Adsorption of phenol from its aqueous solution on to fly ash obeys Freundlich isotherm. At a given temperature, from $10 \mathrm{mg} \mathrm{g}^{-1}$ and $16 \mathrm{mg} \mathrm{g}^{-1}$ aqueous phenol solutions, the concentrations of adsorbed phenol are measured to be $4 \mathrm{mg} \mathrm{g}^{-1}$ and $10 \mathrm{mg} \mathrm{g}^{-1}$, respectively. At this temperature, the concentration (in $\mathrm{mg} \mathrm{g}^{-1}$ ) of adsorbed phenol from $20 \mathrm{mg} \mathrm{g}^{-1}$ aqueous solution of phenol will be ______________.
Use: $\log _{10} 2=0.3$
Explanation:
$\begin{aligned} & \frac{x}{m}=K(C)^{\frac{1}{n}} \\\\ & 4=K(10)^{\frac{1}{n}}........(i) \\\\ & 10=K(16)^{\frac{1}{n}}.......(ii)\end{aligned}$
From (ii) and (i)
$ \begin{aligned} & \log 4-\log 10=\frac{1}{n}(\log 10-\log 16) \\ & 2 \log 2-1=\frac{1}{n}(1-4 \log 2) \\ & 0.6-1=\frac{1}{x}(1-1.2) \\ & -0.4=\frac{-0.2}{n} \\ & n=\frac{2}{4}=\left(\frac{1}{2}\right) \end{aligned} $
$\begin{aligned} \text { Now } 10 & =K(16)^2 \\ K & =\frac{10}{256} \\ \text { So, } \frac{x}{m} & =K(20)^2 \\ \frac{x}{m} & =\frac{10}{256} \times 400 \\ \frac{x}{m} & =15.625\end{aligned}$
or
$ \begin{aligned} & 4=K(10)^2 \\ & K=\frac{4}{100} \\ & \frac{x}{m}=K(20)^2 \\ & \frac{x}{m}=\frac{4}{100} \times 400 \\ & \frac{x}{m}=16 \end{aligned} $
To form a complete monolayer of acetic acid on $1 \mathrm{~g}$ of charcoal, $100 \mathrm{~mL}$ of $0.5 \mathrm{M}$ acetic acid was used. Some of the acetic acid remained unadsorbed. To neutralize the unadsorbed acetic acid, 40 $\mathrm{mL}$ of $1 \mathrm{M} \mathrm{NaOH}$ solution was required. If each molecule of acetic acid occupies $\mathbf{P} \times 10^{-23} \mathrm{~m}^2$ surface area on charcoal, the value of $\mathbf{P}$ is _____.
[Use given data: Surface area of charcoal $=1.5 \times 10^2 \mathrm{~m}^2 \mathrm{~g}^{-1}$; Avogadro's number $\left(\mathrm{N}_{\mathrm{A}}\right)=6.0 \times 10^{23}$ $\left.\mathrm{mol}^{-1}\right]$
Explanation:
Let's first understand and break down the given information. We are provided with the following details to solve for the value of $\mathbf{P}$:
1. Volume of acetic acid solution used: $100 \mathrm{~mL}$ of $0.5 \mathrm{M}$ acetic acid.
2. Volume of $\mathrm{NaOH}$ solution used for neutralization: $40 \mathrm{~mL}$ of $1 \mathrm{M} \mathrm{NaOH}$.
3. Surface area of the charcoal: $1.5 \times 10^2 \mathrm{~m}^2 \mathrm{~g}^{-1}$.
4. Avogadro's number: $\mathrm{N}_{\mathrm{A}}=6.0 \times 10^{23} \mathrm{~mol}^{-1}$.
Let's start with the concentration and volume of acetic acid used:
$ \text{Moles of acetic acid initially} = 0.5 \text{ M} \times \frac{100 \text{ mL}}{1000} = 0.05 \text{ moles} $Next, let's determine the moles of $\mathrm{NaOH}$ used for neutralization:
$ \text{Moles of } \mathrm{NaOH} = 1 \text{ M} \times \frac{40 \text{ mL}}{1000} = 0.04 \text{ moles} $Since $\mathrm{NaOH}$ completely neutralizes the unadsorbed acetic acid:
$ \text{Moles of unadsorbed acetic acid} = 0.04 \text{ moles} $Therefore, the moles of acetic acid adsorbed on the charcoal is given by:
$ \text{Moles of adsorbed acetic acid} = 0.05 - 0.04 = 0.01 \text{ moles} $Now, we need to calculate the total number of molecules of acetic acid adsorbed:
$ \text{Number of molecules} = 0.01 \text{ moles} \times 6.0 \times 10^{23} \text{ molecules/mole} = 6.0 \times 10^{21} \text{ molecules} $Given the surface area of charcoal, we can find the total surface area:
$ \text{Total surface area} = 1.5 \times 10^2 \text{ m}^2 / \text{g} \times 1 \text{ g} = 1.5 \times 10^2 \text{ m}^2 $Finally, we need to determine the area occupied by each molecule of acetic acid:
$ \text{Area per molecule of acetic acid} = \frac{\text{Total surface area}}{\text{Number of molecules}} = \frac{1.5 \times 10^2 \text{ m}^2}{6.0 \times 10^{21} \text{ molecules}} $Let's perform the calculation:
$ \text{Area per molecule of acetic acid} = \frac{1.5 \times 10^2}{6.0 \times 10^{21}} = 0.25 \times 10^{-19} \text{ m}^2 = 2.5 \times 10^{-20} \text{ m}^2 $Since the problem states that the area per molecule is $\mathbf{P} \times 10^{-23} \mathrm{~m}^2$, we equate:
$ 2.5 \times 10^{-20} = \mathbf{P} \times 10^{-23} $Solving for $\mathbf{P}$:
$ \mathbf{P} = \frac{2.5 \times 10^{-20}}{10^{-23}} = 2.5 \times 10^3 = 2500 $Therefore, the value of $\mathbf{P}$ is 2500.
