13.4 Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature
3 min read•june 18, 2024
Gases are fascinating! Their behavior is governed by the , which links microscopic molecular motion to macroscopic properties like and volume. The ties it all together, showing how affects molecular speeds and energies.
Understanding gas kinetics is key to grasping thermodynamics. We'll explore how temperature changes impact molecular motion, energy distribution, and gas properties. We'll also dive into the , which describes the range of molecular speeds in a gas.
Kinetic Theory of Gases
Ideal gas law and molecular properties
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- Ideal gas law expressed in terms of molecular mass and velocity:
- represents pressure of the gas (atmospheres, pascals)
- represents volume of the gas (liters, cubic meters)
- represents number of molecules in the gas (moles, molecules)
- represents mass of a single molecule (grams, kilograms)
- represents root-mean-square (RMS) speed of the molecules (meters per second)
- Ideal gas law relates macroscopic properties (pressure, volume) to microscopic properties (molecular mass, velocity)
- Assumes gas molecules are point particles with no intermolecular forces and perfectly
- : average distance a molecule travels between collisions, affecting gas behavior
Thermal energy and molecular motion
- is total of all molecules in a substance
- Kinetic energy is energy of motion (translational, rotational, vibrational)
- of a gas directly related to average kinetic energy of its molecules
- Higher temperature increases average kinetic energy of molecules
- Lower temperature decreases average kinetic energy of molecules
- Thermal energy depends on number of molecules and their individual kinetic energies
- More molecules or higher individual kinetic energies increase thermal energy
- : random motion of particles suspended in a fluid, resulting from collisions with molecules
Gas molecule kinetic energy calculation
- Average kinetic energy of a gas molecule directly proportional to absolute temperature:
- represents average kinetic energy of a molecule (joules)
- represents Boltzmann constant ( J/K)
- represents absolute temperature (Kelvin)
- Absolute temperature is temperature measured from (0 K or -273.15 ℃)
- At higher temperatures, gas molecules have more kinetic energy on average
- At lower temperatures, gas molecules have less kinetic energy on average
Temperature effects on gas kinetics
- Temperature measures average kinetic energy of molecules in a gas
- Increasing temperature:
- Molecules move faster, increasing their average kinetic energy
- Molecules collide with container walls more frequently and forcefully, increasing pressure
- Decreasing temperature:
- Molecules move slower, decreasing their average kinetic energy
- Molecules collide with container walls less frequently and forcefully, decreasing pressure
- Temperature changes affect molecular motion and macroscopic gas properties (pressure, volume)
Distribution of molecular speeds
- Speeds of molecules in a gas follow Maxwell-Boltzmann distribution
- Distribution is asymmetric with long tail at high speeds and sharp peak at
- Most probable speed () is speed at which largest number of molecules are moving:
- () is arithmetic mean of all molecular speeds:
- Root-mean-square (RMS) speed () is square root of mean of squares of molecular speeds:
- is always greater than average speed
- Distribution of speeds depends on temperature and molecular mass
- Higher temperature broadens distribution and shifts peak to higher speeds
- Higher molecular mass narrows distribution and shifts peak to lower speeds
Energy distribution and molecular properties
- : energy is equally distributed among all in a system at thermal equilibrium
- Degrees of freedom: independent ways a molecule can store energy (e.g., translational, rotational, vibrational)
- : number of particles in one mole of a substance, crucial for relating molecular and macroscopic properties
Key Terms to Review (30)
Absolute zero: Absolute zero is the theoretical lowest temperature possible, where all molecular motion ceases. It is equal to 0 Kelvin or -273.15 degrees Celsius.
Absolute Zero: Absolute zero is the lowest possible temperature on the temperature scale, where the motion of atoms and molecules reaches its minimum. It is the point at which a system reaches its coldest state and has profound implications in the study of temperature and the kinetic theory of gases.
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 pace or rate at which an object moves over a given period, regardless of any changes in speed or direction that may have occurred during the journey.
Avogadro's number: Avogadro's number, approximately $$6.022 \times 10^{23}$$, is the number of particles, typically atoms or molecules, in one mole of a substance. This constant is crucial in relating macroscopic amounts of a substance to its microscopic properties, allowing for the conversion between grams and moles, which is essential in chemistry and physics.
Brownian motion: Brownian motion is the random movement of particles suspended in a fluid (liquid or gas) resulting from their collision with fast-moving atoms or molecules in the fluid. It provides evidence for the existence of atoms and molecules.
Brownian Motion: Brownian motion is the random, erratic movement of particles suspended in a fluid (liquid or gas) resulting from their collision with the fast-moving molecules in the fluid. This phenomenon was first observed by botanist Robert Brown in 1827 while studying pollen grains suspended in water under a microscope.
Critical temperature: Critical temperature is the highest temperature at which a substance can exist as a liquid, regardless of pressure. Beyond this temperature, the substance becomes a supercritical fluid.
Degrees of Freedom: Degrees of freedom is a fundamental concept in physics that refers to the number of independent ways a system can move or change. It is closely related to the number of variables needed to fully describe the state of a system and is particularly important in the context of the kinetic theory of gases and the statistical analysis of data.
Elastic collisions: Elastic collisions are interactions between two or more bodies in which both momentum and kinetic energy are conserved. In these types of collisions, the objects bounce off each other without any loss of kinetic energy, making them ideal for studying fundamental principles of motion and energy transfer. Understanding elastic collisions is crucial for analyzing two-dimensional interactions and comprehending molecular behavior under varying conditions.
