Question

In water vapor at \(25^{\circ} \mathrm{C}\), what is the average speed of a water molecule in meters \(/ \mathrm{sec} ?\) Recall that 1 Joule \(=1 \mathrm{Kg}-\mathrm{m}^{2} / \mathrm{sec}^{2}\) The Boltzmann constant, \(\mathrm{k}=1.3806 \times 10^{-23} \mathrm{~J} / \mathrm{deg}\)

Short answer

The average speed of a water molecule in water vapor at 25°C is approximately 643 m/s.

Step-by-step solution

Gather all the given information and note the desired value

We are given the following information: - The temperature of water vapor is 25°C. - The Boltzmann constant k = \(1.3806 \times 10^{-23} J/deg\). - The formula to calculate the average speed of a water molecule in water vapor - 1 Joule = \(1 kg \cdot m^2 / sec^2\) We need to find the average speed of a water molecule in meters per second.

Convert Celsius to Kelvin

The temperature given is in Celsius. Therefore, we should convert it to Kelvin. The conversion formula from Celsius to Kelvin is \(K = °C + 273.15\), where K is the temperature in Kelvin, and °C is the temperature in Celsius. We have: \(T = 25°C + 273.15 = 298.15 K\)

Calculate the mass of a water molecule

We need to find the mass of a single water molecule. Water (H2O) consists of two hydrogen (H) atoms and one oxygen (O) atom. The atomic weights of H and O are approximately 1u and 16u, respectively (where 1u is equal to \(1.66 \times 10^{-27} kg\)). So, the molecular weight of water (H2O) is: \(2 \times 1u + 1 \times 16u = 18u\) Then, the mass of a single water molecule is: \(m = 18u \times 1.66 \times 10^{-27} kg/u = 2.988 \times 10^{-26} kg\)

Apply the formula to find the average speed of a water molecule

Now, we can use the formula for the average speed (v) of a gas molecule based on its temperature (T) and mass (m). This formula is derived from the equipartition theorem and can be expressed as: \(v = \sqrt{\frac{3kT}{m}}\) Where: - v = average speed of a water molecule - k = Boltzmann constant - T = temperature in Kelvin - m = mass of a water molecule Plugging in the values, we get: \(v = \sqrt{\frac{3 \times 1.3806 \times 10^{-23} J/deg \times 298.15 K}{2.988 \times 10^{-26} kg}}\)

Calculate the average speed

After substituting the values, calculate the average speed: \(v = \sqrt{\frac{3 \times 1.3806 \times 10^{-23} \times 298.15}{2.988 \times 10^{-26}}}\) \(v = \sqrt{\frac{1.234 \times 10^{-20}}{2.988 \times 10^{-26}}}\) \(v = \sqrt{4.133 \times 10^5}\) \(v \approx 643 m/s\) Therefore, the average speed of a water molecule in water vapor at 25°C is approximately 643 m/s.

Key concepts

Average Molecular Speed

The average molecular speed of a gas is an important concept in the kinetic theory of gases. It tells us how fast molecules of a gas are moving on average at a particular temperature. This speed depends on several factors including the temperature and the mass of the molecule.

One of the key equations to calculate this speed is given by:

• \(v = \sqrt{\frac{3kT}{m}}\)

where:

• \(v\) is the average speed we're looking for,
• \(k\) is the Boltzmann constant,
• \(T\) is the temperature in Kelvin,
• \(m\) is the mass of a single water molecule.

The formula comes from the equipartition theorem, which connects the motion of molecules to temperature. It's a neat way to see that molecules move faster as they or the environment gets hotter, and slower if they have more mass. Understanding this can help visualize and predict gas behaviors in various conditions.

Boltzmann Constant

The Boltzmann constant (\(k\)) is a fundamental constant in physics and plays a vital role in the kinetic theory of gases. It essentially connects the macroscopic properties of gases with their microscopic molecular dynamics.

The value of the Boltzmann constant is

• \(k = 1.3806 \times 10^{-23} \mathrm{~J/K}\)

This tiny constant converts energy units like joules into temperature units like Kelvin. It acts as a bridge between the energy of molecules and their temperature.In calculations, the Boltzmann constant is a crucial component of the formula used to find the average molecular speed. It allows us to include temperature as a factor while assessing the kinetic energy of molecules.

By understanding the role of the Boltzmann constant, students can better grasp how temperature and energy relate on a molecular level.

Temperature Conversion

Temperature conversion is often the first step in solving problems related to the kinetic theory of gases. Most scientific calculations require temperature to be in Kelvin. Luckily, converting from Celsius to Kelvin is straightforward.

The conversion formula is simple:

• \(K = °C + 273.15\)

With this, if you're given a temperature in degrees Celsius (°C), you just add 273.15 to get it in Kelvin (K).For instance, if the temperature is given as 25°C, it converts to:

• \(25°C + 273.15 = 298.15 K\)

This measurement in Kelvin can then be used in further calculations, such as determining molecular speed.

Converting temperatures correctly ensures that calculations are accurate and that they're set in the context of a universal temperature scale, which is essential for understanding gas behavior at a molecular level.

Molecular Mass Calculation

Calculating the mass of a molecule is another crucial step in problems dealing with the kinetic theory of gases. In order to use any formula that involves the mass of a molecule, like that calculating the average molecular speed, you need to determine the molecular mass.

Here's how to find it for a water molecule (H₂O):

• Each hydrogen (H) atom has an approximate atomic mass of 1 atomic mass unit (u),
• Oxygen (O) has an approximate atomic mass of 16 u.

Thus, a water molecule's mass is calculated by combining these:

• \(2 \times 1u + 1 \times 16u = 18u\)

To convert this into kilograms, recall that

• 1 atomic mass unit (u) = \(1.66 \times 10^{-27} kg\)

Thus for water:

• \(m = 18u \times 1.66 \times 10^{-27} kg/u = 2.988 \times 10^{-26} kg\)

This mass can then be applied in further calculations for molecular speed and kinetic energy.

Accurate molecular mass calculation is vital in ensuring that kinetic energy calculations, among others, reflect true molecular behavior.