Find the value of log`(underset ("Volume of solute particles in true solution")overset("volume of colloidal particels")(_))` (In case of limiting value)

Answer

To find the value of \( \log\left(\frac{\text{Volume of solute particles in true solution}}{\text{Volume of colloidal particles}}\right) \) in the limiting case, we can follow these steps:

### Step 1: Understand the sizes of the particles
- The diameter of colloidal particles ranges from 1 nm to 1000 nm.
- The diameter of solute particles in a true solution ranges from 0.1 nm to 1 nm.
- For the limiting case, we will take the upper limit of colloidal particles (1 nm) and the lower limit of solute particles (0.1 nm).

### Step 2: Calculate the volumes of the particles
- The volume \( V \) of a spherical particle is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
- In terms of diameter \( d \), the formula becomes:
\[ V = \frac{1}{6} \pi d^3 \]

### Step 3: Calculate the volume of colloidal particles
- For colloidal particles with diameter \( d = 1 \, \text{nm} \):
\[ V_{\text{colloidal}} = \frac{1}{6} \pi (1 \, \text{nm})^3 = \frac{1}{6} \pi \, \text{nm}^3 \]

### Step 4: Calculate the volume of solute particles
- For solute particles with diameter \( d = 0.1 \, \text{nm} \):
\[ V_{\text{solute}} = \frac{1}{6} \pi (0.1 \, \text{nm})^3 = \frac{1}{6} \pi (0.001 \, \text{nm}^3) = \frac{1}{6} \pi \times 10^{-3} \, \text{nm}^3 \]

### Step 5: Set up the ratio of the volumes
- We need to find:
\[ \frac{V_{\text{solute}}}{V_{\text{colloidal}}} = \frac{\frac{1}{6} \pi \times 10^{-3} \, \text{nm}^3}{\frac{1}{6} \pi \, \text{nm}^3} \]
- The \( \frac{1}{6} \pi \) cancels out:
\[ = \frac{10^{-3} \, \text{nm}^3}{1 \, \text{nm}^3} = 10^{-3} \]

### Step 6: Calculate the logarithm
- Now, we take the logarithm of the ratio:
\[ \log\left(\frac{V_{\text{solute}}}{V_{\text{colloidal}}}\right) = \log(10^{-3}) = -3 \]

### Final Answer
Thus, the value of \( \log\left(\frac{\text{Volume of solute particles in true solution}}{\text{Volume of colloidal particles}}\right) \) in the limiting case is:
\[ \boxed{-3} \]