When light with a wavelength of 400 nm falls on the surface of sodium , electrons with a kinetic energy of ` 1.05 xx 10^(5) " J mol"^(-1)` are emitted .
What is the maximum wavelength of light that will cause a photoelectron to be emitted ?

Answer

To solve the problem step by step, we need to find the maximum wavelength of light that can cause a photoelectron to be emitted from sodium. We will use the principles of the photoelectric effect and the relationship between energy, wavelength, and kinetic energy.

### Step 1: Convert the given wavelength to meters
The wavelength of the incident light is given as 400 nm. We need to convert this to meters:
\[ \text{Wavelength} (\lambda) = 400 \, \text{nm} = 400 \times 10^{-9} \, \text{m} \]

**Hint:** Remember that 1 nm = \(10^{-9}\) m.

### Step 2: Calculate the energy of the incident light
The energy of a photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \(h\) (Planck's constant) = \(6.626 \times 10^{-34} \, \text{J s}\)
- \(c\) (speed of light) = \(3 \times 10^{8} \, \text{m/s}\)

Substituting the values:
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J s}) \times (3 \times 10^{8} \, \text{m/s})}{400 \times 10^{-9} \, \text{m}} \]

**Hint:** Make sure to keep track of units while performing calculations.

### Step 3: Calculate the energy value
Calculating the energy:
\[ E = \frac{(6.626 \times 3) \times 10^{-34 + 8}}{400 \times 10^{-9}} = \frac{19.878 \times 10^{-26}}{400 \times 10^{-9}} = 4.9695 \times 10^{-19} \, \text{J} \]

**Hint:** Simplify the exponent carefully during division.

### Step 4: Use the kinetic energy to find the work function
The kinetic energy of the emitted electrons is given as \(1.05 \times 10^{5} \, \text{J/mol}\). To convert this to energy per photon, we divide by Avogadro's number (\(N_A = 6.022 \times 10^{23} \, \text{mol}^{-1}\)):
\[ \text{Kinetic Energy (per photon)} = \frac{1.05 \times 10^{5}}{6.022 \times 10^{23}} \approx 1.74 \times 10^{-19} \, \text{J} \]

**Hint:** Remember that energy per mole must be converted to energy per single photon.

### Step 5: Calculate the work function
According to the photoelectric effect:
\[ E = \phi + KE \]
where \(\phi\) is the work function (minimum energy required to remove an electron). Rearranging gives:
\[ \phi = E - KE \]
Substituting the values:
\[ \phi = 4.9695 \times 10^{-19} \, \text{J} - 1.74 \times 10^{-19} \, \text{J} = 3.2295 \times 10^{-19} \, \text{J} \]

**Hint:** Ensure that you are subtracting the kinetic energy from the total energy correctly.

### Step 6: Calculate the maximum wavelength
The maximum wavelength \(\lambda_{max}\) that can cause photoelectron emission can be found using:
\[ \lambda_{max} = \frac{hc}{\phi} \]
Substituting the values:
\[ \lambda_{max} = \frac{(6.626 \times 10^{-34} \, \text{J s}) \times (3 \times 10^{8} \, \text{m/s})}{3.2295 \times 10^{-19} \, \text{J}} \]

**Hint:** You are calculating the wavelength corresponding to the work function, so ensure you are using the correct energy value.

### Step 7: Calculate the maximum wavelength value
Calculating:
\[ \lambda_{max} = \frac{1.9878 \times 10^{-25}}{3.2295 \times 10^{-19}} \approx 6.15 \times 10^{-7} \, \text{m} = 615 \, \text{nm} \]

**Hint:** Convert meters to nanometers by multiplying by \(10^{9}\).

### Final Answer
The maximum wavelength of light that will cause a photoelectron to be emitted from sodium is approximately **615 nm**.