If the value of Planck's constant were to increase by a factor of two, the size of a hydrogen atom would

Options

A. Decrease by a factor of 4
B. Decrease by a factor of 2
C. Increase by a factor of 2
D. Increase by a factor of 4

Answer

To solve the problem of how the size of a hydrogen atom changes if Planck's constant increases by a factor of two, we can follow these steps:

### Step 1: Understand the relationship between the size of the hydrogen atom and Planck's constant.
The size of a hydrogen atom can be related to its radius, which can be derived from the balance of forces in the atom and the quantization of angular momentum.

### Step 2: Write down the relevant equations.
1. **Force Balance Equation**: The centripetal force is balanced by the electrostatic force:
\[ \frac{mv^2}{r} = \frac{k e^2}{r^2} \]
where \( m \) is the mass of the electron, \( v \) is its velocity, \( r \) is the radius of the atom, \( k \) is Coulomb's constant, and \( e \) is the charge of the electron.

2. **Quantization of Angular Momentum**:
\[ mvr = \frac{nh}{2\pi} \]
where \( n \) is the principal quantum number and \( h \) is Planck's constant.

### Step 3: Combine the equations to derive the radius.
By manipulating these equations, we can express the radius \( r \) in terms of Planck's constant \( h \):
\[ r \propto \frac{h^2}{m k e^2} \]
This shows that the radius of the hydrogen atom is proportional to the square of Planck's constant.

### Step 4: Analyze the effect of increasing Planck's constant.
If Planck's constant \( h \) is increased by a factor of 2, we can denote the new value as \( h_2 = 2h_1 \).

### Step 5: Substitute the new value into the radius formula.
Substituting \( h_2 \) into the proportionality relation:
\[ r_2 \propto (h_2)^2 = (2h_1)^2 = 4h_1^2 \]
Thus, we can express the new radius as:
\[ r_2 = 4 \cdot r_1 \]

### Step 6: Conclusion.
The size of the hydrogen atom, which is represented by its radius, increases by a factor of 4 when Planck's constant is increased by a factor of 2.

### Final Answer:
The size of the hydrogen atom increases by a factor of 4.

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