UP Board Class 12 Physics Chapter 1

Explanation: The question asks what single physical quantity is sufficient to determine the average kinetic energy of molecules in a gas. In Kinetic Theory, the average kinetic energy of gas molecules depends on a fundamental thermodynamic variable that reflects the intensity of Molecular motion. This variable directly relates to how fast particles move on average, regardless of their type or number.

According to Kinetic Theory, the average kinetic energy of gas molecules is proportional to absolute temperature. It does not depend on the number of molecules or pressure alone. Pressure depends on collisions and number density, while kinetic energy is purely linked to Molecular motion. Even if two gases differ in Molecular weight, at the same temperature their average kinetic energy remains identical.

To reason this out, imagine heating a gas. As temperature increases, molecules move faster, increasing their kinetic energy. This relationship is universal for ideal gases and forms the basis of temperature measurement in Thermodynamics.

For example, both Light gases like hydrogen and heavier gases like oxygen have the same average kinetic energy at the same temperature, even though their speeds differ.

In summary, average kinetic energy of gas molecules depends only on absolute temperature and not on other macroscopic variables like pressure or number of molecules.

Explanation: This question compares the root mean square (RMS) speeds of two different gases at the same temperature. RMS speed is a measure of the average speed of gas molecules and depends on both temperature and Molecular Mass. At constant temperature, lighter molecules move faster than heavier ones.

The RMS velocity is given by a relation where it is inversely proportional to the square root of molar Mass. Since hydrogen molecules are much lighter than oxygen molecules, hydrogen will have a significantly higher RMS speed under identical thermal conditions. The relationship involves taking the square root of the ratio of their molar masses.

To analyze step by step, note that oxygen has a much higher Molecular Mass compared to hydrogen. Since velocity varies inversely with the square root of Mass, reducing Mass leads to a sharp increase in speed. This is why lighter gases diffuse faster and move more rapidly.

As an analogy, think of two balls—one heavy and one Light—given the same energy. The lighter one moves faster because it requires less energy to achieve higher speed.

In summary, RMS velocity depends inversely on the square root of Molecular Mass, so lighter gases exhibit much higher speeds than heavier gases at the same temperature.

Explanation: The question compares RMS velocities of two gases under different conditions of temperature and Molecular Mass. RMS velocity depends on both temperature and molar Mass, increasing with temperature and decreasing with increasing Molecular Mass. This relationship helps predict how Molecular motion changes when these parameters vary.

The RMS velocity is proportional to the square root of temperature divided by molar Mass. If temperature decreases, molecular motion slows down. Similarly, if molecular mass increases, molecules become heavier and move more slowly. Both changes act together to reduce the velocity.

Step-by-step, halving the temperature reduces velocity by a factor related to the square root, while doubling the molecular mass further reduces it. Since both factors work in the same direction, the resulting RMS velocity becomes significantly smaller than the original value.

For example, imagine replacing Light particles with heavier ones while also cooling the system—both effects make movement slower.

In summary, RMS velocity decreases when temperature decreases and molecular mass increases, with both effects combining multiplicatively through square root dependence.

Explanation: This question focuses on how RMS speed behaves during an isothermal process. In such a process, temperature remains constant even though pressure and volume may change. RMS speed is directly linked to temperature, making this relationship crucial.

According to Kinetic Theory, RMS speed depends only on absolute temperature and not on pressure or volume. During isothermal compression, although molecules are forced closer together and pressure increases, their average kinetic energy remains unchanged because temperature is constant.

Step-by-step reasoning shows that compression affects collision frequency but not molecular speed. Molecules collide more often due to reduced volume, but their speeds stay the same since energy is unchanged.

As an analogy, consider people moving at the same speed in a crowded room versus a large hall. Their speed doesn’t change, but collisions increase in the smaller space.

In summary, RMS speed depends solely on temperature, so it remains unaffected during isothermal compression despite changes in pressure and volume.

Explanation: The question examines how RMS velocity changes with temperature. RMS velocity is proportional to the square root of absolute temperature, meaning that any change in temperature affects velocity in a predictable way.

When temperature increases, molecular motion becomes more energetic, leading to higher velocities. However, the increase is not linear—it follows a square root relationship. This means doubling temperature does not double velocity; instead, velocity increases by the square root of the temperature ratio.