Equipartition Theorem: The equipartition theorem is a fundamental principle in statistical mechanics that describes the distribution of energy among the various degrees of freedom of a system in thermal equilibrium. It states that the average energy associated with each independent quadratic term in the Hamiltonian of a system is equal to $\frac{1}{2}kT$, where $k$ is the Boltzmann constant and $T$ is the absolute temperature of the system.
Ideal Gas Law: The ideal gas law is a fundamental equation in thermodynamics that describes the relationship between the pressure, volume, amount of substance, and absolute temperature of an ideal gas. It is a crucial concept in understanding the behavior of gases and their applications in various fields.
Internal kinetic energy: Internal kinetic energy is the sum of the kinetic energies of all particles within a system. It plays a crucial role in understanding how energy is distributed and conserved during elastic collisions.
James Clerk Maxwell: James Clerk Maxwell was a Scottish physicist known for his groundbreaking contributions to the field of electromagnetism and kinetic theory. He is most famous for formulating a set of equations that describe how electric and magnetic fields interact, ultimately predicting the existence of electromagnetic waves. His work laid the foundation for modern physics, linking various phenomena across disciplines through a unified theoretical framework.
Joule: A joule is the SI unit of work or energy, equivalent to one newton-meter. It measures the amount of work done when a force of one newton displaces an object by one meter in the direction of the force.
Joule: The joule (J) is the standard unit of energy in the International System of Units (SI). It represents the amount of work done or energy expended when a force of one newton acts through a distance of one meter. The joule is a fundamental unit that connects various topics in physics, from work and energy to thermodynamics and electricity.
Kinetic Energy: Kinetic energy is the energy of motion possessed by an object. It is the energy an object has by virtue of being in motion and is directly proportional to the mass of the object and the square of its velocity. Kinetic energy is a crucial concept in physics, as it relates to the work done on an object, the conservation of energy, and various other physical phenomena.
Kinetic Theory: Kinetic theory is a fundamental concept in physics that explains the behavior of gases and other substances at the atomic and molecular level. It provides a framework for understanding the relationship between the properties of matter, such as pressure, temperature, and volume, and the motion and interactions of the individual particles that make up that matter.
Ludwig Boltzmann: Ludwig Boltzmann was an Austrian physicist who made significant contributions to the field of statistical mechanics, particularly in the understanding of the relationship between the microscopic behavior of atoms and molecules and the macroscopic properties of matter, such as pressure, temperature, and entropy. His work laid the foundation for the statistical interpretation of thermodynamics and the kinetic theory of gases.
Maxwell-Boltzmann Distribution: The Maxwell-Boltzmann distribution is a statistical model that describes the distribution of molecular speeds or kinetic energies in an ideal gas at equilibrium. It is a fundamental concept in the kinetic theory of gases, which explains the macroscopic properties of gases in terms of the microscopic motion and interactions of gas molecules.
Mean free path: The mean free path is the average distance a particle travels between collisions with other particles. This concept is crucial for understanding the behavior of gases and heat transfer in materials, as it influences how particles interact and transfer energy, ultimately affecting pressure, temperature, and conduction.
Most Probable Speed: The most probable speed is the speed at which the greatest number of molecules in a gas have at a given temperature. It represents the peak of the Maxwell-Boltzmann distribution of molecular speeds, which describes the statistical distribution of molecular velocities in a gas at thermal equilibrium.
Pascal: Pascal is a unit of pressure, which is the force applied perpendicular to a surface per unit area. It is a fundamental concept in physics that is closely tied to the study of fluids, gases, and the behavior of materials under stress and strain.
Pressure: Pressure is the force exerted per unit area on a surface. It is a fundamental concept in physics that describes the amount of force applied to a given area, and it plays a crucial role in understanding the behavior of fluids, gases, and various physical systems.
PV = 1/3Nm𝑣̄²: The equation PV = 1/3Nm$\bar{v}^2$ is a fundamental relationship in the kinetic theory of gases, which provides an atomic and molecular explanation for the concepts of pressure and temperature. This equation establishes a connection between the macroscopic properties of a gas, such as pressure (P) and volume (V), and the microscopic properties of the gas molecules, including the number of molecules (N), the average molecular speed ($\bar{v}$), and the mass (m) of the individual molecules.
PV = nRT: PV = nRT is the fundamental equation that describes the relationship between the pressure (P), volume (V), amount of substance (n), absolute temperature (T), and the universal gas constant (R) for an ideal gas. This equation is a central concept in the kinetic theory of gases and provides a framework for understanding the behavior of gases at the atomic and molecular level.
RMS Speed: RMS (Root Mean Square) speed is a statistical measure used in the kinetic theory of gases to describe the average speed of gas molecules or atoms. It represents the square root of the mean of the squares of the individual molecular speeds, providing a way to quantify the thermal motion of particles in a gas.
Root-Mean-Square Speed: The root-mean-square (RMS) speed is a statistical measure that represents the typical or average speed of the individual particles in a gas. It provides a way to characterize the distribution of speeds of the molecules or atoms within a system.
Temperature: Temperature is a physical quantity that measures the average kinetic energy of the particles, such as atoms or molecules, in a substance. It is a fundamental concept that is closely related to the behavior of matter and energy in various contexts, including vectors, scalars, coordinate systems, the ideal gas law, kinetic theory, and phase changes.
Thermal energy: Thermal energy is the internal energy of a system due to its temperature. It arises from the random motions of atoms and molecules within the system.
Thermal Energy: Thermal energy is the total kinetic energy of the random motion of the particles (atoms and molecules) that make up a substance. It is a measure of the internal energy of a system due to the vibration and movement of its atoms and molecules. Thermal energy is a fundamental concept that connects the topics of nonconservative forces, conservation of energy, power, world energy use, temperature, kinetic theory, heat transfer, and the second law of thermodynamics.