2.4 Distribution of Molecular Speeds
2 min read•june 24, 2024
Gases are made up of tiny particles zipping around at different speeds. The describes how fast these particles move in an ideal gas. It's a key concept in understanding gas behavior at the molecular level.
This distribution helps us calculate important speeds like the most probable, average, and root-mean-square speed of gas molecules. Temperature plays a big role, affecting how fast the particles move and how their speeds are spread out.
Distribution of Molecular Speeds in Ideal Gases
Maxwell-Boltzmann distribution of speeds
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- Probability distribution of molecular speeds in an ideal gas at assumes no intermolecular interactions and elastic collisions between molecules (billiard balls)
- gives the probability of finding a molecule with a specific speed
- , where is , is Boltzmann constant, and is absolute temperature (Kelvin)
- Distribution is asymmetric and positively skewed with a long tail at high speeds (right-skewed)
- Most molecules have speeds close to the with fewer molecules at very low or very high speeds (bell curve)
Calculation of molecular speeds
- is the speed at which reaches its maximum value
- is the mean speed of all molecules in the gas
- Root-mean-square (rms) speed is the square root of the average of the squares of molecular speeds
- Relationship between these speeds:
- Most probable speed is lower than average speed which is lower than rms speed
- These speeds are inversely proportional to the square root of molecular mass, affecting the distribution of speeds for different gases
Temperature effects on speed distribution
- As temperature increases, Maxwell-Boltzmann distribution shifts to the right and becomes broader (higher speeds)
- Peak of distribution (most probable speed) moves to higher speeds
- Proportion of molecules with higher speeds increases (more high-speed molecules)
- Average, most probable, and rms speeds all increase with square root of absolute temperature
- Doubling absolute temperature increases these speeds by factor of (41% increase)
- At higher temperatures, distribution of molecular speeds becomes more spread out with larger standard deviation (wider distribution)
- Changes in temperature do not affect shape of distribution, only its position and width (still bell-shaped)
Kinetic Theory of Gases and Thermal Equilibrium
- Kinetic theory of gases explains macroscopic properties of gases using the motion of their constituent particles
- is achieved when a system's temperature is uniform and not changing with time
- is the average distance a molecule travels between collisions, affecting the rate of energy transfer in the gas
Key Terms to Review (17)
Average Speed: Average speed is a measure of the total distance traveled by an object divided by the total time taken to travel that distance. It provides a general indication of the overall pace or rate of motion over a given period, regardless of any changes in speed or direction that may have occurred during the journey.
Effusion: Effusion is the process by which gas particles escape from a container through a tiny opening into a vacuum or a lower pressure area. This phenomenon is significantly influenced by factors such as pressure and temperature, as well as the root mean square (RMS) speed of the gas molecules, which affects how quickly and easily they can pass through the opening.
Gas Diffusion: Gas diffusion is the process by which gas molecules spread from an area of high concentration to an area of low concentration, resulting in a uniform distribution of gas particles over time. This process is essential for understanding how gases behave in different environments and is influenced by factors like temperature, pressure, and the mass of the gas molecules, which are all connected to the distribution of molecular speeds.
James Clerk Maxwell: James Clerk Maxwell was a renowned Scottish physicist who made significant contributions to the understanding of electromagnetism, the nature of light, and the foundations of modern physics. His work laid the groundwork for many of the key concepts and theories that are central to the topics of 2.4 Distribution of Molecular Speeds, 6.4 Conductors in Electrostatic Equilibrium, 11.1 Magnetism and Its Historical Discoveries, 12.3 Magnetic Force between Two Parallel Currents, 13.4 Induced Electric Fields, 13.5 Eddy Currents, and 16.1 Maxwell's Equations and Electromagnetic Waves.
Ludwig Boltzmann: Ludwig Boltzmann was an Austrian physicist known for his foundational contributions to statistical mechanics and thermodynamics, particularly in understanding the behavior of particles in gases. His work established a connection between the microscopic properties of matter and macroscopic observations, illustrating how molecular speeds distribute among particles in a gas and how these distributions relate to entropy on a microscopic scale.
Maxwell-Boltzmann distribution: The Maxwell-Boltzmann distribution describes the distribution of speeds of particles in a gas. It shows that most particles have speeds around an average value, with fewer particles moving much slower or much faster.
Maxwell-Boltzmann Distribution: The Maxwell-Boltzmann distribution is a statistical model that describes the distribution of molecular speeds in an ideal gas at a given temperature. It is a fundamental concept in the study of the molecular model of an ideal gas, pressure, temperature, and the distribution of molecular speeds, as well as the microscopic understanding of entropy.
Maxwell-Boltzmann statistics: Maxwell-Boltzmann statistics is a statistical framework that describes the distribution of speeds among particles in a gas. It provides insights into how these particles behave at a given temperature and plays a crucial role in understanding thermodynamic properties. This statistical approach is particularly relevant for ideal gases, where it helps illustrate the relationship between temperature, energy, and the motion of particles, giving a comprehensive view of molecular speed distribution.
Mean free path: The mean free path is the average distance a particle travels before colliding with another particle. It is crucial for understanding gas behavior under various conditions.
Mean Free Path: The mean free path is the average distance a particle, such as a molecule or an electron, travels between successive collisions or interactions with other particles in a medium. This concept is crucial in understanding the behavior of gases, the conduction of electricity in metals, and the propagation of particles through various materials.
Molecular Mass: Molecular mass, also known as molar mass, is the mass of a single molecule or the average mass of all the molecules in a substance. It is an important property in the study of the distribution of molecular speeds, as it directly impacts the kinetic energy and velocity of individual molecules within a gas or liquid.
Most probable speed: The most probable speed is the speed at which the largest number of gas molecules are moving in a given sample. It is derived from the Maxwell-Boltzmann distribution of molecular speeds.
Most Probable Speed: The most probable speed is the speed at which the largest number of molecules in a gas or liquid have at a given temperature. It represents the peak or mode of the Maxwell-Boltzmann distribution of molecular speeds, which describes the statistical distribution of speeds for the individual molecules in a system.
Peak speed: Peak speed is the most probable speed of particles in a gas at a given temperature, as described by the Maxwell-Boltzmann distribution. It is the speed at which the greatest number of particles are moving in a system.
Probability Density Function: The probability density function (PDF) is a mathematical function that describes the relative likelihood of a continuous random variable taking on a particular value. It is a fundamental concept in probability theory and statistics, and is widely used in various fields, including physics, engineering, and finance.
Thermal equilibrium: Thermal equilibrium is the state in which two or more objects in thermal contact no longer exchange heat, resulting in a uniform temperature throughout the system. This occurs when the temperatures of the objects are equal.
Thermal Equilibrium: Thermal equilibrium is a state in which two or more objects or systems have reached the same temperature and no longer exchange heat energy. This concept is fundamental to understanding temperature, thermometers, heat transfer, and the behavior of thermodynamic systems.