The mass of an electron is 9.11*10^-31kg. If the de Broglie wavelength for an electron in a hydrogen atom is 3.31*10^-10m, how fast is the electron moving relative to the speed of light? The speed of light is 3.00*10^8m/s.
Express your answer numerically as a percentage of the speed of light.
How should this question be worked?
I tried: 0.692% already but it was incorrect.
m=9.11*10^-31 kg
lambda=3.31*10^-10 m
c= 3.00*10^8 m/s
Check you math. I get something like 0.732% if I use 6.626 x 10^-34 J*s for h or 0.733 if I use 6.63 x 10^-34.
thank you..I reworked it & the answer came to: 0.73%.
You should carry it to three places since 9.11, 3.00 and 3.31 have three places.
To find the velocity of an electron relative to the speed of light, you can use the de Broglie wavelength formula:
wavelength = h / (mass * velocity),
where
- wavelength is the de Broglie wavelength,
- h is the Planck's constant (approximately 6.626 × 10^-34 J·s),
- mass is the mass of the electron,
- velocity is the velocity of the electron.
Rearranging the formula, you can solve for velocity:
velocity = (h / (mass * wavelength)).
Now, substitute the given values into the equation:
mass = 9.11 × 10^-31 kg,
wavelength = 3.31 × 10^-10 m,
h = 6.626 × 10^-34 J·s.
Plugging these values in, you get:
velocity = (6.626 × 10^-34 J·s) / ((9.11 × 10^-31 kg) * (3.31 × 10^-10 m)).
Calculate the numerator:
(6.626 × 10^-34 J·s) = 6.626 × 10^-34.
Calculate the denominator:
(9.11 × 10^-31 kg) * (3.31 × 10^-10 m) ≈ 3.01 × 10^-40 kg·m.
Now divide the numerator by the denominator:
velocity ≈ (6.626 × 10^-34) / (3.01 × 10^-40)
The result is approximately 2.20 × 10^6 m/s.
To express the velocity as a percentage of the speed of light, divide the velocity by the speed of light (3.00 × 10^8 m/s) and multiply by 100:
percentage = (velocity / (3.00 × 10^8 m/s)) * 100.
Substitute the values to find the percentage:
percentage ≈ (2.20 × 10^6 m/s / (3.00 × 10^8 m/s)) * 100.
Evaluate the expression:
percentage ≈ 0.733%.
So, the electron is moving at approximately 0.733% of the speed of light.