3. When the spacing between translational energy levels is small compared to the
ID: 1023745 • Letter: 3
Question
3. When the spacing between translational energy levels is small compared to the thermal energy (kBT), classical mechanics is a good approximation for quantum mechanics. Consider whether the translational motion of each particle below can be treated classically. For each system below, calculate the energy of the particle in a box for the n=1 and n=2 states and determine whether E (ie E2 – E1) is smaller or larger than the available thermal energy, kBT. (a) A helium atom in a 1000 Å box at 298 K (L = 10-7 m). (b) A protein with a molecular weight of 50 kDa in a 100 Å box at 298 K (L = 10-8 m). (c) A helium atom in a 1 Å box at 1 K (L = 10-10 m).
3. When the spacing between translational energy levels is small compared to the thermal energy (kBT), classical mechanics is a good approximation for quantum mechanics. Consider whether the translational motion of each particle below can be treated classically. For each system below, calculate the energy of the particle in a box for the n=1 and n=2 states and determine whether E (ie E2 – E1) is smaller or larger than the available thermal energy, kBT. (a) A helium atom in a 1000 Å box at 298 K (L = 10-7 m). (b) A protein with a molecular weight of 50 kDa in a 100 Å box at 298 K (L = 10-8 m). (c) A helium atom in a 1 Å box at 1 K (L = 10-10 m).
3. When the spacing between translational energy levels is small compared to the thermal energy (kBT), classical mechanics is a good approximation for quantum mechanics. Consider whether the translational motion of each particle below can be treated classically. For each system below, calculate the energy of the particle in a box for the n=1 and n=2 states and determine whether E (ie E2 – E1) is smaller or larger than the available thermal energy, kBT. (a) A helium atom in a 1000 Å box at 298 K (L = 10-7 m). (b) A protein with a molecular weight of 50 kDa in a 100 Å box at 298 K (L = 10-8 m). (c) A helium atom in a 1 Å box at 1 K (L = 10-10 m).
Explanation / Answer
The formula for energy of particle in a box is
En = n^2h^2 / 8mL^2 m in Kg
a] Helium atom
m = 4 gms = 0.004 Kg
L =10^-7 ; h =6.625*10^-34
E1 = 1.37*10^-51 Joules
E2 = 5.48*10^-51 Joules
E2 - E1 = 4.11*10^-51 Joules
KbT = 4.11*10^-21 Joules
Thermal energy is greater than E2-E1
b] protein with a molecular weight of 50 kDa in a 100 Å box at 298 K (L = 10-8 m).
50Kda = 8.3*10^-23 Kg
E1 = 6.61 *10^-30 Joules
E2 = 26.44*10^-30 Joules
E2-E1 =19.83*10^-30 Joules
KbT = 1.38*10^-23 * 298 = 4.12*10^-21 joules
Thermal energy greater than E2-E1
c] A helium atom in a 1 Å box at 1 K (L = 10-10 m).
E1 = 1.37*10^-45 Joules
E2 =5.48*10^-45 Joules
E2 - E1 = 4.11*10^-45 Joules
KbT = 1.38*10^-23 JOules
SO thermal energy is more than E2-E1
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