The gravitational force between a H-atom and another particle of mass m will be given by Newton's law: `F=G (M.m)/(r^(2)`, where r is in km and

Options

A. `M = M_("proton") + m_("electron")`
B. `M = M_("proton") + m_("electron")- (B)/(C^(2)) (B = 13.6 eV)`
C. M is not relate to the mass of the hydrogen atom
D. `M = M_("proton") + m_("electron")- (|V|)/(c^(2))|V| =` magnitude of the potential energy of electron in the H-atom

Explanation

To solve the problem of calculating the gravitational force between a hydrogen atom and another particle of mass \( m \), we will follow these steps:

### Step 1: Identify the mass of the hydrogen atom
The mass of a hydrogen atom is primarily determined by its nucleus, which consists of one proton, and it also includes the mass of one electron. Therefore, the effective mass \( M \) of the hydrogen atom can be expressed as:

\[ M = m_p + m_e \]

Where:
- \( m_p \) is the mass of the proton (approximately \( 1.67 \times 10^{-27} \) kg)
- \( m_e \) is the mass of the electron (approximately \( 9.11 \times 10^{-31} \) kg)

### Step 2: Calculate the total mass of the hydrogen atom
Now, we can calculate the total mass of the hydrogen atom:

\[ M = 1.67 \times 10^{-27} \text{ kg} + 9.11 \times 10^{-31} \text{ kg} \approx 1.67 \times 10^{-27} \text{ kg} \quad (\text{since } m_p \gg m_e) \]

### Step 3: Consider the binding energy
When a hydrogen atom is formed, some mass is converted into energy, known as binding energy. According to Einstein's mass-energy equivalence principle, this can be expressed as:

\[ E = mc^2 \]

The binding energy \( B \) of a hydrogen atom is approximately \( 13.6 \) eV. To convert this energy into mass, we use:

\[ m = \frac{B}{c^2} \]

Where \( c \) is the speed of light (\( c \approx 3 \times 10^8 \) m/s). First, we need to convert the binding energy from eV to joules:

\[ B = 13.6 \text{ eV} \times 1.6 \times 10^{-19} \text{ J/eV} \approx 2.18 \times 10^{-18} \text{ J} \]

Now we can calculate the equivalent mass:

\[ m = \frac{2.18 \times 10^{-18} \text{ J}}{(3 \times 10^8 \text{ m/s})^2} \approx \frac{2.18 \times 10^{-18}}{9 \times 10^{16}} \approx 2.43 \times 10^{-35} \text{ kg} \]

### Step 4: Adjust the mass of the hydrogen atom
Now we can adjust the mass of the hydrogen atom to account for the binding energy:

\[ M_{\text{actual}} = M - m \]

Substituting the values we have:

\[ M_{\text{actual}} = (1.67 \times 10^{-27} \text{ kg}) - (2.43 \times 10^{-35} \text{ kg}) \approx 1.67 \times 10^{-27} \text{ kg} \quad (\text{since } m \text{ is negligible}) \]

### Step 5: Calculate the gravitational force
Now we can calculate the gravitational force \( F \) between the hydrogen atom and another particle of mass \( m \) using Newton's law of gravitation:

\[ F = \frac{G M m}{r^2} \]

Where \( G \) is the gravitational constant (\( G \approx 6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2 \)) and \( r \) is the distance between the two masses in meters (remember to convert kilometers to meters).

### Final Calculation
Assuming \( r \) is given in kilometers, convert it to meters:

\[ r \text{ (in meters)} = r \text{ (in kilometers)} \times 1000 \]

Substituting the values into the gravitational force equation will give you the final answer.