Animated Solution for Chemistry - Atomic Structure: The work function of sodium metal is $4.41\times 10^{-19}$ J. If photons of wavelength $300$ nm are incident on the metal, the kinetic energy of the ejected electrons will be $(h=6.63\times 10^{-34}\text{ J s};c=3\times 10^8\text{ m/s})$ ............ $\times 10^{-21}$ J.

Imagine you are trapped inside a deep well, and the only way out is to pay a toll. If someone throws you a bag of coins, you can use some of those coins to pay the toll, and whatever you have left over is yours to keep as you run away. This is exactly what happens in the Photoelectric Effect.

In our problem, the 'well' is the sodium metal, and the 'toll' is the Work Function ($\Phi$). The work function is the minimum energy required to just barely pull an electron out of the metal's surface. Here, $\Phi =4.41\times 10^{-19}$ J.

The 'bag of coins' is the incoming photon of light. The energy of this photon depends on its wavelength. Albert Einstein beautifully summarized this transaction with his famous photoelectric equation:

This means the total energy of the photon ($E$) is used to pay the toll ($\Phi$), and the remaining energy becomes the maximum kinetic energy ($K{E}_{max}$) of the escaping electron.

Before we can find the kinetic energy, we need to know exactly how much energy our incoming photon carries. We are given its wavelength, $\lambda =300$ nm. We can calculate its energy using the formula:

Here, $h$ is Planck's constant ($6.63\times 10^{-34}$ J s) and $c$ is the speed of light ($3\times 10^8$ m/s). A crucial step here is to convert the wavelength from nanometers to meters to keep our units consistent:

Now, let's substitute these values into our energy equation:

Let's simplify the numerator first:

Dividing the numbers and subtracting the exponents gives us:

To make this easier to work with, let's adjust the decimal point to match the power of ten in our work function:

Now that we know the total energy provided by the photon ($6.63\times 10^{-19}$ J) and the toll required to leave the metal ($4.41\times 10^{-19}$ J), we can easily find the kinetic energy of the ejected electron by rearranging Einstein's equation:

Substituting our values:

Since the powers of ten are identical, we can simply subtract the coefficients:

We have the correct kinetic energy, but the question has a specific trap. It asks for the answer in the format of $...\times 10^{-21}$ J.

To convert $2.22\times 10^{-19}$ to a number multiplied by $10^{-21}$, we need to shift the decimal point two places to the right. Mathematically, we are multiplying by $100$ and dividing by $100$:

Therefore, the integer value required to fill in the blank is 222.