A smoke particle undergoing Brownian motion in air has a mass of 2.00 x 10 -5 kg.
a) what is the its average velocity if the temperature is 300 K?
b) what is its average kinetic energy?
The average kinetic energy of a particle or molecule of any mass is
Eav = (3/2) k T
where k is the Boltzmann constant,
1.38^10^-23 J/K, and
T is the temperature in Kelvin
Once you have the average KE, get the average (actually, r.m.s) velocity from
(1/2) M V^2 = Eav
M is the particle mass
a) To find the average velocity of the smoke particle undergoing Brownian motion in air, we can use the root mean square (r.m.s.) velocity formula.
The r.m.s. velocity can be calculated using the formula:
V = √(2Eav/M)
Where V is the r.m.s. velocity, Eav is the average kinetic energy, and M is the particle mass.
First, let's find the average kinetic energy.
Given:
Temperature (T) = 300 K
Boltzmann constant (k) = 1.38 x 10^-23 J/K
Using the formula:
Eav = (3/2)kT
Substituting the given values:
Eav = (3/2)(1.38 x 10^-23 J/K)(300 K)
Eav ≈ 6.21 x 10^-21 J
Now we can calculate the r.m.s. velocity:
V = √(2Eav/M)
Substituting the values:
V = √((2)(6.21 x 10^-21 J)/(2.00 x 10^-5 kg))
V ≈ √(6.21 x 10^-16 J/2.00 x 10^-5 kg)
V ≈ √(3.1 x 10^-11 m^2/s^2)
V ≈ 5.57 x 10^-6 m/s
Therefore, the average velocity of the smoke particle is approximately 5.57 x 10^-6 m/s.
b) To find the average kinetic energy, we can use the formula:
Eav = (3/2)kT
Substituting the values:
Eav = (3/2)(1.38 x 10^-23 J/K)(300 K)
Eav ≈ 6.21 x 10^-21 J
Therefore, the average kinetic energy of the smoke particle is approximately 6.21 x 10^-21 J.