Question

Calculate the average kinetic energies of \(\mathrm{CH}_{4}(g)\) and \(\mathrm{N}_{2}(g)\) molecules at \(273 \mathrm{~K}\) and \(546 \mathrm{~K}\).

Short answer

The average kinetic energies of CH₄(g) and N₂(g) at 273 K and 546 K can be calculated using the formula \(K.E. = \dfrac{3}{2} kT \), where k is Boltzmann's constant (\(1.38 \times 10^{-23} JK^{-1}\)). At 273 K, both CH₄(g) and N₂(g) have an average kinetic energy of \(6.10 \times 10^{-21} J\). At 546 K, the average kinetic energy of both CH₄(g) and N₂(g) is \(1.22 \times 10^{-20} J\).

Step-by-step solution

Calculate the average kinetic energy of CH₄(g) at 273 K

To calculate the average kinetic energy of CH₄(g) at 273 K, we need to use the formula mentioned above and plug in the known values: \(K.E. = \dfrac{3}{2} kT\) \(K.E. = \dfrac{3}{2} (1.38 \times 10^{-23} J K^{-1})(273 K)\) Now, perform the calculation to find the average kinetic energy.

Calculate the average kinetic energy of N₂(g) at 273 K

Similarly, to calculate the average kinetic energy of N₂(g) at 273 K, use the formula and plug in the known values: \(K.E. = \dfrac{3}{2} kT\) \(K.E. = \dfrac{3}{2} (1.38 \times 10^{-23} J K^{-1})(273 K)\) Perform the calculation to find the average kinetic energy.

Calculate the average kinetic energy of CH₄(g) at 546 K

Next, calculate the average kinetic energy of CH₄(g) at 546 K using the formula and known values: \(K.E. = \dfrac{3}{2} kT\) \(K.E. = \dfrac{3}{2} (1.38 \times 10^{-23} J K^{-1})(546 K)\) Perform the calculation to find the average kinetic energy.

Calculate the average kinetic energy of N₂(g) at 546 K

Finally, calculate the average kinetic energy of N₂(g) at 546 K using the formula and known values: \(K.E. = \dfrac{3}{2} kT\) \(K.E. = \dfrac{3}{2} (1.38 \times 10^{-23} J K^{-1})(546 K)\) Perform the calculation to find the average kinetic energy. After completing steps 1-4, you will have the average kinetic energies of CH₄(g) and N₂(g) at 273 K and 546 K.