The correct statement regarding various types of molecular speeds are

Options

A. increasing temperature increasing the fraction of molecules having `U_("mps")`
B. Increasing temperature increases `U_("mps")`
C. In a sample of gas at a given temperature, molecules with extremely low and high speeds are less
D. At the same temperature lighter gaseous having narrow distribution of molecular speeds than heavier gaseous

Answer

To answer the question regarding the correct statements about various types of molecular speeds, we can analyze the information provided in the video transcript step by step.

### Step-by-Step Solution:

1. **Understanding Molecular Speed and Temperature Relationship**:
- When the temperature of a gas increases, the average kinetic energy of the molecules also increases. This leads to an increase in molecular speeds.
- **Hint**: Recall that temperature is a measure of the average kinetic energy of molecules in a substance.

2. **Graph Analysis**:
- A graph can be plotted with the fraction of molecules on the y-axis and molecular speed on the x-axis. As temperature increases (from T1 to T2), the peak of the graph shifts to the right.
- This indicates that at higher temperatures, more molecules have higher speeds.
- **Hint**: Visualize how the distribution curve changes with temperature.

3. **Most Probable Speed (UMBS)**:
- The most probable speed of molecules (UMBS) increases with temperature. This means that as the temperature rises, the speed at which the highest number of molecules are moving also increases.
- **Hint**: Remember that UMBS is the speed at which the maximum number of molecules are found.

4. **Fraction of Molecules at Maximum Speed**:
- Although the most probable speed increases with temperature, the fraction of molecules that have the maximum speed decreases. This is because while more molecules are moving faster, the distribution broadens, leading to fewer molecules at the peak speed.
- **Hint**: Consider how a bell curve spreads out as the average shifts.

5. **Area Under the Curve**:
- The area under the curve in the graph represents the total number of molecules. This area remains constant regardless of temperature, as the total number of gas molecules in the sample does not change.
- **Hint**: Think about conservation of mass in a closed system.

6. **Comparing Gases at the Same Temperature**:
- At the same temperature, lighter gases will have a broader distribution of speeds compared to heavier gases. This means lighter gases will have more molecules at higher speeds.
- **Hint**: Reflect on how mass affects speed according to the kinetic molecular theory.

### Summary of Correct Statements:
- **Correct Statements**:
- Increasing temperature leads to an increase in the most probable speed (UMBS).
- The area under the curve remains constant across different temperatures.

- **Incorrect Statements**:
- The fraction of molecules having maximum speed does not increase with temperature.
- At the same temperature, lighter gases do not have a narrower distribution of speeds compared to heavier gases.

### Final Answer:
The correct statements regarding various types of molecular speeds are:
- Increasing temperature increases the most probable speed (UMBS).
- The area under the curve remains the same at all temperatures, representing the total number of gas molecules.