Nitrogen gas is absorbed on 20% surface sites. On heating N$_2$ gas evolved from sites and was collected at 0.01 atm and 298 K in a container of volume 2.46 cm$^3$. Find out the number of surface sites occupied per molecule of N, if the density of surface sites is $6.023\times10^{14}/\mathrm{cm^3}$ and surface area is 1000 cm$^2$.
Explanation:
Given Data:
P = 0.01 atm
T = 298 K
V = 2.46 cm$^3$
Density of surface sites $=6.023\times10^{14}/\mathrm{cm^3}$
Surface area (A) = 1000 cm$^2$
Goal:
We want to find out how many surface sites are covered by each nitrogen (N$_2$) molecule.
Step 1: Find the Total Number of Surface Sites
We are given the number of sites per cm$^3$ and the total surface area. Multiply these to get the total number of sites:
Total number of surface sites = Density $\times$ Total surface area
$=6.023\times10^{14}\mathrm{~cm}^{-3}\times1000\mathrm{~cm^2}$
$=6.023\times10^{17}$ sites
Step 2: Calculate the Number of Sites Occupied by N$_2$
Nitrogen covers just 20% of the sites. So, multiply the total sites by 20% (or 0.20):
Sites occupied by N$_2$ = $\frac{20}{100}\times6.023\times10^{17}$
$=1.2046\times10^{17}$ sites
Step 3: Find Number of N$_2$ Molecules Released (Using Gas Laws)
To get the number of N$_2$ molecules, first calculate how many moles of N$_2$ gas are collected using the ideal gas law:
$n = \frac{PV}{RT}$
Where R = 0.0821 L-atm/K·mol; V = 2.46 cm$^3$ = 2.46 × 10$^{-3}$ L
Now, plug in the values:
$n = \frac{0.01\,\text{atm} \times 2.46 \times 10^{-3}\,\text{L}}{0.0821\, \text{L-atm/(K·mol)} \times 298\,\text{K}}$
$n = 1.01 \times 10^{-7}$ moles
Now, multiply the number of moles by Avogadro's number (6.023 × 10$^{23}$) to get molecules:
Total N$_2$ molecules released = $n \times N_A$
$= 1.01\times 10^{-7}\times 6.023\times10^{23} = 6.08\times10^{16}$ molecules
Step 4: Number of Surface Sites Occupied by Each N$_2$ Molecule
Now divide the number of sites occupied by the number of N$_2$ molecules:
$\text{Number of sites per N}_2 = \frac{1.2046\times10^{17}}{6.08\times10^{16}} = 1.98 \approx 2$
Final Answer
Each nitrogen molecule occupies about 2 surface sites.
(I) Lyophobic colloids are not formed by simple mixing of dispersed phase and dispersion medium.
(II) For emulsions, both the dispersed phase and the dispersion medium are liquid.
(III) Micelles are produced by dissolving a surfactant in any solvent at any temperature.
(IV) Tyndall effect can be observed from a colloidal solution with dispersed phase having the same refractive index as that of the dispersion medium.
The option with the correct set of statements is :
Among the electrolytes Na$_2$SO$_4$, CaCl$_2$, Al$_2$(SO$_4$)$_3$ and NH$_4$Cl, the most effective coagulating agent for Sb$_2$S$_3$ sol is
Among the following, the surfactant that will form micelles in aqueous solution at the lowest molar concentration at ambient condition is:
Statement 1 : Micelles are formed by surfactant molecules above the critical micellar concentration (CMC).
Statement 2 : The conductivity of a solution having surfactant molecules decreases sharply at the CMC.
The given graphs/data I, II, III and IV represent general trends observed for different physisorption and chemisorption processes under mild conditions of temperature and pressure. Which of the following choice(s) about I, II, III and IV is(are) correct?




Explanation:
The number of moles of acetic acid in 100 mL (before adding charcoal) = 0.05 mol
The number of moles of acetic acid in 100 mL (after adding charcoal) = 0.049 mol
The number of moles of acetic acid adsorbed on the surface of charcoal = 0.05 - 0.049 = 0.001 mol
The number of molecules of acetic acid adsorbed on the surface of charcoal = 0.001 $\times$ 6.02 $\times$ 1023 = 6.02 $\times$ 1020
Given that the surface area of charcoal = 3.01 $\times$ 102
m2
, so the area occupied by single acetic acid molecule on the
surface of charcoal is
= ${{3.01 \times {{10}^2}} \over {6.02 \times {{10}^{20}}}}$ = 5 $\times$ 10-19 m2
