4 explores the idea that chemical reactions are rearrangements of atoms, and students undertake the process of balancing chemical reactions with that standard in seventh grade. The introduction of molar masses in high school makes an explicit connection between the number of atoms, as previously investigated, and the actual masses of the atoms, permitting the collection of more concrete evidence for the conservation of mass in chemical reactions.

CHEM1.PS1.2

Reaction Yield

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Perform stoichiometric calculations involving the following relationships: mole- mole; mass-mass; mole-mass; mole- particle; and mass- particle. Show a qualitative understanding of the phenomenon of percent yield, limiting, and excess reagents in a chemical reaction through pictorial and conceptual examples. (states of matter liquid and solid; excluding volume of gasses). EXPLANATION: This standard builds on the idea that balancing chemical reactions provides evidence for conservation of mass and that the behavior of atoms follows predictable patterns. Students now have the opportunity to utilize this understanding as they perform and evaluate chemical reactions. (The concepts of limiting reagent and percent yield are presented; however, students are not expected to determine these values algebraically. Use of molar volume in stoichiometric processes is beyond the scope of this standard.)

CHEM1.PS1.3

Predicting Reaction Outcomes

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Chemical Reactions in Aqueous Solutions

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Precipitation Reactions

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Oxidation-Reduction Reactions

Oxidation&ndash;Reduction Reactions

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Oxidation Numbers

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Acids, Bases and Neutralization Reactions

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Synthesis and Decomposition Reactions

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Types of Chemical Reactions: Exchange and Reversible

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Use the reactants in a chemical reaction to predict the products and identify reaction classes (synthesis, decomposition, combustion, single replacement, double replacement). EXPLANATION: Chemical reactions are rearrangements of atoms that follow predicable patterns. There are patterns both at the macroscopic level in the behavior of some of the reaction classes, as well as patterns in the rearrangements of the atoms underlying the reaction. Students should be able to predict the products of the reactions, which also requires the ability to recognize the general patterns for each type of reaction.

CHEM1.PS1.4

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Kinetic Molecular Theory and Gas Laws Explain Properties of Gas Molecules

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Basic Postulates of Kinetic Molecular Theory: Particle Size, Energy, and Collision

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Ideal Gas Law

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Kinetic Theory of an Ideal Gas

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Conduct investigations to explore and characterize the behavior of gasses (pressure, volume, and temperature), develop models to represent this behavior, and construct arguments to explain this behavior. Evaluate the relationship (qualitative and quantitatively) at STP between pressure and volume (Boyle's Law), temperature and volume (Charles' Law), temperature and pressure (Gay-Lussac Law) and moles and volume (Avogadro's Law), and evaluate and explain these relationships with respect to kinetic molecular theory. Be able to understand, establish, and predict the relationships between volume, temperature, and pressure using combined gas law both qualitatively and quantitatively. EXPLANATION: When exploring the behavior of gasses, it is important to consider experimental design. Experiments used to show the relationships between these sets of variables should include one independent and one dependent variable. Other variables should be held constant. Pressure should serve as the dependent variable because it cannot be manipulated directly. Avogadro's Law can be used to relate the amount of gas in two different containers in lieu of knowing actual number of gas particles in an individual container. Individual demonstrations can be performed to explore each of the different gas laws.

CHEM1.PS1.5

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Analyze solutions to identify solutes and solvents, quantitatively analyze concentrations (molarity, percent composition, and ppm), and perform separation methods such as evaporation, distillation, and/or chromatography and show conceptual understanding of distillation. Construct argument to justify the use of certain separation methods under different conditions. EXPLANATION: In seventh grade, students developed an understanding that chemical and physical properties of a sample of matter could be used to identify a substance. Building on this understanding, students now explore the use of these physical and chemical properties to separate mixtures. Analysis of the substances is performed to aid in further discussion and facilitate further quantitative analysis.

CHEM1.PS1.7

Gas Laws Behind the Ideal Equation

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