The mass of an electron is 9.11\times 10^{-31}~\rm kg. If the de Broglie wavelength for an electron in a hydrogen atom is 3.31\times 10^{-10}~\rm m, how fast is the electron moving relative to the speed of light? The speed of light is 3.00\times10^8 ~\rm m/s.

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To solve this problem, we need to use the de Broglie wavelength equation and relate it to the speed of the electron.

The de Broglie wavelength (λ) of a particle is given by the equation:

λ = h / p,

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

We can rewrite the equation to solve for momentum:

p = h / λ.

The momentum of an object is defined as the product of its mass (m) and velocity (v):

p = m * v.

Combining these equations, we get:

m * v = h / λ.

We can rearrange this equation to find the velocity of the electron (v):

v = (h / λ) / m.

Now we can substitute the given values into this equation:

v = [(6.626 × 10^(-34) J·s) / (3.31 × 10^(-10) m)] / (9.11 × 10^(-31) kg).

Simplifying this expression:

v = (6.626 × 10^(-34) J·s) / (3.31 × 10^(-10) m) * (1 / 9.11 × 10^(-31) kg).

Canceling and multiplying the appropriate terms:

v = (6.626 / 3.31) * (1 / 9.11) * (10^(-34) J·s / 10^(-10) m / 10^(-31) kg).

Simplifying the exponents:

v = (6.626 / 3.31) * (1 / 9.11) * 10^(34 - 10 + 31) (J·s / m / kg).

Again simplifying:

v = 2 * (1 / 9.11) * 10^(55) (J·s / m / kg).

Now we can substitute the value for the speed of light (c), which is 3.00 × 10^(8) m/s:

v = 2 * (1 / 9.11) * 10^(55) * (3.00 × 10^(8) m/s / 3.00 × 10^(8) m/s).

Simplifying:

v = 2 * (1 / 9.11) * 10^(55) * 1.

Further simplifying:

v = 2 * (1 / 9.11) * 10^(55).

Calculating the value:

v ≈ 2.19 × 10^(54).

Therefore, the electron is moving at approximately 2.19 × 10^(54) times the speed of light.