Name: gas particle speed distributions and temperature
Uploaded: 2020-09-03T13:38:27.000Z
Duration: 3 min 7 s
Channel: JoVE

K12 Standards

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NC K-12 Science Standard Course of Study

9-12

Chemistry

Energy

PS.Chm.7.3

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Kinetic Molecular Theory: Molecular Velocities, Temperature, and Kinetic Energy

모든 기체 입자는 운동 에너지를 가지고 있으며 이것은 킬로그램 단위의 입자 질량과 초당 미터 단위의 속도의 함수입니다. 매번 충돌할 때마다 개별 기체 입자의 속도가 변화됩니다. 따라서 기체 입자의 집합은 실제로 속도 및 운동 에너지의 분포 또는 범위를 가지고 있습니다.

이것은 어떤 순간에 어떤 분자들은 다른 분자들보다 더 느리게 운동하고 있다는 것을 의미합니다. 그러나 평균 운동 에너지는 보존됩니다. 평균 운동 에너지는 속도의 제곱의 평균 또는 평균 제곱 속도와 관련이 있으며 이 두 가지는 모두 주어진 기체에 대해 일정한 온도에서 일정하게 유지됩니다.

이제 기체 일 몰의 평균 운동 에너지는 아보가드로 수, NA를 도입하여 표현합니다. 입자 질량과 몰 당 입자들의 아보가드로 수의 곱은 몰당 킬로그램 단위의 기체의 몰 질량과 같습니다. 기체 일 몰의 평균 운동 에너지가 온도에 정비례한다는 기체 분자 운동론을 상기해 봅시다.

복잡한 유도 과정을 통해 비례 상수는 2분의 3 R 이라는 것을 발견했습니다. 두 방정식을 결합하고 항을 재배열하고 양 변에 제곱근을 취한 결과, 제곱근 평균 제곱 속도는 몰 질량과 기체의 절대 온도에 관련됩니다. 제곱근 평균 제곱 속도는 몰 질량에 반비례 관계가 있고 온도에 비례하는 관계가 있습니다.

헬륨과 아르곤 두 기체의 온도가 같다고 가정해 보겠습니다. 헬륨은 몰 질량이 작기 때문에 이 방정식에 따르면 헬륨이 아르곤보다 제곱근 평균 제곱 속도가 더 빨라야 합니다. 온도가 같은 헬륨, 아르곤, 염소의 3가지 기체에 대한 분자 속도의 분포도에서도 이와 유사한 관측결과를 볼 수 있습니다.

모든 기체가 동일한 평균 운동 에너지를 가지고 있더라도 가장 가벼운 기체인 헬륨은 분자 속도의 범위가 가장 넒음에 따라 제곱근 평균 제곱 속도가 가장 높고 속도 분포가 가장 넓습니다. 서로 다른 온도에서 기체, 말하자면 아르곤의 속도 분포도에 따르면 더 높은 온도에서 제곱근 평균 제곱 속도가 증가하고 속도 분포가 넓어집니다. 간단히 말해서 기체는 높은 온도에서 더 빨리 운동합니다.

예를 들어 뜨거운 음식에서 나오는 향기로운 기체 입자는 차가운 음식에서 나오는 입자보다 더 빨리 운동합니다. 따라서 차가운 음식보다 뜨거운 음식에 더 빨리 눈이 갑니다.

운동 분자 이론: 분자 속도, 온도 및 운동 에너지

운동 분자 이론은 다양한 기체 법칙에 의해 설명되는 거동을 정성적으로 설명합니다. 이 이론의 가정은 이러한 개별 법칙을 도출하기 위해 보다 정량적인 방식으로 적용될 수 있습니다.

종합적으로, 가스 샘플의 분자는 평균 운동 에너지와 평균 속도를 갖습니다. 하지만 개별적으로는 서로 다른 속도로 움직입니다. 분자는 운동량이 보존되는 탄성 충돌을 자주 겪습니다. 충돌하는 분자는 서로 다른 속도로 편향되기 때문에 개별 분자의 속도는 매우 다양합니다. 그러나 수많은 분자와 충돌이 관련되어 있기 때문에 분자 속도 분포와 평균 속도는 일정합니다. 이 분자 속도 분포는 Maxwell-Boltzmann 분포로 알려져 있으며, 주어진 속도를 갖는 대량의 가스 샘플에 있는 분자의 상대적 수를 나타냅니다.

질량(m)과 속도(u)를 갖는 입자의 운동 에너지(KE)는 다음과 같이 계산됩니다.

