The work function for Al, K and Pt is 4.28 eV, 2.30 eV and 5.65 eV respectively. Their respective threshold frequencies would be

Options

A. `PtgtAlgtK`
B. `AlgtPtgtK`
C. `KgtAlgtPt`
D. `AlgtKgtPt`

Explanation

To find the threshold frequencies for Aluminum (Al), Potassium (K), and Platinum (Pt) based on their work functions, we can use the formula that relates the work function (φ) to the threshold frequency (ν₀):

\[ \phi = h \nu_0 \]

Where:
- φ is the work function in electron volts (eV),
- h is Planck's constant (approximately \(4.135667696 \times 10^{-15} \text{ eV s}\)),
- ν₀ is the threshold frequency in hertz (Hz).

### Step 1: Convert Work Function to Threshold Frequency
We can rearrange the formula to solve for the threshold frequency:

\[ \nu_0 = \frac{\phi}{h} \]

### Step 2: Calculate the Threshold Frequencies
Now, we will calculate the threshold frequencies for each metal.

1. **For Potassium (K)**:
- Work function (φ) = 2.30 eV
- Threshold frequency (ν₀) = \(\frac{2.30 \text{ eV}}{4.135667696 \times 10^{-15} \text{ eV s}}\)

\[ \nu_{0(K)} = \frac{2.30}{4.135667696 \times 10^{-15}} \approx 5.56 \times 10^{14} \text{ Hz} \]

2. **For Aluminum (Al)**:
- Work function (φ) = 4.28 eV
- Threshold frequency (ν₀) = \(\frac{4.28 \text{ eV}}{4.135667696 \times 10^{-15} \text{ eV s}}\)

\[ \nu_{0(Al)} = \frac{4.28}{4.135667696 \times 10^{-15}} \approx 1.04 \times 10^{15} \text{ Hz} \]

3. **For Platinum (Pt)**:
- Work function (φ) = 5.65 eV
- Threshold frequency (ν₀) = \(\frac{5.65 \text{ eV}}{4.135667696 \times 10^{-15} \text{ eV s}}\)

\[ \nu_{0(Pt)} = \frac{5.65}{4.135667696 \times 10^{-15}} \approx 1.37 \times 10^{15} \text{ Hz} \]

### Step 3: Order the Threshold Frequencies
Now we can summarize the threshold frequencies:
- ν₀(K) ≈ \(5.56 \times 10^{14} \text{ Hz}\)
- ν₀(Al) ≈ \(1.04 \times 10^{15} \text{ Hz}\)
- ν₀(Pt) ≈ \(1.37 \times 10^{15} \text{ Hz}\)

### Step 4: Establish the Order
The order of threshold frequencies from lowest to highest is:
- K < Al < Pt

### Final Answer
The respective threshold frequencies in increasing order are:
- K < Al < Pt