Root-Mean-Square Velocity of Gases: Videos & Practice Problems

The root mean square (RMS) speed of an ideal gas is a type of average speed that accounts for the squares of individual particle speeds, making it slightly higher than the arithmetic average. The RMS speed can be calculated using the equations: $3^k/m$ or $\sqrt{\frac{3}{M}}$, where k is Boltzmann's constant, T is temperature in Kelvin, and M is molar mass in kg/mol. For hydrogen gas at 27°C, the RMS speed is approximately 1934 m/s, illustrating the high kinetic energy of gas particles.

Root-Mean-Square Speed of Ideal Gases

The root mean square (RMS) speed of an ideal gas is a specific type of average speed that reflects the motion of gas particles. Understanding RMS speed is essential in thermodynamics and kinetic theory, as it provides insight into the behavior of gas molecules at a given temperature. The RMS speed is calculated using the formula:

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

In this equation, \(R\) represents the universal gas constant (approximately 8.314 J/(mol·K)), \(T\) is the absolute temperature in Kelvin, and \(M\) is the molar mass of the gas in kilograms per mole. Alternatively, the RMS speed can also be expressed as:

\[ v_{\text{RMS}} = \sqrt{\frac{3kT}{m}} \]

where \(k\) is Boltzmann's constant and \(m\) is the mass of a single gas particle in kilograms. It is important to note that temperature must always be converted to Kelvin for these calculations.

To illustrate the difference between average speed and RMS speed, consider three values: 5, 11, and 32. The average speed is calculated by summing these values and dividing by the number of values, yielding an average of 16. In contrast, the RMS speed involves squaring each value, averaging those squares, and then taking the square root of that average. This process results in an RMS speed of approximately 19.7, which is typically higher than the average due to the squaring of larger numbers.

For example, to calculate the RMS speed of hydrogen gas at 27 degrees Celsius, first convert the temperature to Kelvin by adding 273, resulting in 300 K. The molar mass of hydrogen is 2 grams per mole, which converts to 0.002 kilograms per mole. Plugging these values into the RMS speed formula gives:

\[ v_{\text{RMS}} = \sqrt{\frac{3 \times 8.314 \times 300}{0.002}} \approx 1934 \text{ m/s} \]

This high speed indicates that hydrogen particles move rapidly, contributing to the low concentration of hydrogen in the atmosphere. Understanding RMS speed is crucial for grasping the kinetic behavior of gases and their properties under various conditions.

What is the temperature of a sample of CO₂ molecules whose rms speed is 300 m/s? The molecular mass for Carbon and Oxygen is 12.01 g/mol and 16 g/mol, respectively.

• A. 158.7 K
• B. 529.2 K
• C. 1.7 × 10⁵ K
• D. 1391 K

Answer: (A)

Finding Pressure from RMS Speed

In this problem, we are tasked with calculating the pressure of an ideal gas contained in a volume of 1.6 liters, with a mass of 200 grams and a molecular mass of 28 grams per mole. The root mean square (RMS) speed of the gas molecules is given as 600 meters per second. To find the pressure, we will utilize the ideal gas law, represented by the equation:

PV = nRT

Here, P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (8.314 J/(mol·K)), and T is the temperature in Kelvin. To isolate P, we rearrange the equation:

P = \frac{nRT}{V}

First, we need to determine the number of moles (n). Using the relationship between mass and molar mass, we can calculate n as follows:

n = \frac{m}{M}

Substituting the values, we have:

n = \frac{200 \text{ grams}}{28 \text{ grams per mole}} = 7.14 \text{ moles}

Next, we need to find the temperature (T). The RMS speed is related to temperature and molar mass through the equation:

v_{RMS} = \sqrt{\frac{3RT}{M}}

Squaring both sides gives:

v_{RMS}^2 = \frac{3RT}{M}

Rearranging for T, we have:

T = \frac{v_{RMS}^2 \cdot M}{3R}

Substituting the known values (with molar mass converted to kilograms):

T = \frac{(600 \text{ m/s})^2 \cdot 0.028 \text{ kg/mol}}{3 \cdot 8.314 \text{ J/(mol·K)}}

Calculating this yields:

T \approx 404.1 \text{ K}

Now, we convert the volume from liters to cubic meters for consistency in units:

V = 1.6 \text{ liters} = 0.0016 \text{ m}^3

With all variables determined, we can now calculate the pressure:

P = \frac{(7.14 \text{ moles}) \cdot (8.314 \text{ J/(mol·K)}) \cdot (404.1 \text{ K})}{0.0016 \text{ m}^3}

Upon performing the calculation, we find:

P \approx 1.5 \times 10^7 \text{ Pascals}

This result represents the pressure of the ideal gas in the container. Understanding the relationships between mass, moles, temperature, and pressure is crucial in solving problems involving ideal gases.

Here’s what students ask on this topic:

What is the root mean square velocity of gases?

The root mean square (RMS) velocity of gases is a measure of the average speed of gas particles, taking into account the squares of their individual speeds. It is slightly higher than the arithmetic average speed because it gives more weight to higher speeds. The RMS velocity is calculated using the formula:

$\sqrt{\frac{3kT}{m}}$ or $\sqrt{\frac{3RT}{M}}$,

where k is Boltzmann's constant, T is the temperature in Kelvin, m is the mass of a gas particle, R is the universal gas constant, and M is the molar mass in kg/mol.

How do you calculate the RMS speed of an ideal gas?

To calculate the RMS speed of an ideal gas, you can use one of the following formulas depending on the given variables:

$\sqrt{\frac{3kT}{m}}$ or $\sqrt{\frac{3RT}{M}}$,

where k is Boltzmann's constant (1.38 × 10^-23 J/K), T is the temperature in Kelvin, m is the mass of a gas particle, R is the universal gas constant (8.314 J/(mol·K)), and M is the molar mass in kg/mol. Ensure the temperature is in Kelvin and the mass is in the correct units.

Why is the RMS speed higher than the average speed?

The RMS speed is higher than the average speed because it takes into account the squares of individual particle speeds. When you square the speeds, higher speeds contribute more significantly to the average. This skews the RMS speed towards higher values compared to the arithmetic average. Mathematically, the RMS speed is calculated as:

$\sqrt{\frac{3kT}{m}}$ or $\sqrt{\frac{3RT}{M}}$,

where the squaring and subsequent square rooting process results in a value that is generally higher than the simple arithmetic mean of the speeds.

What factors affect the RMS speed of gas particles?

The RMS speed of gas particles is affected by two main factors: temperature and the mass of the gas particles. The RMS speed increases with temperature because higher temperatures provide more kinetic energy to the particles. It is inversely proportional to the mass of the gas particles, meaning lighter particles move faster. The formula for RMS speed is:

$\sqrt{\frac{3kT}{m}}$ or $\sqrt{\frac{3RT}{M}}$,

where k is Boltzmann's constant, T is the temperature in Kelvin, m is the mass of a gas particle, R is the universal gas constant, and M is the molar mass in kg/mol.

How does temperature influence the RMS speed of gases?

Temperature has a direct influence on the RMS speed of gases. As temperature increases, the kinetic energy of gas particles also increases, leading to higher RMS speeds. The relationship is given by the formula:

$\sqrt{\frac{3kT}{m}}$ or $\sqrt{\frac{3RT}{M}}$,

where T is the temperature in Kelvin. Higher temperatures result in greater kinetic energy, thus increasing the RMS speed. This is why gases at higher temperatures have faster-moving particles.