Calculating Electron Wavelength from Kinetic Energy
This solution outlines the steps to calculate the de Broglie wavelength ($\lambda$) of an electron using its given mass ($m$) and kinetic energy ($K.E.$).
Step 1: Determine Electron Velocity ($v$)
The relationship between kinetic energy and velocity is $K.E. = \frac{1}{2}mv^2$. We rearrange this to find the velocity:
$ v = \sqrt{\frac{2 \times K.E.}{m}} $
Using the provided values $m = 9.1 \times 10^{-31}\text{ kg}$ and $K.E. = 3.0 \times 10^{-25}\text{ J}$, we calculate $v$.
Step 2: Compute de Broglie Wavelength ($\lambda$)
The de Broglie wavelength formula is $\lambda = \frac{h}{p}$, where momentum $p = mv$ and Planck's constant $h \approx 6.626 \times 10^{-34}\text{ J s}$. Combining these gives:
$ \lambda = \frac{h}{m \sqrt{\frac{2 \times K.E.}{m}}} $
Substituting the known values into this combined formula allows for the calculation of the wavelength in meters, which is then converted to nanometers.
Final Answer
The calculation yields the electron's wavelength. The correct option is identified as B.
The correct option is B.
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