Problem 151 - A Complete Resource Book in Chemistry for JEE Main

Question

Which one of the following statement is not true about the effect of an increase in temperature on the distribution molecular speeds in a gas? (a) the most probable speed increases (b) the fraction of the molecules with the most probable speed increases (c) the distribution becomes broader (d) the area under the distribution curve remains the same as the under the lower temperature

Step-by-Step Solution

Answer

Statement (b) is not true.

Step 1: Understanding the Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the speed distribution of gas molecules at a certain temperature. The distribution curve is asymmetrical and shows how molecular speeds are distributed across different energies.

Step 2: Analyzing the Effect of Temperature Increase

When the temperature of a gas increases, the kinetic energy of its molecules also increases. This causes changes in the Maxwell-Boltzmann speed distribution: the peak position (most probable speed) shifts to a higher speed, and the distribution curve becomes broader and flatter.

Step 3: Evaluating Statement (a): Most Probable Speed

The most probable speed is the speed at which the distribution curve peaks. As temperature increases, the peak speed shifts to the right, indicating an increase in most probable speed. Therefore, statement (a) is true.

Step 4: Evaluating Statement (b): Fraction with Most Probable Speed

With increased temperature, the peak of the Maxwell-Boltzmann distribution curve not only shifts to higher speeds but also becomes lower. This means the fraction of molecules at the most probable speed decreases. Therefore, statement (b) is not true.

Step 5: Evaluating Statement (c): Distribution Broadens

As temperature rises, the range of speeds expands, broadening the distribution curve. This broadening effect is due to the increased kinetic energy spread among gas molecules. So, statement (c) is true.

Step 6: Evaluating Statement (d): Area Under the Curve

The total area under the molecular speed distribution curve represents the total number of molecules. Since the number of molecules doesn't change with temperature, the area remains constant, just the shape changes. Thus, statement (d) is true.

Key Concepts

Temperature Effect on Gases

When the temperature of a gas increases, the behavior of the gas molecules changes significantly. This is mainly due to the increase in kinetic energy associated with higher temperatures. Let's dive into how temperature affects gases:

• An increase in temperature results in an increase in the average kinetic energy of gas molecules. This means that, on average, the molecules move faster at higher temperatures.
• Most notably, the most probable speed, which is the speed at which the largest number of molecules are moving, shifts towards higher values. This means that at a higher temperature, you will find the peak of the speed distribution at a higher speed.
• The distribution of molecular speeds also broadens. This broadening occurs because the range of molecular speeds spreads out as the temperature rises. More molecules achieve higher speeds, so the curve flattens out across a wider speed spectrum.

In conclusion, increasing the temperature of a gas not only speeds up the molecules but also spreads out their speed distribution across a broader range.

Molecular Speed Distribution

The molecular speed distribution of gases is a rich and important concept for understanding the motion of gas molecules. The Maxwell-Boltzmann distribution provides the framework for understanding this:

• It illustrates how many gas molecules have a particular speed at a specific temperature, forming a curve known as the distribution curve.
• This curve is characterized by three key types of speed: average speed, the most probable speed, and root mean square speed. Specifically, the most probable speed is where the distribution curve peaks.
• The area under the curve represents the total number of gas molecules and remains constant, regardless of temperature. However, the shape of the curve changes with temperature adjustments.
• At higher temperatures, the speed distribution broadens and flattens, indicating a wider spread of molecular speeds as more molecules move at differing rates.

By analyzing the molecular speed distribution, scientists can deduce much about the kinetic energy, behavior, and properties of gases at different conditions.

Kinetic Theory of Gases

The kinetic theory of gases offers a deep dive into the molecular world, explaining how macroscopic properties of gases, like pressure and temperature, are due to molecular activity:

• According to this theory, gases consist of a large number of small particles (molecules) that are in constant random motion. These molecules collide with each other and with the walls of their container.
• The kinetic energy of gas molecules is directly proportional to the temperature of the gas. So, if temperature increases, the average kinetic energy also increases, causing molecules to move faster.
• This theory also explains that the pressure exerted by a gas in a container results from collisions of the molecules with the container walls.
• Importantly, the kinetic theory relates temperature, pressure, and volume of a gas, helping to derive laws such as the Ideal Gas Law, which mathematically describe gas behaviors.

Understanding the kinetic theory not only helps in explaining the fundamental behavior of gases but also provides the basis for further thermodynamic studies and applications.