The pH scale is used to measure the acidity or alkalinity of a solution. It is defined as the negative logarithm (base 10) of the hydrogen ion concentration (\([H^+]\)). The formula is:

\[ \text{pH} = -\log_{10}[H^+] \]

This means that the hydrogen ion concentration can be calculated from the pH using the formula:

\[ [H^+] = 10^{-\text{pH}} \]

We are given the pH values for endocytic vesicles and gastric juice. We need to find the hydrogen ion concentration for each and then compare them.

1. Hydrogen Ion Concentration in Endocytic Vesicles:

The pH of endocytic vesicles is 5.2.

Using the formula \([H^+] = 10^{-\text{pH}}\):

\[ [H^+]_{\text{vesicle}} = 10^{-5.2} \]

2. Hydrogen Ion Concentration in Gastric Juice:

The pH of gastric juice is 2.0.

Using the formula \([H^+] = 10^{-\text{pH}}\):

\[ [H^+]_{\text{gastric}} = 10^{-2.0} \]

3. Comparing the Concentrations:

We want to determine how many times lower the \([H^+]\) of the endocytic vesicle is compared to that of gastric juice. This means we need to find the ratio of the gastric juice \([H^+]\) to the endocytic vesicle \([H^+]\).

Ratio = \(\frac{[H^+]_{\text{gastric}}}{[H^+]_{\text{vesicle}}}\)

Substituting the values we calculated:

Ratio = \(\frac{10^{-2.0}}{10^{-5.2}}\)

Using the rule of exponents \(\frac{a^m}{a^n} = a^{m-n}\):

Ratio = \(10^{-2.0 - (-5.2)}\)

Ratio = \(10^{-2.0 + 5.2}\)

Ratio = \(10^{3.2}\)

Now, we calculate the value of \(10^{3.2}\):

\(10^{3.2} \approx 1584.89\)

Rounding this value gives approximately 1585.

This means that the hydrogen ion concentration in gastric juice is approximately 1585 times higher than that in endocytic vesicles.

Conversely, the hydrogen ion concentration in endocytic vesicles is approximately \(\frac{1}{1585}\) times the concentration in gastric juice. This is equivalent to saying that the endocytic vesicle has a \([H^+]\) that is 1585 times lower than that of gastric juice.

Therefore, the endocytic vesicle has a \([H^+]\) that is approximately 1585 times lower than that of gastric juice.

Let's examine the options:

• 15.85 times lower
• 0.1585 times lower
• 158.5 times lower
• 1585 times lower

Our calculated ratio is approximately 1585.

The correct comparison based on our calculation is that the endocytic vesicle's \([H^+]\) is 1585 times lower than gastric juice's \([H^+]\).

Final Answer is obtained by calculating \(10^{(pH_{gastric} - pH_{vesicle})}\). In this case, \(10^{(5.2 - 2.0)} = 10^{3.2} \approx 1585\). This value represents how many times more concentrated the gastric juice is compared to the vesicle in terms of H+. Thus, the vesicle's concentration is this many times lower.