NA = 6.02 x 1023/mol.
R = 8.31 J/(mol K)
a) How many molecules are there?
b) What is the internal energy for the cubic meter of gas?
c) What is the average translational kinetic energy per molecule?
d) To what speed does that correspond?
Question so kindly provided by Joe Kiskis
This is a simple application of PV=nRT, just solving in molecules instead of moles.
n = PV⁄RT
n = 40.1 moles
40.1 moles ⋅ 6.02 x 1023 = 2.41 x 1025 molecules
- U = 1⁄(γ−1) nRT
- We know that γNitrogen = 7/5
- U = 5/2 ⋅ 40.1 moles ⋅ 8.3 J/molK ⋅ 300K
- U = 2.50 x 105J
Notice that the internal energy for this problem is different than the translational energy, which is calculated like so:
- KTR = 3/2 nRT
- KTR = 3/2 ⋅ 40.1 mole ⋅ 8.31 J/molK ⋅ 300K
- Note that we want translational energy per molecule, not total, so divide by our answer from part A.
- KTR⁄molecule = 6.21 x 10-21 J/molecule
Simply apply the energy from the last step to kinetic motion for 1 molecule:
- 6.21 x 10-21 J = 1/2 mv2
- v = 514 m/s