Step-by-step, the temperature increases by a factor of four. Taking the square root of this factor gives the corresponding increase in RMS velocity. This shows how sensitive velocity is to temperature changes.

For example, heating a gas significantly increases molecular motion, but not as drastically as the temperature increase itself.

In summary, RMS velocity increases with the square root of temperature, so large temperature changes produce moderate increases in molecular speed.

Explanation: This question explores the relationship between RMS speed and temperature. RMS speed is proportional to the square root of absolute temperature, so increasing speed requires increasing temperature in a squared manner.

To double the RMS speed, the temperature must increase such that the square root relationship holds. This means the new temperature must be four times the original absolute temperature, since √4 = 2.

Step-by-step, first convert the initial temperature into Kelvin. Then multiply it by four to find the required temperature. Finally, convert back to Celsius if needed to match the options.

As an analogy, think of speed increasing slowly compared to temperature—large temperature changes are needed for noticeable increases in speed.

In summary, doubling RMS speed requires quadrupling the absolute temperature due to the square root dependence of velocity on temperature.

Explanation: This question involves reducing RMS velocity by changing temperature. RMS velocity depends on the square root of absolute temperature, so reducing velocity requires reducing temperature significantly.

To make RMS velocity half, the temperature must be reduced such that the square root relationship holds. This means the final temperature must be one-fourth of the initial absolute temperature.

Step-by-step, convert the initial temperature to Kelvin. Then divide it by four to find the new temperature. Finally, convert back to Celsius to interpret the result properly.

For example, lowering temperature reduces molecular motion, but halving speed requires a much larger drop due to the square root relationship.

In summary, reducing RMS velocity to half requires reducing absolute temperature to one-fourth of its original value.

Explanation: This question compares RMS velocities of two gases at different temperatures. RMS velocity depends on both temperature and molar mass, so equality of velocities requires balancing these two factors.

Hydrogen is much lighter than oxygen, so at the same temperature it moves faster. To make their RMS velocities equal, hydrogen must be at a lower temperature compared to oxygen.

Step-by-step reasoning involves equating the expressions for RMS velocity of both gases. This leads to a relationship between their temperatures and molar masses, allowing calculation of the required temperature.

As an analogy, lighter objects need less energy to move fast, so to match speeds with heavier ones, they must have less energy.

In summary, equal RMS velocities for different gases require adjusting temperature inversely with their molar masses.

Explanation: This question compares RMS velocities of different gases at different temperatures. RMS velocity depends on the square root of temperature and inversely on the square root of molar mass.

Hydrogen is much lighter than oxygen, so it naturally has higher velocity. However, oxygen is considered at a higher temperature, which increases its molecular speed.

Step-by-step, the increase in temperature tends to increase velocity, while higher molecular mass reduces it. Both effects must be combined using the square root relationship to determine the final velocity.

For example, heating a heavier gas can increase its speed, but it may still be slower than a lighter gas at lower temperature.

In summary, RMS velocity depends on both temperature and molecular mass, and their combined effect determines the final speed.

Explanation: This problem compares RMS speeds of two gases and asks for the temperature adjustment required. RMS speed depends on temperature and molecular mass, and equality of speeds requires balancing these factors.

Since hydrogen is lighter, it moves faster at the same temperature. To match oxygen’s speed, hydrogen must be cooled to a lower temperature.

Step-by-step, equate RMS velocity expressions for both gases. This yields a relationship between their temperatures and molar masses, allowing calculation of the required temperature for hydrogen.

As an analogy, a lighter runner must slow down to match the pace of a heavier runner.

In summary, equal RMS speeds require reducing the temperature of the lighter gas in proportion to the ratio of molecular masses.

Explanation: This question compares two gases with similar but not identical molecular masses. RMS velocity depends on both temperature and molar mass, so equal velocities require adjusting temperature accordingly.

Nitrogen and oxygen have comparable molecular masses, so only a small adjustment in temperature is needed to equalize their RMS speeds.

Step-by-step, equate the RMS velocity expressions and solve for the unknown temperature. The relationship involves the ratio of their molar masses and temperatures.

For example, two nearly similar masses require only slight temperature changes to achieve the same speed.

In summary, when molecular masses are close, only a small temperature difference is needed to equalize RMS velocities.