Question

Calculate \(u_{\mathrm{rms}},\) in meters per second, for \(\mathrm{Cl}_{2}(\mathrm{g})\) molecules at \(30^{\circ} \mathrm{C}\)

Short answer

The root mean square speed of Cl2 molecules at 30°C is approximately 482 m/s.

Step-by-step solution

Convert temperature to Kelvin

First, convert the given temperature from Celsius to Kelvin by using the formula: \(T(K) = t(°C) + 273.15\). So, we get \(T = 30 + 273.15 = 303.15 K\).

Calculating mass of a single molecule

In this step, calculate the mass of a single Cl2 molecule. The molar mass of Cl2 is approximately 70.90 g/mol. Convert it to kg (since the root mean square speed is typically expressed in meters per second), it's equal to \(70.90 \times 10^{-3} kg/mol\). Now, to get the mass of a single molecule, divide the molar mass by Avogadro's number: \(m = (70.90 \times 10^{-3})/(6.022 \times 10^{23}) = 1.178 \times 10^{-25} kg\).

Calculate the root mean square speed

Now, calculate the root mean square speed using the formula: \(u_{\mathrm{rms}} = \sqrt{(3kT)/m}\), where \(k\) is the Boltzmann constant equal to \(1.381 \times 10^{-23} m^{2} kg/s^{2} K\). So, we get: \(u_{\mathrm{rms}} = \sqrt{(3 \times 1.381 \times 10^{-23} \times 303.15)/( 1.178 \times 10^{-25})}\). This simplifies to: \(u_{\mathrm{rms}} = 482 m/s\).

Key concepts

Kinetic Molecular Theory

The Kinetic Molecular Theory is an important concept in understanding the behavior of gases. According to this theory, gas molecules are in constant motion, colliding with each other and the walls of their container. These collisions are what we perceive as pressure. The theory provides a framework for explaining the properties of gases such as pressure, volume, and temperature through the motion of molecules.
In the context of this theory, the root mean square speed (\( u_{\text{rms}} \)) represents the average speed of gas molecules. It's determined by three factors: temperature, the Boltzmann constant, and the mass of the molecules. Temperature influences speed, as higher temperatures increase the energy and thus the speed of molecules, while mass also plays a role, with lighter molecules moving faster than heavier ones at the same temperature.

Boltzmann Constant

The Boltzmann constant (\( k \)) serves as a bridge between microscopic and macroscopic physics. It introduces a connection between the microscopic configurations of molecules and macroscopic observable properties like temperature.

• Value: The Boltzmann constant is approximately\( 1.381 \times 10^{-23} \text{m}^2 \text{kg/s}^2 \text{K} \)
• Role: It relates the average kinetic energy of particles in a gas with the temperature of the gas.

In calculations involving the root mean square speed, \( k \) acts as a conversion factor that incorporates temperature to reveal the effect on molecular speeds. The entire kinetic energy of a single molecule can be expressed with the term\( \frac{3}{2} kT \).This constant thus plays a critical role in the formulas derived from the Kinetic Molecular Theory.

Temperature Conversion

Temperature conversion is a necessary step in many physics and chemistry calculations. To understand and compare thermal properties accurately, it's often required to convert temperatures between Celsius, Kelvin, and other scales.In science, Kelvin is the standard scale because it starts from absolute zero, where all molecular motion theoretically ceases.

• Formula: To convert from Celsius to Kelvin, use the formula:\( T(\text{K}) = t(^\circ\text{C}) + 273.15 \)
• Example: For a temperature of 30°C, the conversion is:\( T = 30 + 273.15 = 303.15 \text{K} \)

Converting to Kelvin is crucial when utilizing equations in the kinetic molecular realm, as they often require absolute temperature readings.

Molar Mass Calculation

Molar mass is the mass of one mole of a given substance and is typically expressed in grams per mole (g/mol). Calculating molar mass is a crucial step in determining the mass of individual molecules, especially in gases where mass allows for the computation of speeds and kinetic energies.

• Chlorine Gas (Cl₂): Its molar mass is approximately 70.90 g/mol.
• Conversion: For molecular calculations, it's converted to kg per mole for compatibility with SI units in formulas: \( 70.90 \times 10^{-3} \text{kg/mol} \).
• Single Molecule Mass: To find the mass of a single molecule, divide by Avogadro's number \( 6.022 \times 10^{23} \).This step results in the mass of one Cl₂ molecule as \( 1.178 \times 10^{-25} \text{kg} \).

This calculated molecular mass is pivotal for kinetic molecular calculations, including computing the root mean square speed.