An alpha particle (mass = 6.6 10-24 g) emitted by radium travels at 2.5 107 ± 0.1 107 mi/h.
(a) What is its de Broglie wavelength (in meters)?
(b) What is the uncertainty in its position?
wavelength = h/mv
Don't forget m is in kg.
To find the de Broglie wavelength of an alpha particle, we can use the de Broglie wavelength formula:
λ = h / p
where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle.
(a) First, we need to find the momentum of the alpha particle. The momentum of any particle is given by the product of its mass and velocity:
p = mv
where m is the mass of the particle and v is its velocity.
Given:
Mass of alpha particle (m) = 6.6 x 10^-24 g
Velocity of alpha particle (v) = 2.5 x 10^7 ± 0.1 x 10^7 mi/h
Before we continue, we need to convert the mass and velocity to SI units:
1 g = 10^-3 kg
1 mi = 1.60934 km
1 h = 60 min x 60 s = 3600 s
Converting the units:
Mass of alpha particle (m) = 6.6 x 10^-27 kg
Velocity of alpha particle (v) = (2.5 x 10^7 ± 0.1 x 10^7) mi/h x 1.60934 km/mi x 1000 m/km / 3600 s
Now we can calculate the momentum of the alpha particle:
p = (6.6 x 10^-27 kg) * [(2.5 x 10^7 ± 0.1 x 10^7) mi/h x 1.60934 km/mi x 1000 m/km / 3600 s]
Once you calculate the value of p, you can substitute it into the de Broglie wavelength formula to find the wavelength λ. Remember to use the correct value of Planck's constant, which is h = 6.626 x 10^-34 J·s.
(b) To find the uncertainty in position (Δx), we can use the uncertainty principle:
Δx Δp ≥ h
where Δp is the uncertainty in momentum. We can calculate Δp by using the uncertainty in velocity (Δv):
Δp = m Δv
where Δv is the uncertainty in velocity.
Using the given uncertainty in velocity (0.1 x 10^7 mi/h) and following the same units conversion as before, you can calculate Δp. Then, by substituting that value into the uncertainty principle equation, you can find the uncertainty in position (Δx).