2.2 Pressure, Temperature, and RMS Speed
3 min read•june 24, 2024
Gas behavior is all about molecular motion. Pressure comes from molecules hitting container walls, while temperature relates to their average kinetic energy. The ideal gas law connects these macroscopic properties to the microscopic world of molecules.
Gases in mixtures exert partial pressures, adding up to the total pressure. Molecules constantly collide, with their mean free path and depending on conditions. The describes the range of molecular speeds in a gas.
Kinetic Theory of Gases
Microscopic vs macroscopic gas properties
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- Pressure caused by gas molecules colliding with container walls
- Directly proportional to number of collisions per unit time and average force per collision
- Example: higher pressure in a tire due to more frequent and forceful collisions of air molecules with tire walls
- Temperature related to average kinetic energy of gas molecules
- Higher temperature means higher average kinetic energy and faster moving molecules
- Example: molecules in a hot air balloon move faster than those in a cold room
- Systems in have the same temperature and average kinetic energy
- () measures average speed of gas molecules
- Calculated using , where is Boltzmann constant, is absolute temperature, and is mass of a single molecule
- Example: of nitrogen molecules at room temperature (20℃) is about 511 m/s
- Ideal gas law [PV = nRT](https://www.fiveableKeyTerm:PV_=_nRT) relates pressure (), volume (), number of moles (), gas constant (), and absolute temperature ()
- Describes behavior of ideal gases under various conditions
- Example: increasing temperature of a gas in a closed container increases its pressure
- The number of molecules in one mole of any substance is given by
Partial pressures in gas mixtures
- ###'s_Law_0### states total pressure of a gas mixture is sum of partial pressures of each component gas
- , where are partial pressures of each gas in mixture
- Example: in air, of oxygen is about 21% of total atmospheric pressure
- of a gas in a mixture is pressure gas would exert if it occupied entire volume alone
- Calculated using , where is partial pressure of gas , is of gas , and is total pressure of mixture
- Example: in a 60% nitrogen, 40% oxygen mixture at 1 atm total pressure, partial pressure of nitrogen is 0.6 atm and oxygen is 0.4 atm
- Molar fraction is ratio of number of moles of a particular gas to total number of moles in mixture
- Calculated using , where is number of moles of gas and is total number of moles in mixture
- Example: in a mixture of 3 moles of helium and 1 mole of neon, molar fraction of helium is 0.75 and neon is 0.25
Molecular motion in gases
- Mean free path () is average distance a molecule travels between collisions
- Calculated using , where is diameter of molecule and is number density (molecules per unit volume)
- Example: mean free path of air molecules at room temperature and pressure is about 68 nm
- Collision frequency () is average number of collisions per unit time for a single molecule
- Calculated using , where is diameter of molecule, is number density, and is
- Example: collision frequency of nitrogen molecules at room temperature and pressure is about collisions per second
- Higher pressure and temperature lead to:
- Shorter mean free paths due to more molecules in a given volume
- Higher collision frequencies due to faster moving molecules
- Example: in a high-pressure gas cylinder, molecules have shorter mean free paths and higher collision frequencies compared to the same gas at atmospheric pressure
Statistical distribution of molecular speeds
- The describes the probability distribution of molecular speeds in a gas
- The states that energy is equally distributed among all degrees of freedom in a system
- is the net movement of particles from regions of high concentration to low concentration
- occurs when gas molecules escape through a small hole in a container
Key Terms to Review (32)
$ ext{lambda} = \frac{1}{ ext{sqrt}{2} ext{pi} d^2 n}$: This key term represents the wavelength of a wave, which is the distance between consecutive peaks or troughs of a wave. It is a fundamental concept in the study of wave phenomena, including topics such as pressure, temperature, and root-mean-square (RMS) speed.
$P_{total} = P_1 + P_2 + ... + P_n$: The total pressure in a system is equal to the sum of the individual pressures exerted by each component or part of the system. This term is particularly relevant in the context of 2.2 Pressure, Temperature, and RMS Speed, as it describes how the overall pressure in a system is determined by the contributions of the various pressure sources.
$P_i = x_i P_{total}$: $P_i = x_i P_{total}$ is an equation that describes the relationship between the partial pressure of a gas component ($P_i$) and the total pressure of a mixture of gases ($P_{total}$). The term $x_i$ represents the mole fraction or the proportion of the particular gas component in the mixture.
This equation is particularly relevant in the context of understanding pressure, temperature, and root-mean-square (RMS) speed, as it provides a fundamental way to analyze the behavior of gases and their properties within a mixture.
$v_{rms} = \sqrt{\frac{3kT}{m}}$: The root-mean-square (RMS) speed, denoted as $v_{rms}$, is a statistical measure of the average speed of particles in a gas or liquid. It represents the square root of the average of the squares of the individual particle speeds, and is a key concept in the kinetic theory of gases.
$x_i = \frac{n_i}{n_{total}}$: $x_i = \frac{n_i}{n_{total}}$ is a key term that represents the fraction or proportion of a particular component $n_i$ within a total population $n_{total}$. This term is widely used in various contexts, including the analysis of pressure, temperature, and root-mean-square (RMS) speed in physics.
