Do all the molecules in a 1 -mol sample of \(\mathrm{CH}_{4}(g)\) have the same kinetic energy at \(273 \mathrm{~K}\) ? Do all molecules in a \(1-\mathrm{mol}\) sample of \(\mathrm{N}_{2}(g)\) have the same velocity at \(546 \mathrm{~K}\) ? Explain.

The average kinetic energy of a molecule in a gas sample is directly proportional to the temperature, meaning that increasing the temperature will increase the average kinetic energy of the molecules. The formula to find the average kinetic energy of a molecule can be written as: \[K.E = \dfrac{3}{2} kT\] Where K.E is the average kinetic energy, k is the Boltzmann constant, and T is the temperature in Kelvin. It is essential to note that this formula provides us with the average kinetic energy of the molecules in a sample, not the kinetic energy of each molecule.

Now, we know that the average kinetic energy of the sample depends on the temperature. However, each molecule's kinetic energy can be different due to their unique velocities and collisions that occur within the gas sample. Even though the average kinetic energy at 273 K can be the same for all molecules in a 1-mol sample of CH4(g), individual molecules can have a range of different kinetic energies.

The kinetic energy of a molecule is related to its mass and velocity by the following formula: \[K.E = \dfrac{1}{2} mv^2\] Where K.E is the kinetic energy, m is the mass of the gas molecule, and v is its velocity. In this exercise, we are interested in examining the individual molecular velocities of N2(g) at 546 K.

It is important to remember that molecular velocity, temperature, mass, and kinetic energy are related. Therefore, even if the average kinetic energy of the N2 molecules at 546 K is the same, their individual velocities can vary because of their unique pathway through the sample and collisions with other gas molecules. Hence, all molecules in a 1-mol sample of N2 do not have the same velocity at 546 K. In conclusion, - All molecules in a 1-mol sample of CH4(g) do not have the same kinetic energy at 273 K, even though their average kinetic energy is the same due to the temperature. - All molecules in a 1-mol sample of N2(g) do not have the same velocity at 546 K, even though their average kinetic energy is the same due to the temperature.