Question

The mass of an electron is $9.1\times {{10}^{-31}}kg$. If its K.E. is $3.0\times {{10}^{-25}}J$. Calculate its wavelength.

Answer 1

Hint: From values of KE and mass of electron, we can find the velocity of electron. The equation of KE is $KE=\dfrac{1}{2}m{{V}^{2}}$ . De Broglie equation is used to relate the velocity, mass and wavelength of a particle. The equation is $\lambda =\dfrac{h}{mv}$ .

Complete answer:
Here, we are given the Kinetic energy of the electron. We are also given the weight of an electron. So, from this, we can find the velocity of the electron. From this velocity, we can find the wavelength by the De-Broglie equation.
We, know that kinetic energy is $KE=\dfrac{1}{2}m{{V}^{2}}$ ……….(1)
Where m is the mass of the electron which is $9.1\times {{10}^{-31}}kg$.
KE is given which is $3.0\times {{10}^{-25}}J$. Putting all these values into equation (1), we will get,
     $3.0\times {{10}^{-25}}=\dfrac{1}{2}\times 9.1\times {{10}^{-31}}\times {{V}^{2}}$
So, we can write that ${{V}^{2}}=\dfrac{6.0\times {{10}^{-25}}}{9.1\times {{10}^{-31}}}=0.6593\times {{10}^{6}}$
Thus, $V=\sqrt{0.6593\times {{10}^{6}}}=0.8119\times {{10}^{3}}m{{s}^{-1}}$
Now, we have the velocity of the electron and mass of the electron. So, from this, we can calculate the wavelength of an electron by the de Broglie equation. The equation is shown as below.
     \[\lambda =\dfrac{h}{mv}\] ………..(2)
Here, $\lambda $ is the wavelength, v is velocity and m is mass of the electron, h is the Planck’s constant and its value is $6.62\times {{10}^{-34}}Js$. We can put all the available values into equation (2). So, we will get
     \[\lambda =\dfrac{6.62\times {{10}^{-34}}}{9.1\times {{10}^{-31}}\times 0.8119\times {{10}^{3}}}=8.9625\times {{10}^{-7}}m\]
Thus, we can say that wavelength of the electron whose KE is $3.0\times {{10}^{-25}}J$, will be $8.9625\times {{10}^{-7}}m$

Note:
Do not forget that we must put the weight of the electron in kg units in all the equations used here. Here, we cannot use the formula $E=m{{c}^{2}}$ as c is the velocity of light in vacuum in this equation and we need the velocity in an electron in which the medium is not vacuum.