Animated Solution for Chemistry - Atomic Structure: A 50 watt bulb emits monochromatic red light of wavelength of $795\text{nm}$. The number of photons emitted per second by the bulb is $x\times 10^{20}$. The value of $x$ is ......... . [Given, $h=6.63\times 10^{-34}\text{Js}$ and $c=3.0\times 10^8{\text{ms}}^{-1}$]

Imagine you are standing in a dark room, and suddenly, a 50-watt red light bulb flickers to life. What does 50 watts actually mean? In the realm of physics, power is simply the rate of energy flow. A 50 W bulb is a machine that pumps out 50 Joules of energy every single second.

But here is where the magic of quantum mechanics comes in. This energy doesn't flow like a continuous river. Instead, it rains down in tiny, indivisible packets called photons.

Planck's quantum theory gives us the master key to unlock this problem. The energy of a single photon is inversely proportional to its wavelength, given by the famous equation:

If our bulb emits $n$ photons every second, the total energy emitted per second must be the sum of the energies of all these individual photons. Therefore, we can write our master equation as:

The calculation phase is where many students panic, but let's take a breath and tackle it systematically. We know the total energy $E=50\text{ J}$, Planck's constant $h=6.63\times 10^{-34}\text{ Js}$, the speed of light $c=3\times 10^8\text{ m/s}$, and the wavelength $\lambda =795\times 10^{-9}\text{ m}$.

Substituting these values into our master equation, we get:

Our goal is to isolate $n$. Let's rearrange the terms:

To make this manageable, we must separate the powers of 10 from the numerical coefficients. Let's look at the exponents. In the numerator, we have $10^{-9}$. In the denominator, we have $10^{-34}\times 10^8=10^{-26}$. Bringing the denominator's power up to the numerator gives us $10^{-9-(-26)}=10^{17}$.

Now, our equation looks much cleaner:

Let's multiply the numbers. The numerator becomes $50\times 795=39750$. The denominator becomes $6.63\times 3=19.89$.

When we perform the division, we find:

To match the format requested in the question, we need to convert this into scientific notation. By shifting the decimal point three places to the left, we increase the exponent by 3:

Since the question asks for the nearest integer value for the coefficient, we can safely round $1.998$ to $2$.

Comparing this with the given expression $x\times 10^{20}$, we can clearly see that:

$$x=2$$

This means our 50-watt bulb is firing off a staggering 200 quintillion photons every single second!