Question

If the rms speed of \(\mathrm{NH}_{3}\) molecules is found to be \(0.600 \mathrm{~km} / \mathrm{s}\), what is the temperature (in degrees Celsius)?

Short answer

The temperature is approximately 2819.48°C.

Step-by-step solution

Recall the RMS Speed Formula

The root-mean-square (RMS) speed of gas molecules is given by the formula: \[v_{rms} = \sqrt{\frac{3kT}{m}}\]where \(v_{rms}\) is the RMS speed, \(k\) is the Boltzmann constant \(1.38 \times 10^{-23} \text{ J/K}\), \(T\) is the temperature in Kelvin, and \(m\) is the molar mass of the gas in kilograms per mole.

Convert Molar Mass to Kilograms

Given the molar mass of \(\mathrm{NH}_{3}\) is \(17.03 \text{ g/mol}\), we convert it to kg by dividing by 1000:\[m = \frac{17.03 \text{ g/mol}}{1000} = 0.01703 \text{ kg/mol}\]

Rearrange the RMS Formula to Find Temperature

Rearrange the RMS speed expression to solve for temperature \(T\): \[T = \frac{m v_{rms}^2}{3k}\]Substitute the given RMS speed (converted to \(\text{m/s}\) because \(1 \text{ km/s} = 1000 \text{ m/s}\)) into the equation.

Substitute Known Values and Compute

Substitute the known quantities into the formula:\[T = \frac{0.01703 \times (600)^2}{3 \times 1.38 \times 10^{-23}}\]Calculate to find \(T\) in Kelvin.

Convert Temperature to Celsius

Convert the temperature from Kelvin to Celsius using the relation:\[\text{Temperature in Celsius} = T - 273.15\]Subtract 273.15 from the temperature calculated in Kelvin.

Key concepts

Molar mass

Molar mass is a fundamental concept in chemistry. It represents the mass of a given substance (chemical element or chemical compound) divided by the amount of substance. The units typically used are grams per mole (g/mol).
Understanding molar mass is essential when working with gas calculations, such as finding the root-mean-square (RMS) speed of molecules. For example, the molar mass of ammonia (\(\text{NH}_3\)) is 17.03 g/mol, which is equivalent to 0.01703 kg/mol when converted to kilograms per mole. This conversion is crucial since the RMS speed formula requires the mass of the molecules in terms of kilograms per mole rather than grams. To convert from g/mol to kg/mol:

• Divide the molar mass by 1000.

This simple conversion step helps ensure that all the units used in calculations are consistent, which is vital for obtaining accurate results.

Temperature conversion

Temperature conversion is often necessary when solving physics and chemistry problems. The problem typically arises from needing to switch between the Kelvin and Celsius scales.
The Kelvin scale is largely used in scientific calculations since it starts at absolute zero, making it ideal for thermodynamic equations. However, practical temperature readings are often needed in Celsius. This discrepancy requires conversion between the two scales using a simple adjustment. The formula for conversion is:

• Kelvin to Celsius: \(\text{C} = \text{K} - 273.15\)
• Celsius to Kelvin: \(\text{K} = \text{C} + 273.15\)

For instance, when determining the temperature of a gas in an RMS speed calculation, the result will initially be in Kelvin. To express this in Celsius, one must subtract 273.15 from the Kelvin temperature. This conversion helps contextualize the result into a familiar temperature scale.

Boltzmann constant

The Boltzmann constant \(k\) is a significant constant in physics and chemistry that connects the macroscopic and microscopic physical worlds. Its value is \(1.38 \times 10^{-23} \text{ J/K}\), and it plays a pivotal role in kinetic molecular theory. The Boltzmann constant helps bridge quantities at the molecular or atomic scale with those at the macroscopic level, particularly in the context of temperature.
In the formula for RMS speed, \(v_{rms} = \sqrt{\frac{3kT}{m}}\), the Boltzmann constant helps relate the absolute temperature \(T\) to the average kinetic energy of molecules in a system. This constant is pivotal when dealing with molecular speeds and energies, as it allows for the calculation of macroscopic thermal properties based on microscopic molecular behavior.Remember:

• The Boltzmann constant is necessary for calculating thermal properties of gases.
• It plays a key role in the formula used to determine root-mean-square speeds.

Gas molecules

Gas molecules are tiny particles that are in constant, random motion. This motion is central to understanding the behaviors of gases, as explained by the kinetic molecular theory.
The kinetic molecular theory states that:

• Molecules are in continuous motion and have a range of speeds.
• Collisions between molecules or with the walls of a container are elastic, meaning energy is conserved.
• The average kinetic energy of the gas molecules is directly proportional to the temperature of the gas in Kelvin.

In calculations, attributes of gas molecules, like molecular mass and speed, are often considered to understand gases better. For instance, RMS speed is an important measure that considers these random motions to describe the average speed of molecules within a gas. Understanding how these gas molecules behave under various temperatures and pressures enables scientists and engineers to develop accurate predictions of gases' behavior in different scenarios.

Root-mean-square speed

Root-mean-square speed (RMS speed) is an essential concept in the study of gases. It provides a measure of the speed of particles in a gas, combining their velocities into a single value that represents the average speed of molecules in the system.
The RMS speed is calculated using the formula:\[v_{rms} = \sqrt{\frac{3kT}{m}}\]where:

• \(v_{rms}\) is the root-mean-square speed.
• \(k\) is the Boltzmann constant.
• \(T\) is the temperature in Kelvin.
• \(m\) is the molar mass of the gas in kilograms per mole.

This formula highlights the relationships between molecular speed, temperature, and mass. By analyzing the RMS speed, we can infer how temperature and molar mass influence molecular motion. At higher temperatures, molecules move faster, resulting in a higher RMS speed. Conversely, larger molecules (higher molar mass) move slower if energy is the same, reducing the RMS speed. RMS speed also plays a crucial role in predicting how gas molecules distribute their speeds at a given temperature.