Problem 147 - Chemistry

Question

At what temperature will He atoms have the same \(u_{\mathrm{rms}}\) value as \(\mathrm{N}_{2}\) molecules at \(25^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The temperature at which He atoms will have the same \(u_{rms}\) value as \(\mathrm{N}_{2}\) molecules at \(25^{\circ} \mathrm{C}\) is \(2,725.96^{\circ} \mathrm{C}\).

Step 1: Calculate \(u_{rms}\) for \(\mathrm{N}_{2}\)

First, calculate the root-mean-square speed for N2 using the formula \(u_{rms} = \sqrt{\frac{3kT}{m}}\), with \(T = 25^{\circ}C = 298.16K\), \(k = 1.38x10^{-23} \,J/K-atom\), and \(m = 28.01 \,g/mol, m = \frac{28.01 \times 10^{-3}}{6.02 \times 10^{23}} kg/atom\)

Step 2: Use \(\mathrm{N}_{2}\) \(u_{rms}\) to calculate temperature for He

Now use this \(u_{rms}\) value and the molar mass of helium (4.003 g/mol, so \(m = \frac{4.003 \times 10^{-3}}{6.02 \times 10^{23}} kg/atom\)) in the root-mean-square speed equation and solve for \(T = \frac{m u_{rms}^{2}}{3k}\)

Step 3: Convert the temperature to Celsius

The temperature calculated in Step 2 will be in Kelvin. Convert it to Celsius using the relation \(C = K - 273.16\).

Key Concepts

Helium Atoms

Helium atoms are fascinating because they are one of the lightest and most abundant elements in the universe. Helium, symbolized as He, has an atomic mass of approximately 4.003 grams per mole. This light atomic mass has significant implications in physics, especially in the study of gas behaviors.

A noteworthy property of helium atoms is their small size and low density, which allows them to move at high speeds even at standard room temperatures.

• This is relevant when calculating the root-mean-square speed (\(u_{rms}\)) of helium atoms.
• As **in the given problem**, helium's relative atomic lightness compared to other gases like nitrogen plays a crucial role in determining how temperature affects its speed.

Understanding helium’s behavior is essential in applications ranging from balloons to advanced scientific experiments.

Nitrogen Molecules

Nitrogen makes up approximately 78% of Earth's atmosphere and is symbolized as \(\mathrm{N}_{2}\). This diatomic molecule has a molecular mass of 28.01 grams per mole. Given its prevalence and importance, nitrogen is often studied in chemistry and physics.

In the kinetic theory of gases, nitrogen’s molecular mass affects its root-mean-square speed compared to lighter gases like helium.

• The equation \(u_{rms} = \sqrt{\frac{3kT}{m}}\) allows us to calculate the average speed of nitrogen molecules at a given temperature.
• Because nitrogen molecules are heavier than helium atoms, they move more slowly at equivalent temperatures.

In atmospheric and environmental sciences, understanding nitrogen’s properties helps explain air pressure and gas exchange processes.

Temperature Conversion

Temperature conversion is a fundamental skill in both chemistry and physics. It's crucial when calculating properties like the root-mean-square speed of gases.

In the context of **the problem**, temperature must be consistently measured in Kelvin for the calculations to be accurate.

• Firstly, converting Celsius to Kelvin is straightforward with the formula: \(K = C + 273.16\).
• When the result needs to be in Celsius, the conversion back is \(C = K - 273.16\).

Acquiring proficiency in these conversions ensures that measurements and outcomes remain coherent, aligning accurately with scientific formulas and computations.

Kinetic Theory of Gases

The kinetic theory of gases is a significant principle in physical science, describing the behavior of gases at a molecular level. It asserts that gases consist of numerous small particles in constant and random motion.

This theory helps explain gas properties such as pressure, temperature, and volume.

• The root-mean-square speed (\(u_{rms}\)), for example, stems from this theory and indicates the average speed of gas particles.
• According to the kinetic theory, as temperature increases, particle motion—and thus the \(u_{rms}\)—increases, causing changes in pressure and volume as well.
• The formula \(u_{rms} = \sqrt{\frac{3kT}{m}}\) derives from these principles, illustrating the relationship between temperature and particle speed.

Employing the kinetic theory provides vital insights into gas laws and helps solve practical problems in various scientific disciplines.