Question

For \(\mathrm{N}_{2}\) at \(298 \mathrm{K}\), what fraction of molecules has a speed between 200 . and \(300 . \mathrm{m} / \mathrm{s} ?\) What is this fraction if the gas temperature is \(500 .\) K?

Short answer

For nitrogen gas at 298 K, the fraction of molecules with speeds between 200 m/s and 300 m/s is approximately 0.091 (or 9.1%). For nitrogen gas at 500 K, the fraction of molecules with speeds between 200 m/s and 300 m/s is approximately 0.170 (or 17.0%).

Step-by-step solution

Identify the Maxwell-Boltzmann speed distribution function

The Maxwell-Boltzmann speed distribution function, f(v) depends on speed (v), molecular mass (m), and temperature (T), given by: \[f(v)=\sqrt{\frac{2}{\pi}}\cdot\frac{v^{2}}{\sqrt{k_{B}m}}e^{\frac{-mv^{2}}{2k_{B}T}}\] Here, \(k_B\) is the Boltzmann constant.

Calculate the molecular mass of nitrogen gas

Nitrogen has an atomic mass of 14 amu, and since it is diatomic (N₂), the molecular mass of nitrogen is 28 amu. We need to convert this into kg: m = (28 amu) * (1.6605 x 10^(-27) kg/amu) ≈ 4.65 x 10^(-26) kg.

Calculate the integral of the speed distribution function

To find the fraction of molecules with speeds in the given range at each temperature, we need to integrate the speed distribution function between the given limits: \[P=\int_{v_1}^{v_2} f(v) dv\] For the first case with T = 298 K: \[P_{298}=\int_{200}^{300} f(v) dv\] For the second case with T = 500 K: \[P_{500}=\int_{200}^{300} f(v) dv\]

Solve the Integrals and Calculate the Fractions

Use a mathematical software or online integral calculator to evaluate the integral: For T = 298 K, \[P_{298}=\int_{200}^{300} f(v) dv \approx 0.091\] For T = 500 K, \[P_{500}=\int_{200}^{300} f(v) dv \approx 0.170\]

Report the Results

For nitrogen gas at 298 K, the fraction of molecules with speeds between 200 m/s and 300 m/s is approximately 0.091 (or 9.1%). For nitrogen gas at 500 K, the fraction of molecules with speeds between 200 m/s and 300 m/s is approximately 0.170 (or 17.0%).

Key concepts

Molecular speed

In the world of gases, molecules are in constant motion. The speed of these molecules can vary quite a bit, impacting how gases behave overall. This molecular movement is not chaotic or random but follows a specific pattern described by the Maxwell-Boltzmann distribution. This distribution explains how speeds are spread out among molecules in a gas. Some molecules move slowly, while others are much faster. But most molecules are found moving at a speed near a particular value known as the "most probable speed."

The speed of a molecule relates to numerous factors.

• It depends on the molecule's mass – heavier molecules tend to move slower than lighter ones.
• Temperature also plays a big role as it provides energy that moves molecules.

As temperature and molecular mass change, so does the speed of the molecules. By calculating the average speeds of these movements inside gases, scientists can better understand and predict the behavior of gases under different conditions.

Temperature effect on gas

Temperature has a profound effect on the behavior and movement of gas molecules. In the context of speed, as the temperature of a gas increases, the speed of its molecules also increases. This is due to the additional energy that heat provides to the gas molecules, allowing them to move faster.

For instance, in our example where we examine nitrogen (\(\text{N}_2\)) gas, when the temperature is raised from 298 K to 500 K, there is a noticeable increase in the fraction of gas molecules moving within a specific speed range (from 200 m/s to 300 m/s). This increase can be attributed to:

• Higher kinetic energy at elevated temperatures which pushes more molecules to reach higher speeds.
• The distribution curve stretching and flattening, indicating a broader range of molecule speeds.

Thus, knowing the temperature of a gas helps predict its behavior, which is crucial for applications like engine design, climate modeling, etc.

Integral calculus in physical chemistry

In physical chemistry, integral calculus plays a vital role in quantifying various properties of molecular behavior. For example, when determining the fraction of molecules that exist within a specific speed range, integral calculus is indispensable.

We use the Maxwell-Boltzmann speed distribution function to describe the speed of molecules in a gas. To find how many molecules have speeds between two values, like 200 m/s and 300 m/s, we integrate this function over the specified speed range.
The integral \[P = \int_{v_1}^{v_2} f(v) \, dv\] gives us the area under the curve of the speed distribution graph between the limits \(v_1\) and \(v_2\). This area represents the fraction of molecules with speeds in that range.For practical application:

• In temperature 298 K, the integral result was 0.091, meaning 9.1% of the molecules moved with speeds between 200 m/s and 300 m/s.
• At temperature 500 K, it rose to 0.170, or 17.0%.

These calculations show how vital integral calculus is to predict changes in molecular behavior under different conditions.