The root mean square speed is a quantity used to describe the motion of particles in a gas and it is the square root of the average square velocity of all the particles. It can be used to find the pressure, volume and temperature of a gas.
Equations
description
equation
Pressure and volume of an ideal gas
pV=31Nmc2
Mean kinetic energy
21mc2=23kT
Constants
constant
symbol
value
Boltzmannconstant
k
1.38×10−23JK−1
Variable definitions
quantity name
symbol
derived units
si base units
pressure
p
Pa
kgm−1s−2
volume
V
m3
m3
temperature
T
K
K
numberofatoms
N
mass
m
kg
kg
velocity
c
ms−1
ms−1
The root mean square speed
The root mean square speed cr.m.s. (r.m.s. speed) is a quantity used to describe the motion of particles and it is essentially the average speed of each particle.
To calculate it you need to square the velocity c of each particle in a gas and then find the average known as the mean square speed c2. The square root of this quantity is the r.m.s. speed:
c2=cr.m.s.
Note: The bar (c) means average.
The reason that r.m.s speed is used, rather than the velocity, is because velocity is a vector. There could be both negative and positive values for velocity, but this is not the case for speed as it is a scalar.
Therefore, if you added all of the velocities together, they will cancel out to zero, because of the large number of particles you are taking an average from. So, by squaring the velocities, this removes the signs, and gives the squared speed.
Pressure and volume of an ideal gas
The mean square speed can be used to find the volume and pressure of an ideal gas with N particles of mass m through the equation:
pV=31Nmc2
This equation is significant as it describes the properties of a macroscopic object based on the properties of its constituent microscopic parts.
Mean kinetic energy of an ideal gas
By substituting the equation pV=NkT into the expression above, you can find that the temperature of an ideal gas depends on the mean kinetic energy of the particles with no regard for their number.
Equate the two equations:
31Nmc2=NkT
Cancel N on both sides, divide by 2 and multiply by 3:
21mc2=23kT
Where the mean kinetic energy of the particles Ek=21mc2.
Example
Calculate the temperature of helium gas when its atoms have an r.m.s. speed of 1.36×103ms−1. A helium atom has a mass of 6.64×10−27kg.
Firstly, write down what you know:
cr.m.s.=1.36×103ms−1m=6.64×10−27kg
Write down the equation that relates temperature and the mean kinetic energy and rearrange it for T:
21mc2=23kT
T=3kmc2
The mean square speed is simply:
c2=cr.m.s2
Substitute all the values into the equation:
T=3×1.38×10−236.64×10−27×(1.36×103)2
Calculate the temperature, and give your answer to the lowest number of significant figures given in the question:
T=297K
The temperature of the helium gas is 297K.
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FAQs - Frequently Asked Questions
What is the root mean square speed?
The root mean square speed is a quantity used to describe the motion of particles in a gas and it is the square root of the average square velocity of all the particles.
How do you find the root mean square velocity of the particles in a gas?
To find the root mean square velocity you first square the velocity of all the particles in a gas and calculate their average. Then you take the square root of this average.
What does the temperature of an ideal gas depend on?
The temperature of an ideal gas depends on the pressure and volume of the gas, along with the mean kinetic energy of all the particles in the gas.