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Chapter 5: Problem 109

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

Expert verified
At 273K, the average kinetic energies of both CH4(g) and N2(g) are \(KE_{avg(CH_4, 273K)} = KE_{avg(N_2, 273K)} = 5.60 \times 10^{-21}\) J. At 546K, their average kinetic energies are \(KE_{avg(CH_4, 546K)} = KE_{avg(N_2, 546K)} = 1.12 \times 10^{-20}\) J.

Step by step solution

01

Identify the given information

We are given the following information: - The temperatures: 273K and 546K - The Boltzmann constant: \(1.38 \times 10^{-23}~ J/mol~K\) We will use this information to find the average kinetic energies of CH4 and N2 molecules.
02

Calculate the average kinetic energy at 273K

Using the formula for the average kinetic energy, we will find the avg. kinetic energy for both CH4 and N2 molecules at 273K: \(KE_{avg}=\dfrac{3}{2}kT\) For CH4(g) at 273K: \(KE_{avg(CH_4, 273K)}=\dfrac{3}{2}(1.38 \times 10^{-23}~ J/mol~K)(273~K) = \) For N2(g) at 273K: \(KE_{avg(N_2, 273K)}=\dfrac{3}{2}(1.38\times10^{-23}~ J/mol~K)(273~K) = \) Since both molecules are at the same temperature (273K), their average kinetic energies will be the same.
03

Calculate the average kinetic energy at 546K

Using the same formula, we will find the average kinetic energy for CH4 and N2 molecules at 546K: \(KE_{avg}=\dfrac{3}{2}kT\) For CH4(g) at 546K: \(KE_{avg(CH_4, 546K)}=\dfrac{3}{2}(1.38\times10^{-23}~ J/mol~K)(546~K) = \) For N2(g) at 546K: \(KE_{avg(N_2, 546K)}=\dfrac{3}{2}(1.38\times10^{-23}~ J/mol~K)(546~K) = \) As in step 2, since both molecules are at the same temperature (546K), their average kinetic energies will be the same.
04

Finalize the answers

We have calculated the average kinetic energies of CH4 and N2 gas molecules at given temperatures. At 273K: - CH4(g): \(KE_{avg(CH_4, 273K)} = \) J - N2(g): \(KE_{avg(N_2, 273K)} = \) J At 546K: - CH4(g): \(KE_{avg(CH_4, 546K)} = \) J - N2(g): \(KE_{avg(N_2, 546K)} = \) J

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