Chapter 1, Problem 1B.7E

(a)

Interpretation Introduction

Interpretation:

The energy of a sodium atom when it generates a photon of wavelength $589nm$ has to be calculated.

Concept Introduction:

Energy of a photon can be expressed mathematically as given below.

  $E=h\nu =\frac{hc}{\lambda }$

Where, h is the Planck’s constant, $\nu$ is the frequency of radiation, $c$ is the speed of light and $\lambda$ is the wavelength of the radiation.

Answer to Problem 1B.7E

The energy of a sodium atom when it generates a photon of wavelength $589nm$ is $3.37\times 10^{-19}J.$

Explanation of Solution

Given that, the wavelength is $589nm.$  The corresponding energy of a sodium atom when it generates a photon can be calculated as given below.

  $\begin{array}{l}E=h\nu =\frac{hc}{\lambda }\\ \\ E=\frac{\left(6.626\times {10}^{-34}J.s\right)\times \left(3\times {10}^{8}m/s\right)}{589\times {10}^{-9}m}\\ \\ E=3.37\times 10^{-19}J\\ \\ \left[\because 1nm={10}^{-9}m\right]\end{array}$

Therefore, the energy of a sodium atom when it generates a photon of wavelength $589nm$ is $3.37\times 10^{-19}J.$

(b)

Interpretation Introduction

Interpretation:

The energy of $5.00mg$ sodium atoms emitting light at wavelength $589nm$ has to be calculated.

Concept Introduction:

Refer to part (a).

Answer to Problem 1B.7E

The energy of $5.00mg$ sodium atoms emitting light at wavelength $589nm$ is $44.1J.$

Explanation of Solution

Calculation of number of sodium atoms present in $5.00mg$ of sodium atoms:

  $\begin{array}{l}1moleofNa=23.0g\\ \\ 1g=1000mg\\ \\ 1moleofNacontains6.022\times {10}^{23}atoms.\\ \\ So,5.00mg\times \frac{1g}{1000mg}\times \frac{1mole}{23.0g}\times \frac{6.022\times {10}^{23}atoms}{1mole}=1.31\times 10^{20}atoms\end{array}$

The energy of one sodium atom is $3.37\times 10^{-19}J.$  Then the energy of $1.31\times 10^{20}atoms$ of sodium can be calculated as given below.

  $E=\left(1.31\times {10}^{20}\right)\times \left(3.37\times {10}^{-19}J\right)=44.1J$

Therefore, the energy of $5.00mg$ sodium atoms emitting light at wavelength $589nm$ is $44.1J.$

(c)

Interpretation Introduction

Interpretation:

The energy of $1.00mole$ sodium atoms emitting light of wavelength $589nm$ has to be calculated.

Concept Introduction:

Refer to part (a).

Answer to Problem 1B.7E

The energy of $1.00mole$ sodium atoms emitting light of wavelength $589nm$ is $2.03\times 10^{5}J.mo{l}^{-1}.$

Explanation of Solution

The energy of one sodium atom is $3.37\times 10^{-19}J.$

The energy of $1.00mole$ sodium atoms emitting light of wavelength $589nm$ can be calculated as shown below.

  $\begin{array}{l}E\left(permoleofphotons\right)=N_{A}E\left(1photon\right)\\ \\ E\left(permoleofphotons\right)=\left(6.022\times {10}^{23}mo{l}^{-1}\right)\times \left(\text{3}{\text{.37×10}}^{\text{-19}}\text{J}\right)\\ \\ E\left(permoleofphotons\right)=2.03\times 10^{5}J.mo{l}^{-1}\end{array}$

Therefore, the energy of $1.00mole$ sodium atoms emitting light of wavelength $589nm$ is $2.03\times 10^{5}J.mo{l}^{-1}.$