<img alt="Eq1" src="https://www.jove.com/files/core/CDNSource/lessons/1043/1043_Equations-01.png" style="height: 40px;" />

질량을 킬로그램으로 표현하고 속도를 초당 미터로 표현하면 에너지 값이 줄(J = kg·m^2/s^2) 단위로 표시됩니다. 많은 수의 가스 분자를 처리하기 위해 속도와 운동 에너지 모두에 대한 평균을 사용합니다. KMT에서 입자의 제곱 평균 제곱 속도(u_rms)는 n = 입자 수인 속도의 제곱 평균의 제곱근으로 정의됩니다.

<img alt="Eq2" src="https://www.jove.com/files/core/CDNSource/lessons/1043/1043_Equations-02.png" style="height: 40px;" />

입자 1몰에 대한 평균 운동 에너지 KE_avg는 다음과 같습니다.

<img alt="Eq3" src="https://www.jove.com/files/core/CDNSource/lessons/1043/1043_Equations-03.png" style="height: 40px;" />

여기서 M은 kg/mol 단위로 표현된 몰 질량입니다. 가스 분자 1몰의 KE_avg도 가스 온도에 정비례하며 다음 방정식으로 설명할 수 있습니다.

<img alt="Eq4" src="https://www.jove.com/files/core/CDNSource/lessons/1043/1043_Equations-04.png" style="height: 40px;" />

여기서 R은 기체 상수이고 T는 켈빈 온도입니다. 이 방정식에 사용될 때 기체 상수의 적절한 형태는 8.314 J/mol⋅K(8.314 kg·m^2/s^2·mol·K)입니다. KE_avg에 대한 이 두 가지 개별 방정식을 결합하고 재배열하여 분자 속도와 온도 사이의 관계를 얻을 수 있습니다.

<img alt="Eq5" src="https://www.jove.com/files/core/CDNSource/lessons/1043/1043_Equations-05.png" style="height: 40px;" />

<img alt="Eq6" src="https://www.jove.com/files/core/CDNSource/lessons/1043/1043_Equations-06.png" style="height: 40px;" />

가스의 온도가 증가하면 KE_avg가 증가하고 더 많은 분자가 더 빠른 속도를 가지며 더 적은 수의 분자가 더 낮은 속도를 가지며 분포는 전체적으로 더 빠른 속도, 즉 오른쪽으로 이동합니다. 온도가 감소하면 KE_avg가 감소하고 더 많은 분자가 더 낮은 속도를 가지며 더 적은 수의 분자가 더 높은 속도를 가지며 분포는 전체적으로 더 낮은 속도, 즉 왼쪽으로 이동합니다.

주어진 온도에서 모든 가스는 분자에 대해 동일한 KE_avg를 갖습니다. 가스의 분자 속도는 분자 질량과 직접적인 관련이 있습니다. 더 가벼운 분자로 구성된 가스는 더 많은 고속 입자와 더 높은 u_rms를 가지며 상대적으로 더 높은 속도에서 최고조에 달하는 속도 분포를 갖습니다. 더 무거운 분자로 구성된 가스는 더 많은 저속 입자, 더 낮은 u_rms 및 상대적으로 낮은 속도에서 최고조에 달하는 속도 분포를 갖습니다.

이 문서는 에서 발췌되었습니다 <a href="https://openstax.org/books/chemistry-2e/pages/9-5-the-kinetic-molecular-theory">Openstax, Chemistry 2e, Section 9.5: Kinetic-Molecular Theory.</a>

운동 분자 이론은 다양한 기체 법칙에 의해 설명되는 행동을 질적으로 설명합니다. 이 이론의 가정은 다음과 같이 적용될 수 있습니다.

Education

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KEY TERMS AND CONCEPTS

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gas particle speed distributions and temperature

Gas particle speed distributions and temperature are key ideas in the kinetic molecular theory. This theory explains the gas laws by describing how gas molecules move, collide, and share energy.

Gas molecules do not all move at the same speed. A gas sample has an average speed and an average kinetic energy, but individual molecules move at different speeds. Their collisions are elastic, which means momentum is conserved. Because so many molecules collide, the overall speed distribution stays constant.

That distribution is called the Maxwell-Boltzmann distribution. It shows the relative number of molecules in a gas sample at each speed. Kinetic energy depends on both mass and speed, and it is measured in joules when mass is in kilograms and speed is in meters per second. For large numbers of molecules, chemists use averages for speed and kinetic energy.

