Which of the following gas will have the highest value for translational K.E. per g, at the same temperature? (a) methane (b) helium (c) nitrogen (d) same for all

At the heart of understanding translational kinetic energy of particles in a gas is Boltzmann's constant (denoted by the symbol \(k\)). This constant provides a bridge between macroscopic and microscopic physics, which is a key aspect of kinetic theory and statistical mechanics. When dealing with individual molecules, \(k\) plays a pivotal role in showing how the energy of particles relates to the temperature of the system.

Boltzmann's constant has a value of approximately \(1.38 \times 10^{-23} J/K\) per particle per degree Kelvin. In the context of gases, \(k\) links the average kinetic energy of a single particle to the absolute temperature. The formula \(\frac{3}{2}kT\) signifies that each particle has a kinetic energy proportional to the temperature. The constant serves as a proportionality factor, defining how much kinetic energy particles have at a certain temperature.

Understanding \(k\) is fundamental because it shows that temperature is indeed a measure of the average kinetic energy per molecule in a system. By knowing the temperature, one can infer the average energy that molecules are vibrating or moving with, which subsequently describes the system's thermal energy state.

The universal gas constant (which is represented by the symbol \(R\)) is another significant value used in the study of gases. This constant integrates the Avogadro’s number and Boltzmann’s constant to work at the level of moles, a standard chemical unit. \(R\) historically represents the constant volume heat capacity of an ideal gas, but it also appears in the ideal gas law (\(PV = nRT\)), binding together the pressure (\(P\)), volume (\(V\)), and temperature (\(T\)) in terms of the number of moles (\(n\)).

The value of \(R\) is approximately \(8.314 J/(mol \cdot K)\) and is key to calculating the energy per mole of gas. For translational kinetic energy per mole at a given temperature, the equation is \(\frac{3}{2}RT\). By comparing \(k\) and \(R\), we can see the similarity in their functions, with \(R\) being suited for macroscopic samples of a gas, where total moles are considered. Knowing \(R\) allows chemists and physicists to calculate various properties of gases, including translational kinetic energy per mole.

Molar mass is a crucial concept when comparing the translational kinetic energy of gases on a per-gram basis. Essentially, the molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is the sum of the atomic masses of all the atoms in a molecular formula.

The molar mass allows us to link the microscopic properties of individual molecules to their macroscopic quantities by giving us a way to quantify the amount of substance. In our given context, by knowing the molar mass of each gas, we can determine the translational kinetic energy per gram since this energy is inversely related to molar mass if temperature is constant. A lower molar mass means that there are more moles of gas per gram, and since the energy is the same per mole at a constant temperature, this results in greater energy per gram.

A practical example is comparing helium and nitrogen. With helium having a significantly lower molar mass than nitrogen, it means that per gram, helium has more moles than nitrogen. Consequently, at the same temperature, helium would have higher translational kinetic energy per gram, illustrating why understanding molar mass is important in various scientific applications, from determining gas behavior to calculating chemical reactions.