Question

The temperature in the stratosphere is \(-23^{\circ} \mathrm{C}\). Calculate the root-mean-square speeds of \(\mathrm{N}_{2}, \mathrm{O}_{2},\) and \(\mathrm{O}_{3}\) molecules in this region.

Short answer

The rms speeds are: \( \mathrm{N}_2: 477 \, \text{m/s}, \mathrm{O}_2: 457 \, \text{m/s}, \mathrm{O}_3: 367 \, \text{m/s}.\)

Step-by-step solution

Convert Temperature to Kelvin

The root-mean-square (rms) speed of a gas is calculated using the temperature in Kelvin. Convert \(-23^{\circ} \mathrm{C}\) to Kelvin:\[T(K) = T(^{\circ}\mathrm{C}) + 273.15 = -23 + 273.15 = 250.15 \, \mathrm{K}.\]

Identify the Molar Masses

Determine the molar masses of the gases:- \(\mathrm{N}_2\): 28.02 g/mol- \(\mathrm{O}_2\): 32.00 g/mol- \(\mathrm{O}_3\): 48.00 g/mol.

Calculate RMS Speed Formula

The formula for root-mean-square speed is:\[v_{\text{rms}} = \sqrt{\frac{3RT}{M}},\]where \(R\) is the gas constant \(8.314 \, \text{J mol}^{-1} \text{K}^{-1}\) and \(M\) is the molar mass in kilograms.

Calculate RMS Speed of \(\mathrm{N}_2\)

Convert molar mass to kg/mol: \(28.02 \, \text{g/mol} = 0.02802 \, \text{kg/mol}\).Plug values into the formula:\[v_{\text{rms}} = \sqrt{\frac{3 \times 8.314 \times 250.15}{0.02802}} \approx 477 \text{ m/s}.\]

Calculate RMS Speed of \(\mathrm{O}_2\)

Convert molar mass to kg/mol: \(32.00 \, \text{g/mol} = 0.03200 \, \text{kg/mol}\).Plug values into the formula:\[v_{\text{rms}} = \sqrt{\frac{3 \times 8.314 \times 250.15}{0.03200}} \approx 457 \text{ m/s}.\]

Calculate RMS Speed of \(\mathrm{O}_3\)

Convert molar mass to kg/mol: \(48.00 \, \text{g/mol} = 0.04800 \, \text{kg/mol}\).Plug values into the formula:\[v_{\text{rms}} = \sqrt{\frac{3 \times 8.314 \times 250.15}{0.04800}} \approx 367 \text{ m/s}.\]

Key concepts

Root-Mean-Square Speed

Root-mean-square speed, often abbreviated as RMS speed, is a metric used in physics to determine how fast molecules are moving within a gas. This is crucial because it helps us understand the kinetic energy of the gas and how gases behave under different conditions. When calculating RMS speed, we use the formula:\[v_{\text{rms}} = \sqrt{\frac{3RT}{M}},\]where:

• \(v_{\text{rms}}\) is the root-mean-square speed,
• \(R\) is the gas constant (8.314 J mol^-1 K^-1),
• \(T\) is the temperature in Kelvin, and
• \(M\) is the molar mass of the gas in kilograms per mole (kg/mol).

This formula tells us that the speed depends on three things: the type of gas, its temperature, and its molar mass. Faster molecules have higher RMS speeds, which generally occurs at higher temperatures or with lighter gases.

Temperature Conversion to Kelvin

When working with gas laws, always convert temperatures from Celsius to Kelvin. Kelvin is the SI unit for temperature and it begins at absolute zero, the point where all molecular motion stops.The conversion formula is given by:\[T(K) = T(^{\circ}\mathrm{C}) + 273.15 \]This formula is straightforward:

• Add 273.15 to the Celsius temperature.
• In our exercise, we converted \(-23^{\circ} \mathrm{C}\) to Kelvin, which resulted in 250.15 K.

This enhances calculation accuracy for kinetic energy and molecular speed, as gases behave predictably in Kelvin temperatures.

Molar Mass Calculation

Molar mass is essential to finding the RMS speed of gas molecules. It represents the mass of one mole of a given substance, usually expressed in grams per mole (g/mol).For calculating RMS speed, we need the molar mass in kg/mol:

• To convert from g/mol to kg/mol, simply divide by 1000.
• For example, the molar mass of nitrogen (\(\mathrm{N}_2\)) is 28.02 g/mol, which converts to 0.02802 kg/mol.

Knowing the molar masses of gases like \(\mathrm{N}_2\), \(\mathrm{O}_2\), and \(\mathrm{O}_3\) is crucial, as they have significant impacts on the calculations of molecular speeds.

Gas Constant

The gas constant \(R\) is a fundamental constant in the RMS speed calculation, linking the macroscopic parameters of pressure, volume, and temperature with the molecular motion of particles.The standard value used here is:\[ R = 8.314 \, \text{J mol}^{-1} \text{K}^{-1} \]This value is essential because it ensures that all parameters in the RMS speed formula (\(v_{\text{rms}} = \sqrt{\frac{3RT}{M}}\)) relate back to the energy and movement of the particles at a microscopic level, providing consistency across different gases and conditions.

Molecular Speed Calculation

With all components ready, the calculation of molecular speed using the root-mean-square formula becomes straightforward. After converting the temperature to Kelvin, determining the correct molar mass in kilograms, and using the gas constant, you can plug these values into the RMS speed formula.Let's take the example of nitrogen gas (\(\mathrm{N}_2\)):

• Temperature: 250.15 K
• Molar Mass: 0.02802 kg/mol
• Gas Constant: 8.314 J mol^-1 K^-1
• Use the formula: \(v_{\text{rms}} = \sqrt{\frac{3 \times 8.314 \times 250.15}{0.02802}}\)
• The calculation yields an RMS speed of approximately 477 m/s.

Repeatedly using this method for other gases, such as \(\mathrm{O}_2\) and \(\mathrm{O}_3\), allows for similar calculations tailored to their respective properties.