Question

The molecular speed \(U_{\text {rins }}\) of gas is (1) Independent of temperature (2) Proportional to the absolute temperature (3) Proportional to the square root of absolute temperature (4) Proportional to the square of absolute temperature

Short answer

(3) Proportional to the square root of absolute temperature

Step-by-step solution

Understand the Relationship

First, understand that molecular speed of a gas is related to its temperature. This relationship can be studied using the kinetic theory of gases.

Kinetic Theory of Gases

According to the kinetic theory of gases, the average kinetic energy of gas molecules is directly proportional to the absolute temperature. Therefore, the kinetic energy can be expressed as: \[ KE = \frac{1}{2} m U_{\text{rms}}^2 \text{where } m \text{ is the mass of the gas molecule and } U_{\text{rms}} \text{ is the root-mean-square speed.} \]

Temperature to Speed Relationship

Rearrange the formula for kinetic energy to find the expression for the root-mean-square speed: \[ U_{\text{rms}} = \sqrt{\frac{2 \cdot KE}{m}}\]Since average kinetic energy \( KE \) is proportional to temperature \( T \), we can write: \[ KE \propto T \]Thus, \[ U_{\text{rms}} \propto \sqrt{T} \]This shows that the root-mean-square speed is proportional to the square root of absolute temperature.

Conclusion

Based on the derived relationship, the molecular speed \( U_{\text{rms}} \) of gas is proportional to the square root of absolute temperature.

Key concepts

kinetic theory of gases

The kinetic theory of gases is a fundamental theory that helps us understand how gases behave. It explains the macroscopic properties of gases, like pressure and temperature, in terms of microscopic particles in constant, random motion. According to this theory, gas molecules are always moving and colliding with each other and the walls of their container.
Because these collisions are perfectly elastic, there is no loss of energy when they collide. This means the total kinetic energy of gas molecules is conserved. The theory also states that the average kinetic energy of gas molecules is directly proportional to the absolute temperature of the gas. This simple, elegant theory lays the groundwork for understanding many properties of gases, including molecular speed.

root-mean-square speed

Root-mean-square (RMS) speed is one way to express the speed of gas molecules. It is defined as the square root of the average of the squares of individual speeds of the gas molecules. Mathematically, it is given by:
\[ U_{\text{rms}} = \sqrt{\frac{1}{N} \sum_{i=1}^{N} U_i^2} \]
where:\[ N \] is the total number of gas molecules and \[ U_i \] is the speed of the individual gas molecule.
In simpler terms, RMS speed helps to provide a representative speed of all the gas molecules, taking into account their various individual speeds.
According to the kinetic theory of gases, the RMS speed can also be related to temperature using the formula:
\[ U_{\text{rms}} = \sqrt{\frac{3kT}{m}} \]
where:\[ k \] is the Boltzmann constant, \[ T \] is the absolute temperature, and \[ m \] is the mass of one gas molecule. This formula shows how RMS speed varies based on the temperature and mass of the gas molecules.

temperature dependence

Temperature greatly influences the speed of gas molecules. According to the kinetic theory of gases, as the temperature of a gas increases, the average kinetic energy of the gas molecules does too. This means that at higher temperatures, gas molecules move faster. Conversely, when the temperature drops, the molecules slow down.
Specifically, the relationship between temperature and RMS speed is given by the formula:
\[ U_{\text{rms}} = \sqrt{\frac{3kT}{m}}\]
This implies that RMS speed is proportional to the square root of the temperature. Simply put, if you double the temperature (measured in Kelvin), the RMS speed will increase by a factor of \[ \sqrt{2} \].
Thus, the molecular speed \( U_{\text{rms}} \) is not only dependent on temperature, but it is directly proportional to the square root of the absolute temperature, making temperature a crucial factor in determining how fast gas molecules move.