Question

Calculate the root-mean-square speed of ammonia \(\left(\mathrm{NH}_{3}\right)\) molecules at \(56.0^{\circ} \mathrm{C}\). An atom of nitrogen has a mass of \(2.33 \times 10^{-26} \mathrm{~kg}\) and an atom of hydrogen has a mass of \(1.67 \times 10^{-27} \mathrm{~kg}\)

Short answer

The root-mean-square speed of the ammonia molecule at \(56.0^{\circ}\) is approximately \(665 m/s\)

Step-by-step solution

Convert Temperature from Celsius to Kelvin

To convert temperature from Celsius to Kelvin, the following formula is used: \[T(K) = T(^{\circ}C) + 273.15\]. Substitute the given temperature (56.0°C) to get \[T = 56.0 + 273.15 = 329.15 K\]

Calculate Mass of One Ammonia Molecule

One molecule of ammonia (NH3) has one Nitrogen atom and three Hydrogen atoms. The total mass is hence the sum of the mass of each atom. The given masses are Nitrogen: \(2.33 \times 10^{-26}\) kg and Hydrogen: \(1.67 \times 10^{-27}\) kg. So total mass is \[m = 2.33 \times 10^{-26} kg + 3 \times 1.67 \times 10^{-27} kg = 2.33 \times 10^{-26} kg + 5.01 \times 10^{-27} kg = 2.83 \times 10^{-26} kg\]

Calculate Root-Mean-Square Speed

Now we have the temperature in Kelvin, the Boltzmann constant, and the mass of one ammonia molecule. We can substitute these into the formula \[v_{rms}=\sqrt{\frac{3kT}{m}}\]. Using the Boltzmann constant \(k = 1.38 \times 10^{-23} m^{2} kg s^{-2} K^{-1}\) , we get \[v_{rms}=\sqrt{\frac{3 \times 1.38 \times 10^{-23} m^{2} kg s^{-2} K^{-1} \times 329.15 K}{2.83 \times 10^{-26} kg}} = 664.97 m/s\]

Key concepts

Root-Mean-Square Speed

The root-mean-square speed (\[v_{rms}\] ) is a measure of the speed of particles in a gas. It represents how fast the molecules are moving on average. This concept is important when we want to determine the speed of gases like ammonia at different temperatures. The formula to find the root-mean-square speed is \[v_{rms}=\sqrt{\frac{3kT}{m}}\], where \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of a single molecule.Using root-mean-square speed helps us understand the impact of temperature and molecular mass on the speed of gases:

• Higher temperatures increase kinetic energy, thus increasing \(v_{rms}\).
• Heavier molecules move slower, so more mass results in a slower \(v_{rms}\).

The computation of \(v_{rms}\) demonstrates how gas particles move and behave, essential to grasping the Kinetic Theory of Gases.

Ammonia Molecule

An ammonia molecule, symbolically represented as \(NH_3\), is composed of one nitrogen (\(N\)) atom and three hydrogen (\(H\)) atoms. The understanding and calculation of the mass of an ammonia molecule is crucial for considering its motion in gaseous form.To calculate the mass of \(NH_3\), we sum the atomic masses of its constituent atoms:

• Nitrogen atom: \(2.33 \times 10^{-26}\,\text{kg}\)
• Hydrogen atom: \(1.67 \times 10^{-27}\,\text{kg}\) (and 3 such atoms)

Therefore, the total mass of the molecule is computed as:\[ m = 2.33 \times 10^{-26} \text{ kg} + 3 \times 1.67 \times 10^{-27} \text{ kg} = 2.83 \times 10^{-26} \text{ kg} \]This total molecular mass is integral for employing the root-mean-square speed equation to explore how ammonia molecules behave under varying conditions.

Boltzmann Constant

The Boltzmann constant \(k\) is a pivotal element in the world of thermodynamics and statistical mechanics. It bridges the microscopic and macroscopic worlds, providing a measure for the energy distribution of particles in matter. Its value is \(1.38 \times 10^{-23}\,\text{m}^2 \text{kg s}^{-2} \text{K}^{-1}\).In the context of gases:

• It is used in calculations to determine the average kinetic energy of particles.
• Helps ascertain the root-mean-square speed of molecules.

In the formula for \(v_{rms}\), the Boltzmann constant plays the role of ensuring that the correct scaling of energy is applied in relation to temperature and molecular mass.Understanding this constant allows us to predict and model the behavior of particles based on temperature and molecular dynamics.

Temperature Conversion

Temperature is a fundamental physical quantity that signifies the degree of heat, and different scales can be used to express it. Commonly, Celsius (°C) and Kelvin (K) are utilized in scientific calculations.For molecular speed and kinetic energy evaluations, the Kelvin scale is preferred as it starts from absolute zero, the theoretically coldest possible temperature. The conversion formula to shift from Celsius to Kelvin is:\[T(K) = T(^{\circ}C) + 273.15\]Converting 56.0°C would therefore become\[T = 56.0 + 273.15 = 329.15\,\text{K}\]This conversion to Kelvin is vital because gas laws and kinetic equations assume absolute temperature scales. Understanding this conversion ensures accurate calculations and insights into thermodynamic behaviors.