Chapter 14, Problem 59E

Interpretation Introduction

(a)

Interpretation:

The molar hydroxide ion concentration with $\text{pH}=4.\text{55}$ is to be calculated.

Concept introduction:

The $\text{pH}$ scale tells about the strength of an acid and a base. It measures the concentration of the hydrogen ions. There is an inverse relationship between the $\text{pH}$ value and the concentration of the hydrogen ion. Higher the concentration of hydrogen ions, lesser the value of $\text{pH}$. There is an inverse relationship between the $\text{pOH}$ value and the concentration of the hydroxide ion. Higher the concentration of hydroxide ions, lesser the value of $\text{pOH}$.

Answer to Problem 59E

The molar hydroxide ion concentration with $\text{pH}=4.\text{55}$ is $3.6\times {10}^{-10}M$.

Explanation of Solution

The relationship between $\text{pH}$ and hydrogen ion concentration is equal to $10$ raise to the power negative value of $\text{pH}$. It is represented as shown below.

$\left[{\text{H}}^{+}\right]={10}^{-\text{pH}}$ …(1)

The given value of $\text{pH}$ is $4.\text{55}$.

Substitute the value of $\text{pH}$ in equation (1) as shown below.

$\left[{\text{H}}^{+}\right]={10}^{-4.\text{55}}$

Therefore,

$\begin{array}{rll}\left[{\text{H}}^{+}\right]& =& {10}^{-\left(5-0.45\right)}\\ & =& {\text{10}}^{0.45}\times {10}^{-5}M\\ & =& \text{2}\text{.8}\times {10}^{-5}M\end{array}$

The relation between hydrogen ion concentration and hydroxide ion concentration is shown below.

$\left[{\text{H}}^{+}\right]\left[{\text{OH}}^{-}\right]={10}^{-14}$ …(2)

Substitute the value of $\left[{\text{H}}^{+}\right]$ in equation (2) as shown below.

$\begin{array}{l}\left[{\text{H}}^{+}\right]\left[{\text{OH}}^{-}\right]={10}^{-14}\\ \left(\text{2}\text{.8}\times {10}^{-5}M\right)\left[{\text{OH}}^{-}\right]={10}^{-14}\end{array}$

Therefore,

$\begin{array}{rll}\left[{\text{OH}}^{-}\right]& =& \frac{{10}^{-14}}{\text{2}\text{.8}\times {10}^{-5}M}\\ & =& 3.6\times {10}^{-10}M\end{array}$

Therefore, the calculated molar hydroxide ion concentration with $\text{pH}=4.\text{55}$ is calculated as $3.6\times {10}^{-10}M$.

Conclusion

The calculated molar hydroxide ion concentration with $\text{pH}=4.\text{55}$ is calculated as $3.6\times {10}^{-10}M$.

Interpretation Introduction

(b)

Interpretation:

The molar hydroxide ion concentration with $\text{pH}=\text{5}\text{.20}$ is to be calculated.

Concept introduction:

The $\text{pH}$ scale tells about the strength of an acid and a base. It measures the concentration of the hydrogen ions. There is an inverse relationship between the $\text{pH}$ value and the concentration of the hydrogen ion. Higher the concentration of hydrogen ions, lesser the value of $\text{pH}$. There is an inverse relationship between the $\text{pOH}$ value and the concentration of the hydroxide ion. Higher the concentration of hydroxide ions, lesser the value of $\text{pOH}$.

Answer to Problem 59E

The molar hydrogen ion concentration with $\text{pH}=\text{5}\text{.20}$ is $1.6\times {10}^{-9}M$.

Explanation of Solution

The relationship between $\text{pH}$ and hydrogen ion concentration is equal to $10$ raise to the power negative value of $\text{pH}$. It is represented as shown below.

$\left[{\text{H}}^{+}\right]={10}^{-\text{pH}}$ …(1)

The value of $\text{pH}$ is $\text{5}\text{.20}$.

Substitute the value of $\text{pH}$ in equation (1) as shown below.

$\begin{array}{rll}\left[{\text{H}}^{+}\right]& =& {10}^{-\text{5}\text{.20}}\\ & =& {10}^{-\left(6.0-0.8\right)}M\end{array}$

Therefore,

$\begin{array}{rll}\left[{\text{H}}^{+}\right]& =& {10}^{-\left(6.0-0.80\right)}\\ & =& {\text{10}}^{0.8}\times {10}^{-6}M\\ & =& \text{6}\text{.3}\times {10}^{-6}M\end{array}$

The relation between hydrogen ion concentration and hydroxide ion concentration is shown below.

$\left[{\text{H}}^{+}\right]\left[{\text{OH}}^{-}\right]={10}^{-14}$ …(2)

Substitute the value of $\left[{\text{H}}^{+}\right]$ in equation (2) as shown below.

$\begin{array}{l}\left[{\text{H}}^{+}\right]\left[{\text{OH}}^{-}\right]={10}^{-14}\\ \left(\text{6}\text{.3}\times {10}^{-6}M\right)\left[{\text{OH}}^{-}\right]={10}^{-14}\end{array}$

Therefore,

$\begin{array}{rll}\left[{\text{OH}}^{-}\right]& =& \frac{{10}^{-14}}{\text{6}\text{.3}\times {10}^{-6}M}\\ & =& 1.6\times {10}^{-9}M\end{array}$

Therefore, the calculated molar hydroxide ion concentration with $\text{pH}=\text{5}\text{.20}$ is calculated as $1.6\times {10}^{-9}M$.

Conclusion

The molar hydroxide ion concentration with $\text{pH}=\text{5}\text{.20}$ is calculated as $1.6\times {10}^{-9}M$.