The work functions `(W_(0))` of K, Na, Li, Mg and Cu are 2.25,2.30,2.42,3.70 and 4.80 eV respectively. How many of these metals do not undergo photoelectric effect when a radiation of wavelength 450 nm is allowed to fall on them? `(1eV=1.602xx10^(-19)J)`

Options

A. 2
B. 1
C. 3
D. 5

Explanation

To determine how many of the given metals do not undergo the photoelectric effect when exposed to radiation of wavelength 450 nm, we will follow these steps:

### Step 1: Calculate the Energy of the Incident Radiation
The energy of the incident radiation can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h = 6.626 \times 10^{-34} \, \text{J s} \) (Planck's constant)
- \( c = 3.0 \times 10^{8} \, \text{m/s} \) (speed of light)
- \( \lambda = 450 \, \text{nm} = 450 \times 10^{-9} \, \text{m} \)

Substituting the values:
\[ E = \frac{(6.626 \times 10^{-34}) \times (3.0 \times 10^{8})}{450 \times 10^{-9}} \]
Calculating this gives:
\[ E \approx 4.417 \times 10^{-19} \, \text{J} \]

### Step 2: Convert Energy from Joules to Electron Volts
To convert the energy from Joules to electron volts, we use the conversion factor \( 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \):
\[ E (\text{eV}) = \frac{4.417 \times 10^{-19} \, \text{J}}{1.602 \times 10^{-19} \, \text{J/eV}} \approx 2.75 \, \text{eV} \]

### Step 3: Compare the Energy with Work Functions
Now we will compare the calculated energy of the incident radiation (2.75 eV) with the work functions of the metals:
- K: 2.25 eV
- Na: 2.30 eV
- Li: 2.42 eV
- Mg: 3.70 eV
- Cu: 4.80 eV

### Step 4: Determine Which Metals Do Not Undergo the Photoelectric Effect
The photoelectric effect will occur if the energy of the incident radiation is greater than or equal to the work function of the metal. Thus, we check:
- K: \( 2.75 \, \text{eV} > 2.25 \, \text{eV} \) (photoelectric effect occurs)
- Na: \( 2.75 \, \text{eV} > 2.30 \, \text{eV} \) (photoelectric effect occurs)
- Li: \( 2.75 \, \text{eV} > 2.42 \, \text{eV} \) (photoelectric effect occurs)
- Mg: \( 2.75 \, \text{eV} < 3.70 \, \text{eV} \) (no photoelectric effect)
- Cu: \( 2.75 \, \text{eV} < 4.80 \, \text{eV} \) (no photoelectric effect)

### Step 5: Count the Metals That Do Not Undergo the Photoelectric Effect
From the comparison, we find that magnesium (Mg) and copper (Cu) do not undergo the photoelectric effect.

### Final Answer
Thus, the total number of metals that do not undergo the photoelectric effect is **2**.

---