My teacher gave me a quick trick to convert wavelength of radiation into energy of quanta(photon) while studying Planck’s Theory for College entrance prep.
$E$(in eV) $= {1240 \over λ( nm )}$
and
$E$(in eV) $= {12400 \over λ (angstrom) }$
But here’s what made sense to me, and many of my friends:
Since 1 nanometre = 10 angstroms
So,
$E$(in eV) $= {1240 \over λ( nm ) }$ $= {1240 \over λ (10 angstroms) }$ $= {1240 \over 10λ( angstroms ) }$$= {124 \over λ( angstroms ) }$
Which is actually incorrect. Even though I used a conversion similar to
5 meters = 5 (100 centimetres) = 500 centimeters
Which is obviously correct.
Why is this the case? Please give me the shortest possible and most powerful explanation that actually goes into my mind, not an essay. I am baffled by this. Half my mind agrees, while the other disagrees. And please don’t say that the trick is nonsense(it actually works).
Planck’s theory Quick review from NCERT textbook.
Obviously, this issue is not limited to this trick, it could have popped up in many different cases. Give specific and general explanation both, please.
I think writing things like $\lambda(nm), \lambda(\text{angstrom})$ is causing you to misunderstand your own notation. Let's do it properly from the beginning. We have: \begin{align} E &= \dfrac{hc}{\lambda} \end{align} Now, divide both sides by $1\text{eV}$, and plug in the values for $h,c$ and keep track of all units: \begin{align} \dfrac{E}{1\text{eV}} &= \dfrac{hc}{1\text{eV}}\cdot \dfrac{1}{\lambda} = (1.24\times10^{-6} \text{m}) \cdot\dfrac{1}{\lambda} \tag{$*$} \end{align} Now, we can always write \begin{align} \lambda &= \lambda_{m} \cdot (1\text{ m}) = \lambda_{nm} \cdot (1\text{ nm}) = \lambda_{ang} \cdot (1 \, \text{ang}), \tag{$**$} \end{align} for some real numbers $\lambda_m, \lambda_{nm}, \lambda_{ang}$. Now, if we plug $(**)$ into $(*)$ then we find \begin{align} \dfrac{E}{1\text{eV}} &= \dfrac{1.24 \times 10^{-6} \text{m}}{\lambda_m \cdot (1\text{m})} = \dfrac{1.24 \times 10^{-6} \text{m}}{\lambda_{nm} \cdot (1\text{nm})} = \dfrac{1.24 \times 10^{-6} \text{m}}{\lambda_{ang} \cdot (1\text{ang})} \\\\ &= \dfrac{1.24 \times 10^{-6}}{\lambda_m}\,\,\,\, = \dfrac{1.24 \times 10^{3}}{\lambda_{nm}} \, \, \, \, \,\,\,\,= \dfrac{1.24 \times 10^{4}}{\lambda_{ang}} \end{align}
Here is where I think you were making the big mistake: it is true that $1 \text{nm} = 10 \text{ ang}$, however, we have $\lambda_{nm} = 10^{-1} \lambda_{ang}$ (if your units get scaled in one way, then the coefficients have to get scaled in the opposite way in order to keep the product same). This is what I meant above when I said you misunderstood your own notation; you were probably thinking of your notation $\lambda(nm)$ as being a number $\lambda$ multiplied by $1\text{ nm}$, when it is really $\lambda_{nm} = \dfrac{\lambda}{1 \text{ nm}}$.
A remark about terminology:
The general rule is that if you have a physical quantity like energy $E$, and you say "energy in Joules", then what you mean is $\dfrac{E}{1\text{J}}$. If you say "energy in micro electron-volt" what you mean is $\dfrac{E}{1 \mu\text{eV}}$. So, for example, whenever you wrote $\lambda(angstrom)$ in your post, what you should have written is $\lambda = \frac{\lambda}{1 \text{ ang}} \cdot (1 \text{ ang})$, and this is exactly what I called $\lambda_{ang} \cdot (1 \text{ ang})$ in my answer above.