Answer
The hydrogen ion concentration of the patient’s blood is about $1.58\times {{10}^{-8}}$ moles per liter.
Work Step by Step
To calculate the hydrogen ion concentration of the patient’s blood at this deadly point, substitute $\text{pH}=7.8$ in the formula $\text{pH}=-\log \left[ {{\text{H}}^{\text{+}}} \right]$ and solve for $\left[ {{\text{H}}^{\text{+}}} \right]$ as follows:
$\begin{align}
& \text{pH}=-\log \left[ {{\text{H}}^{\text{+}}} \right] \\
& 7.8=-\log \left[ {{\text{H}}^{\text{+}}} \right] \\
\end{align}$
Divide by $-1$ on both sides:
$\begin{align}
& 7.8\left( -1 \right)=-\log \left[ {{\text{H}}^{\text{+}}} \right]\left( -1 \right) \\
& -7.8=\log \left[ {{\text{H}}^{\text{+}}} \right]
\end{align}$
The common logarithm has a base of 10, so the above equation can be written as :
$-7.8={{\log }_{10}}\left[ {{\text{H}}^{\text{+}}} \right]$
Now convert it into an exponential equation by using the formula ${{\log }_{a}}x=m\rightarrow{{a}^{m}}=x$:
$\begin{align}
& -7.8={{\log }_{10}}\left[ {{\text{H}}^{\text{+}}} \right] \\
& \left[ {{\text{H}}^{\text{+}}} \right]={{10}^{-7.8}} \\
& ={{10}^{-7.8}}
\end{align}$
${{10}^{-7.8}}\approx 1.58\times {{10}^{-8}}$.
