Question

Wavenumber for a radiation having 5800 Å wavelength is $x \times 10^4$ cm$^{-1}$. The value of $x$ is _______.

Answer 1

1.72

Answer 2

The wavenumber ($\tilde{\nu}$) is the reciprocal of the wavelength ($\lambda$). The formula is: $\tilde{\nu} = \frac{1}{\lambda}$ The given wavelength is $\lambda = 5800 \, \text{\AA}$.

We need to convert the wavelength to centimeters, as the wavenumber is given in cm$^{-1}$. The conversion factor between Ångström (Å) and centimeter (cm) is: $1 \, \text{\AA} = 10^{-10} \, \text{m}$ $1 \, \text{cm} = 10^{-2} \, \text{m}$ So, $1 \, \text{\AA} = \frac{10^{-10}}{10^{-2}} \, \text{cm} = 10^{-8} \, \text{cm}$.

Now, convert the given wavelength to centimeters: $\lambda = 5800 \, \text{\AA} = 5800 \times 10^{-8} \, \text{cm}$ $\lambda = 5.8 \times 10^3 \times 10^{-8} \, \text{cm} = 5.8 \times 10^{-5} \, \text{cm}$.

Now, calculate the wavenumber in cm$^{-1}$: $\tilde{\nu} = \frac{1}{\lambda} = \frac{1}{5.8 \times 10^{-5} \, \text{cm}} = \frac{10^5}{5.8} \, \text{cm}^{-1}$ To calculate the value, we can write $\frac{10^5}{5.8}$ as $\frac{100000}{5.8} = \frac{1000000}{58}$. Performing the division: $1000000 \div 58 \approx 17241.3793...$ So, $\tilde{\nu} \approx 17241.3793 \, \text{cm}^{-1}$.

The problem states that the wavenumber is $x \times 10^4 \, \text{cm}^{-1}$. We have $\tilde{\nu} \approx 17241.3793 \, \text{cm}^{-1}$. We need to find the value of $x$ such that $x \times 10^4 = 17241.3793$. $x = \frac{17241.3793}{10^4} = 1.72413793...$ Rounding to two decimal places gives $1.72$.

Therefore, the value of $x$ is approximately 1.72.