Concept Explanation: Root Mean Square Speed and Average Translational Kinetic Energy

In kinetic theory of gases, for an ideal gas at equilibrium:

Step 1: Define Root Mean Square Speed (U_rms)

U_rms is the square root of the average of the squares of the speeds of molecules. It is given by:

$U_{rms}=\sqrt{\frac{3RT}{M}}$

where R is the gas constant, T is the absolute temperature, and M is the molar mass.

Step 2: Define Average Translational Kinetic Energy (ε_av)

ε_av is the average kinetic energy per molecule due to translational motion. It is given by:

${\epsilon }_{av}=\frac{3}{2}kT$

where k is the Boltzmann constant.

Step 3: Analyze Each Statement

Statement 1: ε_av at a given temperature does not depend on its molecular mass.

From the equation ${\epsilon }_{av}=\frac{3}{2}kT$, ε_av depends only on temperature T, not on molecular mass. So, this statement is correct.

Statement 2: U_rms is inversely proportional to the square root of its molecular mass.

From $U_{rms}=\sqrt{\frac{3RT}{M}}$, U_rms ∝ $\sqrt{\frac{1}{M}}$. So, it is inversely proportional to √M. This statement is correct.

Statement 3: U_rms is doubled when its temperature is increased four times.

From U_rms ∝ √T, if T becomes 4T, U_rms becomes √(4T) = 2√T, i.e., doubled. So, this statement is correct.

Statement 4: ε_av is doubled when its temperature is increased four times.

From ε_av ∝ T, if T becomes 4T, ε_av becomes 4 times, not doubled. So, this statement is incorrect.

Final Answer

The correct statements are 1, 2, and 3.

Related Topics

• Kinetic Theory of Gases
• Ideal Gas Law
• Maxwell-Boltzmann Distribution
• Molar Heat Capacities

Important Formulae

Root Mean Square Speed: $U_{rms}=\sqrt{\frac{3RT}{M}}$

Average Translational Kinetic Energy (per molecule): ${\epsilon }_{av}=\frac{3}{2}kT$

Average Translational Kinetic Energy (per mole): $E_{av}=\frac{3}{2}RT$

Relation between R and k: R = N_Ak, where N_A is Avogadro's number.