The ratio between kinetic enegry and total enegry of the electrons of hydrogen atom according to Bohr's model is

Options

A. `2:1`
B. `1:1`
C. `1:-1`
D. `1:2`

Explanation

To find the ratio between the kinetic energy (KE) and total energy (TE) of the electrons in a hydrogen atom according to Bohr's model, we can follow these steps:

### Step 1: Understanding the Energies in Bohr's Model
In Bohr's model of the hydrogen atom, the total energy (TE) of the electron in a particular orbit is given by the formula:
\[ TE = -\frac{Z^2 \cdot k \cdot e^4 \cdot m}{2 \cdot \hbar^2 \cdot n^2} \]
where:
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)),
- \( k \) is Coulomb's constant,
- \( e \) is the charge of the electron,
- \( m \) is the mass of the electron,
- \( \hbar \) is the reduced Planck's constant,
- \( n \) is the principal quantum number (n = 1, 2, 3,...).

### Step 2: Kinetic Energy of the Electron
The kinetic energy (KE) of the electron in the orbit can be expressed as:
\[ KE = \frac{1}{2}mv^2 \]
According to Bohr's model, the centripetal force required for the electron to move in a circular orbit is provided by the electrostatic force of attraction between the positively charged nucleus and the negatively charged electron. Therefore, we can equate the centripetal force to the electrostatic force:
\[ \frac{mv^2}{r} = \frac{k \cdot e^2}{r^2} \]
From this, we can derive the expression for kinetic energy:
\[ KE = \frac{k \cdot e^2}{2r} \]

### Step 3: Relating Kinetic Energy and Total Energy
From the expressions derived, we can find the relationship between KE and TE. In Bohr's model, it is known that:
\[ TE = KE + PE \]
where \( PE \) (potential energy) is given by:
\[ PE = -\frac{k \cdot e^2}{r} \]
Thus, we can express total energy as:
\[ TE = KE - \frac{k \cdot e^2}{r} \]

### Step 4: Finding the Ratio
From the above equations, we can substitute \( PE \) into the total energy equation:
\[ TE = KE - 2KE = -KE \]
This implies:
\[ KE = -\frac{1}{2} TE \]
Now, we can find the ratio of kinetic energy to total energy:
\[ \frac{KE}{TE} = \frac{KE}{-2KE} = -\frac{1}{2} \]

### Final Answer
Thus, the ratio between the kinetic energy and total energy of the electrons of a hydrogen atom according to Bohr's model is:
\[ \frac{KE}{TE} = -\frac{1}{2} \]