Root mean squared speed

• Root-mean-square speed


  A type of average speed of gas particles, calculated by squaring the speeds, taking the mean, and then taking the square root
• A gas consists of a gazillion number of gas particles each going at its own speed and velocity
• Calculating the root-mean-square speed

  1. Square the speeds

  2. Take the mean

  3. Take the square root
• Why calculate root-mean-square speed instead of other averages

  Because kinetic energy is proportional to the square of the speed, not higher powers
• Mean square speed


  The average of the squares of the speeds, before taking the square root
• To calculate the average kinetic energy


  Use the root-mean-square speed in the formula 1/2 mv^2
• Root mean square speed (rms)


  Measure of the speed of gas molecules in a sample
• Significance of rms speed in kinetic theory

  • Represents the average speed of gas molecules at a given temperature
• Calculating rms speed of gas molecules

  1. v_rms = √(3kT/m)

  2. Where: k = Boltzmann constant, T = Temperature, m = Mass of gas molecules
• As temperature increases

  The average kinetic energy and speed of gas molecules increase
• At the same temperature

  Lighter gas molecules have higher rms speeds compared to heavier molecules
• Deriving the expression for rms speed

  Equate average kinetic energy of gas molecules to kinetic energy of a single molecule and solve for v_rms
• v_rms

  Directly proportional to the square root of the average kinetic energy of gas molecules
• At constant temperature

  v_rms is inversely proportional to the square root of the mass of the molecules
• As pressure increases (assuming constant temperature)


  v_rms increases
• Experimental method to determine rms speed

  Measure the rate of effusion or diffusion of the gas through a small opening or porous barrier and use appropriate equations to calculate v_rms
• Assumptions in kinetic theory (point masses with no volume, perfectly elastic collisions) may not hold true in all cases, especially at high pressures or low temperatures
• Kinetic theory does not account for intermolecular forces or molecular interactions, affecting v_rms in real gases
• The formula for calculating root mean square speed is v = sqrt(v^2/n)
• Root mean square speed is the square root of the average value of the velocity squared.
• Root mean square speed can be calculated using the equation v = sqrt(3RT/M), where R is the universal gas constant, T is absolute temperature, M is molar mass.
• Particles move randomly in different directions, resulting in a net flow of particles in any given direction being equal to zero.
• The root mean square velocity of particles in a gas depends on their mass and temperature according to the relationship v = sqrt(3RT/M).