Answer:
The mean kinetic energy of a molecule can be calculated using the equation:
KE = (3/2) * k * T
where KE is the mean kinetic energy, k is the Boltzmann constant (1.38 x 10^-23 J/K), and T is the temperature in Kelvin.
a) To calculate the mean kinetic energy of a molecule in the thermosphere at a temperature of 1000 K, we can use the equation above:
KE = (3/2) * (1.38 x 10^-23 J/K) * (1000 K)
Simplifying the equation, we find:
KE = 2.07 x 10^-20 J
So, the mean kinetic energy of a molecule in the thermosphere at a temperature of 1000 K is 2.07 x 10^-20 J.
b) The root mean square (r.m.s.) speed of a gas molecule can be calculated using the equation:
v = sqrt((3 * k * T) / m)
where v is the r.m.s. speed, k is the Boltzmann constant, T is the temperature in Kelvin, and m is the molar mass of the gas molecule.
(i) To calculate the r.m.s. speed of hydrogen at a temperature of 1000 K, we can use the equation above with the molar mass of hydrogen (0.0020 kg/mol):
v = sqrt((3 * (1.38 x 10^-23 J/K) * (1000 K)) / 0.0020 kg/mol)
Simplifying the equation, we find:
v ≈ 2655 m/s
So, the r.m.s. speed of hydrogen at a temperature of 1000 K is approximately 2655 m/s.
(ii) To calculate the r.m.s. speed of helium at a temperature of 1000 K, we can use the equation above with the molar mass of helium (0.040 kg/mol):
v = sqrt((3 * (1.38 x 10^-23 J/K) * (1000 K)) / 0.040 kg/mol)
Simplifying the equation, we find:
v ≈ 627 m/s
So, the r.m.s. speed of helium at a temperature of 1000 K is approximately 627 m/s.
In summary:
a) The mean kinetic energy of a molecule in the thermosphere at a temperature of 1000 K is 2.07 x 10^-20 J.
b) The r.m.s. speed of hydrogen at a temperature of 1000 K is approximately 2655 m/s, and the r.m.s. speed of helium at the same temperature is approximately 627 m/s.
Step-by-step explanation: