Question

Wavenumber for a radiation having $5800 \, \mathring{A}$ wavelength is $x \times 10 \, \text{cm}^{-1}$. The value of $x$ is _________

Answer 1

Correct Answer: 1724

We are given the formula for wavenumber:

$$ \bar{v} = \frac{1}{\lambda} \Rightarrow \frac{1}{5800 \times 10^{-8}} \, \text{cm}^{-1} $$

Let's simplify the equation step by step:

$$ \Rightarrow \frac{10^8}{5800} \Rightarrow \frac{10^6}{58} $$

$$ \Rightarrow \frac{100000}{58} \times 10 $$

$$ \Rightarrow 1724.13 \times 10 $$

$$ \Rightarrow 1724 $$

Thus, the wavenumber is 1724 cm^-1.

Answer 2

The wavenumber ($\tilde{\nu}$) is given by:
\[ \tilde{\nu} = \frac{1}{\lambda}, \]
where $\lambda$ is the wavelength in cm.
Step 1: Convert wavelength to cm
\[ \lambda = 5800 \, \text{\AA} = 5800 \times 10^{-8} \, \text{cm}. \]
Step 2: Calculate wavenumber
\[ \tilde{\nu} = \frac{1}{5800 \times 10^{-8}} = \frac{1}{5.8 \times 10^{-5}} = 17241 \, \text{cm}^{-1}. \]
Step 3: Express as $x \times 10 \, \text{cm}^{-1}$
\[ 17241 \, \text{cm}^{-1} = 1724 \times 10 \, \text{cm}^{-1}. \]