Root Mean Square Speed: Videos & Practice Problems

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 Formula

The root mean square speed (RMS speed) is a crucial concept in understanding the motion of gas molecules. It provides a way to calculate the average speed of gas particles in a sample. The formula for calculating the root mean square speed is given by:

\( v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \)

In this equation, \( v_{\text{rms}} \) represents the root mean square speed, \( R \) is the ideal gas constant, which has a value of \( 8.314 \, \text{J/(mol·K)} \), \( T \) is the absolute temperature measured in Kelvin, and \( M \) is the molar mass of the gas expressed in kilograms per mole (kg/mol). It is important to note that the molar mass should not be in grams per mole for this calculation.

The root mean square speed is particularly useful when analyzing gases like chlorine or oxygen, as it allows for the comparison of molecular speeds under varying conditions of temperature and molar mass. Understanding this relationship is essential for applications in fields such as chemistry and physics, where gas behavior is studied.

RMS Speed Example

To calculate the root mean square speed (\(v_{\text{rms}}\)) of ammonia (NH₃) molecules at a temperature of 50 degrees Celsius, we utilize the formula:

\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]

In this equation, \(R\) is the ideal gas constant, \(T\) is the absolute temperature in Kelvin, and \(M\) is the molar mass of the gas in kilograms per mole.

First, we convert the temperature from degrees Celsius to Kelvin:

\[ T = 50 + 273.15 = 323.15 \, \text{K} \]

Next, we identify the appropriate value for \(R\). Since we are working with speed, we use:

\[ R = 8.314 \, \text{J/(mol·K)} \]

We also need the molar mass of ammonia. The molar mass of NH₃ is calculated as follows:

• Nitrogen (N): 14.007 g/mol
• Hydrogen (H): 1.008 g/mol × 3 = 3.024 g/mol
• Total: 14.007 + 3.024 = 17.031 g/mol

To convert this to kilograms per mole, we divide by 1000:

\[ M = \frac{17.031 \, \text{g/mol}}{1000} = 0.017031 \, \text{kg/mol} \]

Now, substituting the values into the \(v_{\text{rms}}\) formula:

\[ v_{\text{rms}} = \sqrt{\frac{3 \times 8.314 \, \text{J/(mol·K)} \times 323.15 \, \text{K}}{0.017031 \, \text{kg/mol}}} \]

Next, we simplify the units. The joules can be expressed in terms of base units:

\[ \text{J} = \text{kg} \cdot \text{m}^2/\text{s}^2 \]

Thus, when substituting, the units of kilograms cancel out, moles cancel out, and kelvins cancel out, leaving us with:

\[ \text{m}^2/\text{s}^2 \]

Taking the square root of this gives us the final unit of:

\[ \text{m/s} \]

After performing the calculation, we find:

\[ v_{\text{rms}} \approx 687.9 \, \text{m/s} \]

It is important to note that while the temperature was given with one significant figure, for accuracy in calculations, we used four significant figures in our final answer. Rounding to one significant figure would yield 700 m/s, which is less precise than the calculated value.

In summary, understanding the relationships between temperature, molar mass, and the ideal gas constant is crucial for accurately determining the root mean square speed of gas molecules.

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

Answer: (C)

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

Answer: (C)

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.