$z = \sqrt{2} \pi d^2 n v_{rms}$: This term represents the expression for the root-mean-square (RMS) speed of gas molecules in a container, which is a fundamental concept in the study of pressure, temperature, and the kinetic theory of gases. It provides a measure of the average speed of the gas molecules and is a key factor in understanding the relationships between pressure, temperature, and the behavior of gases.
Avogadro's number: Avogadro's number is a fundamental constant in chemistry and physics, defined as approximately 6.022 x 10^23, which represents the number of atoms, molecules, or particles in one mole of a substance. This number serves as a bridge between the macroscopic world we observe and the microscopic world of atoms and molecules, enabling calculations involving quantities of gas, relationships between pressure and temperature, and understanding molecular behavior in ideal gas scenarios.
Celsius Scale: The Celsius scale is a temperature scale that measures temperature based on the freezing and boiling points of water. It is widely used in scientific and everyday applications to quantify temperature in degrees Celsius (°C).
Collision Frequency: Collision frequency is the rate at which particles or molecules collide with each other within a system. It is a fundamental concept in the understanding of pressure, temperature, and root-mean-square (RMS) speed in the context of kinetic theory of gases.
Dalton: The dalton, also known as the unified atomic mass unit (u), is the standard unit of mass that quantifies mass on an atomic or molecular scale. It is defined as one twelfth of the mass of a carbon-12 atom.
Dalton's Law: Dalton's Law is a fundamental principle in the field of physics that describes the behavior of gases. It states that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of each individual gas present in the mixture, assuming the gases do not interact chemically with each other.
Dalton’s law of partial pressures: Dalton's law of partial pressures states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of each individual gas. Each gas in a mixture acts independently and contributes to the total pressure proportionally to its amount.
Diffusion: Diffusion is the process by which molecules move from an area of high concentration to an area of low concentration, resulting in the even distribution of particles in a medium. This movement is driven by the kinetic energy of the particles, which causes them to spread out and mix with other substances. In the context of pressure, temperature, and RMS speed, diffusion is influenced by these factors, as higher temperatures and lower pressures can increase the rate of molecular movement and therefore enhance the diffusion process.
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.
Equipartition theorem: The equipartition theorem states that each degree of freedom in a system at thermal equilibrium contributes an average energy of $\frac{1}{2}k_BT$ per particle, where $k_B$ is Boltzmann's constant and $T$ is the temperature.
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 quadratic degree of freedom of a system in thermal equilibrium is equal to $\frac{1}{2}k_BT$, where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature.
Escape velocity: Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a massive body without further propulsion. It depends on the mass and radius of the body being escaped from.
Kelvin Scale: The Kelvin scale is a temperature scale that defines the absolute zero point as the lowest possible temperature. It is the base unit of temperature in the International System of Units (SI) and is used to measure the thermal energy of a system.
Kinetic Theory of Gases: The kinetic theory of gases is a model that explains the macroscopic properties of gases, such as pressure, temperature, and root mean square (RMS) speed, by considering the microscopic behavior of gas molecules. It treats gas molecules as tiny, spherical particles in constant random motion, colliding with each other and the walls of the container.
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.
Mean free time: Mean free time is the average time interval between successive collisions of a gas particle. It quantifies how frequently particles collide in a given medium.
Molar Fraction: Molar fraction, also known as mole fraction, is a measure of the relative amount of a specific component in a mixture. It represents the ratio of the number of moles of a particular substance to the total number of moles in the entire mixture, and is a dimensionless quantity that provides information about the composition of a system.
Partial pressure: Partial pressure is the pressure exerted by a single component of a mixture of gases. It is proportional to its mole fraction in the gas mixture and the total pressure of the mixture.
Partial Pressure: Partial pressure is the pressure exerted by a specific gas in a mixture of gases. It is the portion of the total pressure that can be attributed to a particular gas component within a given volume or system.
PV = nRT: The equation PV = nRT is known as the ideal gas law, which relates the pressure (P), volume (V), and temperature (T) of an ideal gas to the amount of substance in moles (n) and the ideal gas constant (R). This fundamental equation connects these variables, indicating how gases behave under varying conditions. Understanding this relationship is crucial for grasping concepts like gas pressure, temperature scales, and the speed of gas molecules.
RMS speed: RMS speed, or root mean square speed, is a statistical measure used to quantify the average speed of particles in a gas. It is especially important in kinetic theory, where it relates to the temperature and pressure of a gas. The RMS speed provides insights into the energy of the gas molecules and their motion, making it a key concept when studying the behavior of gases under different conditions.
Root-mean-square (rms) speed: The root-mean-square (rms) speed is the measure of the average speed of particles in a gas, calculated as the square root of the average of the squares of individual particle speeds. It is an important parameter in understanding kinetic theory and temperature relations.
Root-mean-square speed: Root-mean-square speed is a statistical measure of the average speed of particles in a gas, calculated as the square root of the average of the squares of individual speeds. This concept helps to relate molecular motion to properties like pressure and temperature, illustrating how gas behavior can be understood in terms of kinetic energy and temperature relationships.
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.
Vapor pressure: Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature. It indicates the tendency of a substance to evaporate.