The average translational energy and the rms speed of...

The average translational energy and the rms speed of molecules in a sample of oxygen gas at $300 \mathrm{~K}$ are $6.21 \times 10^{-21} \mathrm{~J}$ and $484 \mathrm{~m} / \mathrm{s}$ respectively. The corresponding values at $600 \mathrm{~K}$ are nearly (assuming ideal gas behaviour): (a) $12.42 \times 10^{-21} \mathrm{~J}, 968 \mathrm{~m} / \mathrm{s}$ (b) $8.78 \times 10^{-21} \mathrm{~J}, 684 \mathrm{~m} / \mathrm{s}$ (c) $6.21 \times 10^{-21} \mathrm{~J}, 968 \mathrm{~m} / \mathrm{s}$ (d) $12.42 \times 10^{-21} \mathrm{~J}, 684 \mathrm{~m} / \mathrm{s}$

The average translational energy and the rms speed of molecules in a sample of oxygen gas at $300 \mathrm{~K}$ are $6.21 \times 10^{-21} \mathrm{~J}$ and $484 \mathrm{~m} / \mathrm{s}$ respectively. The corresponding values at $600 \mathrm{~K}$ are nearly (assuming ideal gas behaviour):
(a) $12.42 \times 10^{-21} \mathrm{~J}, 968 \mathrm{~m} / \mathrm{s}$
(b) $8.78 \times 10^{-21} \mathrm{~J}, 684 \mathrm{~m} / \mathrm{s}$
(c) $6.21 \times 10^{-21} \mathrm{~J}, 968 \mathrm{~m} / \mathrm{s}$
(d) $12.42 \times 10^{-21} \mathrm{~J}, 684 \mathrm{~m} / \mathrm{s}$

Key Concepts

Ideal Gas Behavior

Ideal gas behavior assumes that the gas molecules are point particles that do not exert forces on each other except during elastic collisions. This model allows for straightforward mathematical descriptions of gas properties, such as the direct proportionality between temperature and average kinetic energy, and the dependence of rms speed on the square root of temperature. These simplifications make it easier to predict how gases behave under changes in temperature.

Temperature Dependence of Translational Kinetic Energy

In an ideal gas, the average translational kinetic energy of the molecules is directly proportional to the absolute temperature. This means that if the temperature is doubled, the average kinetic energy is also doubled. This linear dependence is a direct consequence of the equipartition theorem and is fundamental in kinetic theory.

Root Mean Square (rms) Speed

The rms speed is defined as the square root of the mean of the squares of the speeds of the gas molecules. It is given by the formula v_rms = sqrt(3kBT/m), where m is the molecular mass. This shows that the rms speed is proportional to the square root of the temperature, meaning that if the absolute temperature is increased, the rms speed increases by the square root of the factor by which the temperature is increased.

Equipartition Theorem

The equipartition theorem states that energy is evenly distributed among all of the quadratic degrees of freedom of a system. In the context of an ideal gas, it implies that the average translational kinetic energy per molecule is directly proportional to the temperature, typically represented as (3/2)kBT for a three-dimensional gas. This relationship is central to understanding how energy changes with temperature.

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