Root Mean Square Speed - Online Tutor, Practice Problems & Exam Prep

Understanding the velocity of gas molecules is essential in the study of gases. The root mean square speed (RMS speed) is a key concept that provides insight into the average velocity of gas particles. To calculate RMS speed, use the formula:

$\mathrm{RMS}=\sqrt{\frac{3RT}{M}}$

where 'R' is the ideal gas constant (8.314 J/(mol·K)), 'T' is the absolute temperature in Kelvin, and 'M' is the molar mass of the gas expressed in kilograms per mole (kg/mol). This equation is pivotal in kinetic molecular theory, as it relates the macroscopic properties of gases, such as temperature and pressure, to the microscopic motion of gas molecules. Understanding RMS speed is fundamental for grasping gas behavior under various conditions.

Root Mean Square Speed

So here we're going to say that the root mean square seed formula is used to determine the velocity of one type of gas molecules. O if I want to find the root mean square speed of chlorine gas or oxygen gas, this is the formula I would use. So root mean square speed is abbreviated as the RMS equals.

Now the term root is used in naming this so root would mean square root. So it equals 3M. Here M stands for the molar mass of the gas, not in grams per mole, but kilograms per mole. And here are. Remember, R's value can change if we're talking about speed, velocity, or energy.

Since we're talking about speed here, it becomes 8.314 Joules over moles times K and then finally temperature will be, as always, in units of Kelvin. O. Here. This represents our root mean square seed formula when looking at one particular type of gas molecule.

Root Mean Square Speed Example 1

Here we have to calculate the root mean square speed of NH₃ molecules at 50�C. All right, so NH₃ is ammonia. And remember root mean square speed, so V_RMS equals 3⁢RT/m. Here we're going to say we have temperature of 50�C, so we're going to add 273.15 to that. That gives us 323.15 Kelvin.

Remember, because we're dealing with seed, the R we use is 8.314. Also remember that when we talk about Joules, Joules are kilograms times meters squared over second squared O. Here, if we're dealing with our R constant, which is in joules over moles times K, we can substitute this in for joules, so that becomes kilograms times meters squared over second squared times moles times K. That is all of our units that we have here.

Null times K temperature 323.15 K divided by the molar mass of the gas in kilograms, not grams per mole. So NH₃ itself with its one nitrogen and its three hydrogens has a mass of 17.034g per mole. But if we want to do it in kilograms per mole, remember we have 10 to the three grams and then one kilogram grams cancel out. So this is 0.017034 kilograms per mole. So then that is 0.017034 kilograms per mole here.

Then we look and see what units cancel out. So kilograms cancel out with kilograms, moles cancel out with moles, kelvins cancel out with kelvins. So what we have left is meters squared over seconds squared, and if we're taking the square root, that's going to just leave us with meters per second. Here, 50 has one sig fig, but here having one sig fig is not great in terms of calculating our root mean square speed.

So here we're going to disregard sig figs, at least from what's given to us within the question, we're just going to go with 687.9 meters per second. So here I'm using four significant figures because one significant figure is not accurate enough. If we went by one sig fig, this would round up to 700 meters per second, which is a bit different from our actual answer. All right, so just remember the units needed in order to isolate your root mean square speed.

Determine which gas would have a root mean square speed of 515.59 m/s at 405 K.

• A. Cl₂
• B. CO₂
• C. F₂
• D. NH₃
• E. CH₄

The root mean square speed of gas molecules is 283.0 m/s at a given temperature T when the recorded molar mass is 42.0 g/mol. What would be the root mean square speed for a gas with a molar mass of 152.0 g/mol?

• A. 103.7 m/s
• B. 128.2 m/s
• C. 148.8 m/s
• D. 177.2 m/s
• E. 205.1 m/s

Here’s what students ask on this topic:

What is the root mean square speed?

The root mean square speed (RMS speed) is a measure of the speed of particles in a gas. It is defined as the square root of the average of the squares of the speeds of each particle in the gas. Mathematically, it's represented as:

$v_{\text{rms}}=\sqrt{\frac{v^1+v^2+\dots +v^n}{n}}$

where v1, v2, ..., vn are the speeds of individual particles, and n is the total number of particles.

In the context of kinetic theory of gases, the RMS speed is particularly useful because it is directly related to the temperature of the gas and the mass of the particles. The formula for RMS speed in terms of these quantities is:

$v_{\text{rms}}=\sqrt{\frac{3kT}{m}}$

Here, k is the Boltzmann constant, T is the absolute temperature of the gas, and m is the mass of a single particle of the gas. The RMS speed gives you an idea of how fast the particles are moving on average, which is important for understanding properties like pressure and diffusion in gases.

How can I find the root mean square speed?

To find the root mean square (RMS) speed of a collection of gas particles, you can use the formula:

$v_{\text{rms}}=\sqrt{\frac{3kT}{m}}$

Here's what each symbol represents:

• vrms is the root mean square speed you're looking for.
• k is the Boltzmann constant (1.38 × 10^-23 J/K).
• T is the absolute temperature in Kelvin.
• m is the mass of a single gas particle in kilograms.

To solve for vrms, follow these steps:

1. Ensure you have the absolute temperature (T) in Kelvin. If you have the temperature in Celsius, convert it by adding 273.15.
2. Determine the mass of a single gas particle (m). If you're given the molar mass (M) in grams per mole, convert it to kilograms per mole by dividing by 1000, then divide by Avogadro's number (6.022 × 10²³ mol^-1) to get the mass of a single particle.
3. Plug the values into the formula and solve.

Remember, the RMS speed gives you a measure of the average speed of the gas particles, taking into account their kinetic energy and temperature.

How do you calculate root mean square speed?

To calculate the root mean square (rms) speed of a group of particles, which is a measure of the speed of particles in a gas that's useful in the context of kinetic theory and thermodynamics, you can use the following formula:

$v{\mathrm{rms}}_{}=\sqrt{\frac{3kT}{m}}$

Here's what each symbol represents:

• v_rms is the root mean square speed.
• k is the Boltzmann constant (1.38 × 10^-23 J/K).
• T is the absolute temperature in Kelvin.
• m is the mass of a single particle in kilograms.

To calculate v_rms, follow these steps:

1. Ensure that the temperature (T) is in Kelvin. If it's given in degrees Celsius, convert it to Kelvin by adding 273.15.
2. Use the molar mass of the gas if m is not provided. Convert it to kilograms by dividing by Avogadro's number (6.022 × 10²³ mol^-1) if necessary.
3. Insert the values for k, T, and m into the formula.
4. Calculate the value inside the square root, then take the square root to find the rms speed.

Remember, the rms speed provides an average measure of particle speed, and not the speed of any specific particle.