Question

Calculate the rms speed of \(\mathrm{Br}_{2}\) molecules at \(23^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm} .\) What is the rms speed of \(\mathrm{Br}_{2}\) at \(23^{\circ} \mathrm{C}\) and \(2.00 \mathrm{~atm} ?\)

Short answer

The rms speed of Br2 is approximately 215.12 m/s at both 1.00 atm and 2.00 atm.

Step-by-step solution

Understanding the Problem

To find the root mean square (rms) speed of Br2 molecules, we need to use the formula for rms speed in terms of temperature and molar mass. The formula is: \(v_{\text{rms}} = \sqrt{\dfrac{3RT}{M}}\), where \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass.

Convert Temperature to Kelvin

Convert the given temperature from Celsius to Kelvin because the gas constant requires temperature in Kelvin. Use the formula: \(T(K) = T(^{\circ}C) + 273.15\). Thus, the temperature is \(23 + 273.15 = 296.15\, K\).

Identify Constants and Molar Mass

The universal gas constant \(R\) is \(8.314\, \text{J/(mol}\cdot\text{K)}\). The molar mass of \(\mathrm{Br}_2\) is approximately \(159.808\, \text{g/mol}\), which converts to \(0.159808\, \text{kg/mol}\) for use in the formula.

Calculate RMS Speed at 1.00 ATM

Use the formula \(v_{\text{rms}} = \sqrt{\dfrac{3RT}{M}}\). Substitute \(R = 8.314\, \text{J/(mol}\cdot\text{K)}\), \(T = 296.15\, \text{K}\), and \(M = 0.159808\, \text{kg/mol}\). Calculate: \(v_{\text{rms}} = \sqrt{\dfrac{3 \times 8.314 \times 296.15}{0.159808}}\).

Compute RMS Speed

Perform the calculation: \(v_{\text{rms}} \approx \sqrt{\dfrac{7395.9672}{0.159808}} \approx \sqrt{46256.0714} \approx 215.12\, \text{m/s}\). Thus, the rms speed at 1.00 atm is approximately \(215.12\, \text{m/s}\).

Verify the Dependence on Pressure

The rms speed formula \(v_{\text{rms}} = \sqrt{\dfrac{3RT}{M}}\) does not explicitly depend on pressure. Therefore, for the same temperature, the rms speed remains the same regardless of changes in pressure. Thus, the rms speed at 2.00 atm will also be \(215.12\, \text{m/s}\).

Key concepts

Root Mean Square Speed

Root mean square speed (rms speed) is a measure of the average speed of particles in a gas, calculated from the velocities of the molecules. If you're wondering why it's not just called average speed, think of it as a way to account for both the directional motion and the number of molecules. The root mean square speed is determined by using the formula:\[ v_{\text{rms}} = \sqrt{\dfrac{3RT}{M}} \]This formula includes the gas constant \( R \), the temperature \( T \) in Kelvin, and the molar mass \( M \) in kilograms per mole.
Remember, rms speed is crucial for understanding how fast molecules in a sample move at a given temperature. This speed affects how often molecules collide, which is vital for reactions and behavior of gases.

Molecular Speed

Molecular speed refers to the speed at which individual gas molecules move. While rms speed gives us a particular average, molecular speeds can vary greatly. In gases, molecules are constantly in motion, colliding with each other, and bouncing off the walls of their container. These speeds depend on:

• Temperature: Higher temperatures increase molecular speeds.
• Molar Mass: Heavier molecules move more slowly than lighter ones at the same temperature.

It's fascinating to note that at any given temperature, lighter molecules like hydrogen will have faster molecular speeds compared to heavier ones like bromine.

Gas Laws

The behavior of gases is described by the gas laws, which link together pressure, volume, temperature, and the number of particles. Although the exercise deals with rms speed, which doesn't explicitly depend on pressure, understanding gas laws help in knowing how gases behave under different conditions. The key gas laws include:

• Boyle's Law: Pressure inversely relates to volume for a constant temperature and gas amount.
• Charles's Law: Volume directly relates to temperature for a constant pressure.
• Avogadro's Law: Volume directly relates to the number of gas molecules at constant temperature and pressure.

These provide a foundation for understanding more complex gas behaviors and calculations.

Molecular Mass

Molecular mass, which is crucial when calculating rms speed, is the mass of a molecule. It affects how quickly a molecule moves within a gas at a given temperature. The larger the molecular mass, the slower the molecule's speed will be.For \( \text{Br}_2 \), the molecular mass is approximately 159.808 grams per mole. However, for calculations involving rms speed, it's essential to convert this to kilograms per mole, giving us 0.159808 kg/mol. This conversion is crucial because the other constants used in the calculation, like the gas constant \( R \), are in terms of energy, which requires mass to be in kilograms.

Temperature Conversion

Temperature conversion, particularly from Celsius to Kelvin, is a staple in chemistry calculations. Many equations in thermodynamics require the temperature to be input in Kelvin.The conversion formula is straightforward:\[ T(K) = T(^{\circ}C) + 273.15 \]
For example, to convert 23°C to Kelvin, we add 273.15, resulting in 296.15 K. This step ensures your calculations align with the units required by the formulas.
Understanding and mastering temperature conversion is vital for correctly calculating molecular velocities and other temperature-dependent phenomena.