Q21.148CP - Chemistry: Molecular Nature Of Matter And Change

Question

Even though the toxicity of cadmium has become a concern, nickel-cadmium (nicad) batteries are still used commonly in many devices. The overall cell reaction is:

$\mathbf{Cd}\mathbf{\left(}\mathbf{s}\mathbf{\right)}\mathbf{+}\mathbf{2}\mathbf{NiO}\mathbf{\left(}\mathbf{OH}\mathbf{\right)}\mathbf{\left(}\mathbf{s}\mathbf{\right)}\mathbf{+}\mathbf{2}{\mathbf{H}}_{\mathbf{2}}\mathbf{O}\mathbf{\left(}\mathbf{l}\mathbf{\right)}\mathbf{\to }\mathbf{2}\mathbf{Ni}\mathbf{\left(}\mathbf{OH}\mathbf{\right)}\mathbf{\left(}\mathbf{s}\mathbf{\right)}\mathbf{+}\mathbf{Cd}\mathbf{(}\mathbf{OH}{\mathbf{)}}_{\mathbf{2}}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$

A certain nicad battery weighs  $\mathbf{18}\mathbf{.}\mathbf{3}$g and has a capacity of $\mathbf{300}$mAh (that is, the cell can store charge equivalent to a current of  $\mathbf{300}$mA flowing for 1 h). (a) What is the capacity of this cell in coulombs? (b) What mass of reactants is needed to deliver  $\mathbf{300}$mAh? (c) What percentage of the cell mass consists of reactants?

Step-by-Step Solution

Answer

(a) The capacity of the cell in Columbus is obtained as: $1,080\mathrm{C}$.

(b) The mass of reactants required are obtained as: 

$\begin{array}{c}\mathrm{Cd} = 0.629\mathrm{g}\\ \mathrm{Ni}\left(\mathrm{OH}\right) = 1.028\mathrm{g}\\ {\mathrm{H}}_2\mathrm{O}=0.202 \mathrm{g}\\ \mathrm{Total}=1.859 \mathrm{g}\end{array}$

(c) The percentage of cell mass having the reactants is obtained as: $10.2\%$.

Step 1: Define chemical reaction

Chemical synthesis or, alternatively, chemical breakdown into two or more separate chemicals occurs when one component interacts with another to generate a new material. These processes are known as chemical reactions, and they are generally irreversible until followed by other chemical reactions.

Step 2: Evaluating the capacity

The reaction is:

$\mathrm{Cd}\left(\mathrm{s}\right)+2\mathrm{NiO}\left(\mathrm{OH}\right)\left(\mathrm{s}\right)+2{\mathrm{H}}_2\mathrm{O}\left(\mathrm{l}\right)\to 2\mathrm{Ni}\left(\mathrm{OH}\right)\left(\mathrm{s}\right)+\mathrm{Cd}(\mathrm{OH}{)}_2\left(\mathrm{s}\right)$

The $300$mAh capacity of the 18.3gcell implies it can operate at $300$mA for an hour.

To calculate the cell's capacity, multiply the duration by the current in mA.

$1\mathrm{hr}\times \frac{60\min }{1\mathrm{hr}}\times \frac{60\mathrm{s}}{1\min }\times \frac{0.300\mathrm{C}}{1\mathrm{s}}=1,080\mathrm{C}$

Therefore, the capacity is: $1,080\mathrm{C}$.

Step 3: Evaluating the mass of reactants required

The half responses are as follows:

$\begin{array}{c}\mathrm{Oxidation}:\mathrm{Cd}\to {\mathrm{Cd}}^{2+}+2{\mathrm{e}}^{-}\\ \mathrm{Reduction}:2\mathrm{Ni}\left(\mathrm{OH}\right)+2{\mathrm{H}}_2\mathrm{O}+2{\mathrm{e}}^{-}\to \mathrm{Ni}\left(\mathrm{OH}\right)+2\mathrm{O}{\mathrm{H}}^{-}\end{array}$

Two electrons are involved in each half-reaction. To calculate the moles of electrons, divide the charge by Faraday's constant to get the mass of the reactants.

$1,080\mathrm{C}\times \frac{{\mathrm{mole}}^{-}}{96,500\mathrm{C}}=0.0112{\mathrm{mole}}^{-}$

To find the mass of each reagent, convert the moles of electron to moles of reagent and multiply by the molar masses.

The mass Cd is:

$\begin{array}{c}\mathrm{gm} \mathrm{of} \mathrm{Cd}= 0.0112 {\mathrm{mole}}^{-}\times \frac{1 \mathrm{molCd}}{2 {\mathrm{mole}}^{-}}\times \frac{112.4 \mathrm{g}}{1 \mathrm{molCd}}\\ = 0.629\end{array}$

The mass NiO(OH) is:

$\begin{array}{c}\mathrm{gm} \mathrm{of} \mathrm{Ni}\left(\mathrm{OH}\right)= 0.0112{\mathrm{mole}}^{-}\times \frac{1 \mathrm{molNiO}\left(\mathrm{OH}\right)}{1 {\mathrm{mole}}^{-}}\times \frac{91.70 \mathrm{g}}{1 \mathrm{molNiO}\left(\mathrm{OH}\right)}\\ = 1.028\end{array}$

The mass is:

$\begin{array}{c}{\mathrm{gmofH}}_2\mathrm{O}= 0.0112{\mathrm{mol}}^{-}\times \frac{1 {\mathrm{molH}}_2\mathrm{O}}{1{\mathrm{mole}}^{-}}\times \frac{18.01 \mathrm{g}}{1 {\mathrm{molH}}_2\mathrm{O}}\\ = 0.202 \mathrm{g}\end{array}$

$\begin{array}{c}{\mathrm{mass}}_{\mathrm{total}}= {\mathrm{mass}}_{\mathrm{Cd}}+{\mathrm{mass}}_{{\mathrm{H}}_2\mathrm{O}}+{\mathrm{mass}}_{\mathrm{NiO}\left(\mathrm{OH}\right)}\\ = 0.629+1.028+0.202\\ = 1.859 \mathrm{g}\end{array}$

Therefore, the mass is:

$\begin{array}{c}\mathrm{Cd}=0.629\mathrm{g}\\ \mathrm{Ni}\left(\mathrm{OH}\right)=1.028\mathrm{g}\\ {\mathrm{H}}_2\mathrm{O}=0.202 \mathrm{g}\\ \mathrm{Total}=1.859 \mathrm{g}\end{array}$

Step 4: What percentage of the cell mass consists of reactants?

Multiply the mass ratio of the reactants over the total mass by  to obtain the ratio.

$\frac{1.859}{18.3}\times 100 = 10.2 \mathrm{percent}$

Therefore, the cell mass contains: $10.2\%$.