Animated Solution for Chemistry - Ionic Equilibrium: Hydrogen ion concentration in mol/L in a solution of $\text{pH}=5.4$ will be

Imagine you are working in a chemistry lab, and you dip a digital pH meter into a beaker containing an unknown solution. The screen lights up and displays a pH of exactly 5.4. Our mission is to translate this simple number into the actual physical reality of the solution: exactly how many hydrogen ions ($H^{+}$) are swimming around in one liter of this liquid?

To bridge the gap between the pH reading and the concentration, we must rely on the fundamental definition of pH. The pH is defined as the negative base-10 logarithm of the hydrogen ion concentration:

Since our goal is to find the concentration $[H^{+}]$, we need to rearrange our master equation. By taking the antilogarithm on both sides, we can isolate the hydrogen ion concentration. The equation elegantly transforms into:

Now, we substitute our given pH value of 5.4 into the equation:

Here lies the true challenge of the problem. How do we evaluate a negative decimal exponent without reaching for a calculator?

To solve $10^{-5.4}$, we employ a brilliant mathematical trick. We split the negative decimal exponent into two parts: a negative integer and a positive decimal.

Notice that $-5.4$ is mathematically identical to $-6+0.6$. By applying the fundamental laws of exponents ($x^{a+b}=x^a\cdot x^b$), we can rewrite our expression:

Now, the problem is much more manageable. The term $10^{-6}$ is already in standard scientific notation. We just need to evaluate $10^{0.6}$, which is simply the antilog of 0.6.

If you recall your standard logarithm values, you know that ${\log }_{10}2\approx 0.301$. Using log properties, ${\log }_{10}4=2{\log }_{10}2\approx 0.602$.

Since 0.6 is incredibly close to 0.602, we can confidently state that:

Finally, we substitute this value back into our split expression. Our hydrogen ion concentration becomes:

This perfectly matches option (a). This splitting technique is a powerful tool in your mathematical arsenal, allowing you to conquer any decimal pH problem with ease!