Chlorophyll present in green leaves of plants absorbs light at 4.620 × 10¹4 Hz. Calculate the wavelength of ra

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Q. Chlorophyll present in green leaves of plants absorbs light at 4.620 \times 10^14 Hz. Calculate the wavelength of radiation in nanometer. Which part of the electromagnetic spectrum does it belong to?

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Concept understanding — Electromagnetic Wave Wavelength

What is Wavelength? The Intuition First

Imagine dropping a pebble into a still pond. Ripples spread outward — evenly spaced circles. The distance between two consecutive crests (the highest points) is what we call wavelength. It's the "repeat length" of the wave.

Now, electromagnetic waves are not water waves. They don't need a medium. But the same idea holds: an EM wave is a travelling disturbance in electric and magnetic fields. As it moves, the fields oscillate — they go up, down, up, down. The wavelength ( $\lambda$ , Greek letter lambda) is the distance over which the wave's shape repeats.

Note

In a vacuum, all EM waves travel at the same speed: $c = 3 \times 10^8 \ \text{m/s}$ . What changes from one wave to another is the wavelength (and its partner, frequency).

The Precise Definition

For a sinusoidal electromagnetic wave travelling in one direction, the electric field at a fixed instant of time looks like a sine curve in space. The wavelength $\lambda$ is the spatial distance between two successive points that are in phase — for example, from one crest to the next crest, or from one trough to the next trough.

Mathematically, if the electric field at position $x$ and time $t$ is given by

E(x,t) = E_0 \sin(kx - \omega t)

then the wave number $k$ is related to wavelength by

k = \frac{2\pi}{\lambda}

So $\lambda$ is the distance needed for the argument $kx$ to change by $2\pi$ — one full cycle of the sine wave.

The Fundamental Relationship

The wavelength, frequency $f$ , and speed $c$ are tied together by a simple equation:

c = f \lambda

This is the wave equation for EM waves in vacuum. It means:

If the wavelength is long, the frequency is low.
If the wavelength is short, the frequency is high.
The product is always $c$ , a constant.

Tip

To remember: think of a marching band. If soldiers take long strides (large $\lambda$ ), they take fewer steps per second (low $f$ ). If they take short, quick steps (small $\lambda$ ), they take many steps per second (high $f$ ). The speed of the band is fixed — stride length × steps per second.

The Electromagnetic Spectrum

Wavelength is what separates different kinds of EM radiation. Here's the spectrum from longest to shortest wavelength:

Type of EM Wave	Approximate Wavelength Range
Radio waves	$> 0.1 \ \text{m}$ (up to km)
Microwaves	$1 \ \text{mm}$ to $0.1 \ \text{m}$
Infrared	$700 \ \text{nm}$ to $1 \ \text{mm}$
Visible light	$400 \ \text{nm}$ to $700 \ \text{nm}$
Ultraviolet	$10 \ \text{nm}$ to $400 \ \text{nm}$
X-rays	$0.01 \ \text{nm}$ to $10 \ \text{nm}$
Gamma rays	$< 0.01 \ \text{nm}$

Watch out

A common mistake: thinking that wavelength is the "size" of the wave. It's not. It's the repeat distance. A radio wave can have a wavelength of 1 km, but its amplitude (the strength of the field) might be tiny. Wavelength and amplitude are independent properties.

Why Wavelength Matters …

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The frequency of light absorbed by chlorophyll is $4.620 \times 10^{14}\ \text{Hz}$ . Using $c = \lambda \nu$ , the wavelength is $649.4\ \text{nm}$ , placing it in the red region of the visible spectrum.

The key idea here is the direct relationship between frequency and wavelength for any electromagnetic wave: they are inversely proportional, linked by the constant speed of light. Chlorophyll absorbs light to drive photosynthesis, and the specific frequency given tells us exactly which colour of light it captures — and that colour is red.

Let’s work through it step by step.

Recall the fundamental wave equation. For any electromagnetic wave in vacuum (or air, to excellent approximation), the speed $c$ is constant at $3.00 \times 10^8\ \text{m/s}$ . The relationship is:

c = \lambda \nu

where $\lambda$ is the wavelength (in metres) and $\nu$ is the frequency (in hertz, or $\text{s}^{-1}$ ). We want $\lambda$ , so rearrange:

\lambda = \frac{c}{\nu}

Plug in the numbers. The frequency is $\nu = 4.620 \times 10^{14}\ \text{Hz}$ . So:

\lambda = \frac{3.00 \times 10^8\ \text{m/s}}{4.620 \times 10^{14}\ \text{s}^{-1}}

Do the division. First handle the powers of ten:

\frac{10^8}{10^{14}} = 10^{-6}

Then the coefficient:

\frac{3.00}{4.620} \approx 0.6494

So:

\lambda \approx 0.6494 \times 10^{-6}\ \text{m} = 6.494 \times 10^{-7}\ \text{m}

Convert to nanometres. Since $1\ \text{nm} = 10^{-9}\ \text{m}$ , multiply by $10^9$ : $\lambda = 6.494 \times 10^{-7}\ \text{m} \times \frac{10^9\ \text{nm}}{1\ \text{m}} = 649.4\ \text{nm}$ …

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Q.Chlorophyll present in green leaves of plants absorbs light at 4.620 × 10^14 Hz. Calculate the wavelength of radiation in nanometer. Which part of the electromagnetic spectrum does it belong to?

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