Root mean square speed
Encyclopedia
Root-mean-square speed is the measure of the speed of particles in a gas that is most convenient for problem solving within the kinetic theory
of gases. It is defined as the square root of the average velocity-squared of the molecules in a gas. It is given by the formula
where vrms is the root mean square
of the speed, Mm is the molar mass
of the gas, R is the molar gas constant, and T is the temperature
in kelvin
. Although the molecules in a sample of gas have an average kinetic energy (and therefore an average speed) the individual molecules move at various speeds, i.e. they exhibit a distribution of speeds. Some move fast, others relatively slowly. Collisions change individual molecular speeds but the distribution of speeds remains the same. This equation is derived from kinetic theory of gases using Maxwell–Boltzmann distribution function. The higher the temperature, the greater the mean velocity will be. This works well for both nearly ideal, atomic gases like helium
and for molecular gas
es like diatomic oxygen
. This is because despite the larger internal energy in many molecules (compared to that for an atom), 3RT/2 is still the mean translational
kinetic
energy.
This can also be written in terms of the Boltzmann constant (k) as
where m is the mass of one molecule of the gas.
This can be derived with energy methods:
where Ek is the kinetic energy
and N is the number of gas molecules.
Given that v2 ignores direction, it is logical to assume that the formula can be extended to the entire sample, replacing m with the entire sample's mass, equal to the molar mass times the number of moles n yielding
Therefore
which is equivalent.
The same result is obtained by solving the Gaussian integral
containing the Maxwell speed distribution
, p(v):
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...
of gases. It is defined as the square root of the average velocity-squared of the molecules in a gas. It is given by the formula
where vrms is the root mean square
Root mean square
In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...
of the speed, Mm is the molar mass
Molar mass
Molar mass, symbol M, is a physical property of a given substance , namely its mass per amount of substance. The base SI unit for mass is the kilogram and that for amount of substance is the mole. Thus, the derived unit for molar mass is kg/mol...
of the gas, R is the molar gas constant, and T is the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
in kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
. Although the molecules in a sample of gas have an average kinetic energy (and therefore an average speed) the individual molecules move at various speeds, i.e. they exhibit a distribution of speeds. Some move fast, others relatively slowly. Collisions change individual molecular speeds but the distribution of speeds remains the same. This equation is derived from kinetic theory of gases using Maxwell–Boltzmann distribution function. The higher the temperature, the greater the mean velocity will be. This works well for both nearly ideal, atomic gases like helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...
and for molecular gas
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
es like diatomic oxygen
Oxygen
Oxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...
. This is because despite the larger internal energy in many molecules (compared to that for an atom), 3RT/2 is still the mean translational
Translation (physics)
In physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...
kinetic
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
energy.
This can also be written in terms of the Boltzmann constant (k) as
where m is the mass of one molecule of the gas.
This can be derived with energy methods:
where Ek is the kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
and N is the number of gas molecules.
Given that v2 ignores direction, it is logical to assume that the formula can be extended to the entire sample, replacing m with the entire sample's mass, equal to the molar mass times the number of moles n yielding
Therefore
which is equivalent.
The same result is obtained by solving the Gaussian integral
Gaussian integral
The Gaussian integral, also known as the Euler-Poisson integral or Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.It is named after the German mathematician and...
containing the Maxwell speed distribution
Maxwell Speed Distribution
Classically, an ideal gas' molecules bounce around with somewhat arbitrary velocities, never interacting with each other. In reality, however, an ideal gas is subjected to intermolecular forces. It is to be noted that the aforementioned classical treatment of an ideal gas is only useful when...
, p(v):
