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

Short answer

Statement (b) is not true.

Step-by-step solution

Understanding the Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds of molecules in a gas. It is characterized by a curve where the x-axis represents molecular speed and the y-axis represents the probability density of those speeds.

Effect of Increasing Temperature

An increase in temperature causes the molecules in a gas to have higher average kinetic energy. This affects the shape and position of the Maxwell-Boltzmann distribution curve.

Analyzing the Most Probable Speed

As temperature increases, the most probable speed (the speed at the peak of the distribution) increases. This is because molecules have more kinetic energy, shifting the peak to the right.

Evaluating the Fraction of Molecules at Most Probable Speed

While the position of the most probable speed increases, the fraction of molecules possessing exactly the most probable speed decreases because the curve becomes broader and lower at its peak.

Considering Curve Broadness and Area Under the Curve

The distribution becomes broader with increased temperature, spreading the molecular speeds over a wider range. Despite changes in shape, the total area under the curve represents the total number of molecules, which remains constant regardless of temperature.

Identifying the Incorrect Statement

The statement that the fraction of molecules with the most probable speed increases (option b) is not true. In reality, as temperature increases, the peak becomes lower and the fraction with that exact speed decreases.

Key concepts

Temperature Effect on Molecular Speed

The effect of temperature on molecular speed is significant and directly influences the behavior of gases. As the temperature increases, the molecules in a gas gain more kinetic energy. This increase in energy impacts how fast the molecules move.
- **Increased Average Speed**: With a rise in temperature, the average speed of the molecules goes up. This means that on average, gas molecules are moving faster when it's hotter. - **Broader Speed Range**: The distribution of molecular speeds becomes wider as temperature rises, allowing molecules to achieve a greater variety of speeds.
Ultimately, at higher temperatures, not only do gas molecules move faster on average, but they also spread out over a wider range of speeds. This reflects a key characteristic of gases influenced by temperature.

Kinetic Molecular Theory

The kinetic molecular theory is a fundamental framework that explains the behaviors of gases and their molecules in motion. This theory posits that gas molecules move in constant, random motion and that this motion is influenced by several factors, including temperature.
- **Energy and Motion**: According to the theory, the kinetic energy of gas molecules is directly proportional to the temperature. As temperature goes up, so does the kinetic energy. - **Elastic Collisions**: Gas molecules are in perpetual motion and when they collide, these collisions are elastic. This means they don't lose energy during the collisions—only transfer it. - **Pressure and Volume Relations**: The theory also explains how pressure and volume relate to molecular speed and temperature. As molecules move faster (due to higher temperature), they hit container walls more often and harder, which can affect pressure if the volume is constant.
In sum, the kinetic molecular theory provides valuable insights into how changes in temperature can alter the motion, speed, and energy of gas molecules.

Molecular Speed Distribution

Molecular speed distribution is described by the Maxwell-Boltzmann distribution, which provides a graphical representation of how speeds of molecules in a gas are spread at a given temperature.
- **Curve Characteristics**: The distribution is depicted as a curve where the x-axis represents molecular speed and the y-axis the probability density of those speeds. It shows how likely it is for molecules to be traveling at particular speeds. - **Effect of Temperature**: As temperature increases, the peak of this distribution curve shifts to higher speeds, indicating that more molecules are likely moving faster. Additionally, the curve flattens and broadens, meaning the speeds of molecules are more spread out across a range. - **Area Consistency**: Importantly, the area under the distribution curve remains the same across different temperatures since it represents the total number of molecules, which doesn't change based on temperature.
Understanding molecular speed distribution helps relate temperature changes to specific behaviors of gas molecules, such as their speed variety and concentration.