To find the velocity of the electron in a hydrogen atom, we can use the relationship between the centripetal force and the electrostatic force acting on the electron due to the proton. Here’s a step-by-step solution:
### Step 1: Understand the Forces Acting on the Electron
In a hydrogen atom, the electron orbits the proton due to the electrostatic force of attraction between them. This force provides the necessary centripetal force for the electron's circular motion.
### Step 2: Write the Equation for Centripetal Force
The centripetal force (\(F_c\)) required to keep the electron in circular motion is given by:
\[
F_c = \frac{mv^2}{r}
\]
where:
- \(m\) is the mass of the electron,
- \(v\) is the velocity of the electron,
- \(r\) is the radius of the orbit.
### Step 3: Write the Equation for Electrostatic Force
The electrostatic force (\(F_e\)) between the electron and proton is given by Coulomb's law:
\[
F_e = \frac{k \cdot |q_1 \cdot q_2|}{r^2}
\]
where:
- \(k\) is Coulomb's constant (\(8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\)),
- \(q_1\) and \(q_2\) are the charges of the proton and electron (both have a magnitude of \(e = 1.6 \times 10^{-19} \, \text{C}\)),
- \(r\) is the separation distance (radius of the orbit).
### Step 4: Set the Forces Equal
Since the centripetal force is provided by the electrostatic force, we can set them equal to each other:
\[
\frac{mv^2}{r} = \frac{k \cdot e^2}{r^2}
\]
### Step 5: Rearrange the Equation
Rearranging the equation to solve for \(v\):
\[
mv^2 = \frac{k \cdot e^2}{r}
\]
\[
v^2 = \frac{k \cdot e^2}{m \cdot r}
\]
\[
v = \sqrt{\frac{k \cdot e^2}{m \cdot r}}
\]
### Step 6: Substitute the Known Values
Now we can substitute the known values:
- \(k = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\)
- \(e = 1.6 \times 10^{-19} \, \text{C}\)
- \(m = 9.1 \times 10^{-31} \, \text{kg}\)
- \(r = 0.53 \times 10^{-10} \, \text{m}\)
Substituting these values into the equation:
\[
v = \sqrt{\frac{(8.99 \times 10^9) \cdot (1.6 \times 10^{-19})^2}{(9.1 \times 10^{-31}) \cdot (0.53 \times 10^{-10})}}
\]
### Step 7: Calculate the Velocity
Calculating the above expression:
1. Calculate \(e^2\):
\[
e^2 = (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \, \text{C}^2
\]
2. Calculate \(k \cdot e^2\):
\[
k \cdot e^2 = (8.99 \times 10^9) \cdot (2.56 \times 10^{-38}) = 2.303 \times 10^{-28} \, \text{N m}^2
\]
3. Calculate \(m \cdot r\):
\[
m \cdot r = (9.1 \times 10^{-31}) \cdot (0.53 \times 10^{-10}) = 4.823 \times 10^{-40} \, \text{kg m}
\]
4. Finally, calculate \(v\):
\[
v = \sqrt{\frac{2.303 \times 10^{-28}}{4.823 \times 10^{-40}}} \approx 2.2 \times 10^6 \, \text{m/s}
\]
### Final Answer
The velocity of the electron in a hydrogen atom is approximately:
\[
v \approx 2.2 \times 10^6 \, \text{m/s}
\]
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