An alpha particle has a mass of 6.6*10-24 g travels at 3.4*107 +/- 0.1*107 mi/h

a. What is the DeBroglie wavelength (in meters)?
b. What is the uncertainty in position?

We Know that :

According to debroglies wavelength :
λ = h / mv
= 6.627 x 10 -34 J -s / 6.6 x10-24 x 10-3 kg / 1 g x 3.4x 107 mile / hr x 5 793 638.4 m / s / 1 mile / hr
= 5.09 x 10-22 m

To find the DeBroglie wavelength (λ) of an alpha particle, we can use the DeBroglie equation:

λ = h / p

where λ is the wavelength, h is the Planck's constant (6.626 x 10^-34 J.s), and p is the momentum of the particle.

a. To find the DeBroglie wavelength (λ) of the alpha particle, we need to find its momentum first. The momentum (p) can be calculated using the formula:

p = m * v

where m is the mass of the alpha particle and v is its velocity.

Given:
Mass (m) = 6.6 x 10^-24 g
Velocity (v) = 3.4 x 10^7 +/- 0.1 x 10^7 mi/h

First, we need to convert the mass from grams to kilograms:
Mass (m) = 6.6 x 10^-24 g = 6.6 x 10^-27 kg

Next, we need to convert the velocity from miles per hour (mi/h) to meters per second (m/s):
Velocity (v) = 3.4 x 10^7 mi/h = 3.4 x 10^7 mi/h * 1.60934 km/mi * 1000 m/km * 1/3600 h/s ≈ 1.52 x 10^4 m/s

Now, we can calculate the momentum (p) of the alpha particle:
p = m * v = 6.6 x 10^-27 kg * 1.52 x 10^4 m/s ≈ 1.00 x 10^-22 kg.m/s

Finally, we can use the DeBroglie equation λ = h / p to calculate the DeBroglie wavelength:

λ = 6.626 x 10^-34 J.s / 1.00 x 10^-22 kg.m/s ≈ 6.63 x 10^-12 meters

Therefore, the DeBroglie wavelength of the alpha particle is approximately 6.63 x 10^-12 meters.

b. The uncertainty in position (Δx) can be calculated using the uncertainty principle:

Δx * Δp ≥ h / (4π)

where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and h is Planck's constant.

Given:
Δp = m * Δv

We have already calculated p as 1.00 x 10^-22 kg.m/s. Since we are given an uncertainty (± 0.1 x 10^7 mi/h) for the velocity, we need to calculate the uncertainty in momentum (Δp).

Δv = 0.1 x 10^7 mi/h * 1.60934 km/mi * 1000 m/km * 1/3600 h/s ≈ 4.68 x 10^2 m/s

Δp = m * Δv = 6.6 x 10^-27 kg * 4.68 x 10^2 m/s ≈ 3.08 x 10^-24 kg.m/s

Now we can calculate the uncertainty in position (Δx):

Δx * Δp ≥ h / (4π)
Δx * 3.08 x 10^-24 kg.m/s ≥ 6.626 x 10^-34 J.s / (4π)

Solving for Δx:
Δx ≥ (6.626 x 10^-34 J.s / (4π * 3.08 x 10^-24 kg.m/s))

Finally, we can substitute the values and calculate the uncertainty in position:

Δx ≥ 6.77 x 10^-11 meters

Therefore, the uncertainty in position of the alpha particle is approximately 6.77 x 10^-11 meters.