The root mean square velocity, or u rms, is one way to describe average molecular speed. It is the square root of the average of the squared velocities. The average kinetic energy for one mole of gas also depends on molar mass, and it can be written in terms of the gas constant and the kelvin temperature. In this equation, the correct gas constant is 8.314 J/mol·K.

These relationships show how speed changes with temperature. When temperature increases, average kinetic energy increases. More molecules move at higher speeds, fewer move at lower speeds, and the distribution shifts to the right. When temperature decreases, average kinetic energy decreases, more molecules move slowly, and the distribution shifts to the left.

At the same temperature, gases have the same average kinetic energy, even if their molecules have different masses. Molecular mass still affects speed. Lighter gases have more high-speed particles and a higher u rms, while heavier gases have more low-speed particles and a lower u rms. Their speed distributions peak at different velocities, with lighter gases peaking at higher speeds and heavier gases peaking at lower speeds.

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Stoichiometry of Gases with the Ideal Gas Law

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Use models to explain how changes in energy affect the arrangement and movement of the particles in solids, liquids, and gases, as well as the relative strengths of their intermolecular forces.

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Chemical changes and their accompanying ...

First Law of Thermodynamics

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Quantifying Heat

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Quantifying Work

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Using Differential Scanning Calorimetry to Measure Changes in Enthalpy

Source: Laboratory of Dr. Terry Tritt &#8212; Clemson University
Differential Scanning Calorimetry (DSC) is a method of thermodynamic analysis based on ...

Hess's Law

There are two ways to determine the amount of heat involved in a chemical change: measure it experimentally, or calculate it from other experimentally ...

Standard Enthalpy of Formation

Enthalpy changes are typically tabulated for reactions in which both the reactants and products are at the same conditions. A standard state is a commonly ...

Enthalpies of Reaction

Hess&rsquo;s law can be used to determine the enthalpy change of any reaction if the corresponding enthalpies of formation of the reactants and products ...

Use mathematics and computational thinking to execute simple calorimetric calculations based on the Law of Conservation of Energy.

PS.Chm.7.2

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Through experiments, scientists established the mathematical relationships between pairs of variables, such as pressure and temperature, pressure and ...

Applications of the Ideal Gas Law: Molar Mass, Density, and Volume

The volume occupied by one mole of a substance is its molar volume. The ideal gas law, PV = nRT, &nbsp;suggests that the volume of a given quantity of gas ...

Mixtures of Gases: Dalton's Law of Partial Pressures and Mole Fractions

Unless individual gases chemically react with each other, the individual gases in a mixture of gases do not affect each other&rsquo;s pressure. Each gas ...

Chemical Stoichiometry and Gases: Using Ideal Gas Law to Determine Moles

Chemical stoichiometry describes the quantitative relationships between reactants and products in chemical reactions.

In addition to measuring quantities ...

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

The ideal-gas equation, which is empirical, describes the behavior of gases by establishing relationships between their macroscopic properties. For ...

Kinetic Molecular Theory and Gas Laws Explain Properties of Gas Molecules

The test of the kinetic molecular theory (KMT) and its postulates is its ability to explain and describe the behavior of a gas. The various gas laws ...

The kinetic molecular theory qualitatively explains the behaviors described by the various gas laws. The postulates of this theory may be applied in a ...

Behavior of Gas Molecules: Molecular Diffusion, Mean Free Path, and Effusion

Although gaseous molecules travel at tremendous speeds (hundreds of meters per second), they collide with other gaseous molecules and travel in many ...

Real Gases: Effects of Intermolecular Forces and Molecular Volume Deriving Van der Waals Equation

Thus far, the ideal gas law, PV = nRT, has been applied to a variety of different types of problems, ranging from reaction stoichiometry and empirical and ...

Ideal Gas Law

Source: Laboratory of Dr. Andreas Z&#252;ttel - Swiss Federal Laboratories for Materials Science and Technology
The ideal gas law describes the behavior ...

Derivation of the Ideal Gas Law
Gases are a fundamental state of matter. A gas is a collection of molecules that have a significant distance between their ...

Source: Smaa Koraym at Johns Hopkins University, MD, USA

	Preparation of the LaboratoryExpand
	Here, we show the laboratory preparation for 10 students ...

Source: Smaa Koraym at Johns Hopkins University, MD, USA

	Determining the Mass of the Unknown LiquidExpand
	According to the ideal gas law for a given ...

Use mathematics and computational thinking to explain the relationships among pressure, temperature, volume, and quantity of gas, both qualitatively and quantitatively.

Understand the relationship among pressure, temperature, volume, and phase.

Gas Particle Speed Distributions and Temperature

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