For Freundlich adsorption isotherm, a plot of log(x/m) (y-axis) and log p (x-axis) gives a straight

Q: Why do we take the log of the Freundlich isotherm?

Taking the logarithm converts the exponential relationship into a linear one ($y = mx + c$). This makes it incredibly easy to plot experimental data as a straight line and directly read the constants $K$ and $n$ from the intercept and slope.

Q: What exactly does $x/m$ represent?

It represents the extent of adsorption. Specifically, $x$ is the mass of the adsorbate (the gas being adsorbed), and $m$ is the mass of the adsorbent (the solid surface). So, $x/m$ is the mass of gas adsorbed per unit mass of the solid.

Q: Why is the slope $1/n$ and not $n$?

Because in the original Freundlich equation, the pressure $p$ is raised to the power of $1/n$. When you apply the logarithm property $\log(a^b) = b \log a$, the exponent $1/n$ comes down to become the coefficient of $\log p$, which acts as the slope.

Q: How did we know $K=3$ without a calculator?

The problem states the intercept is $0.4771$, which corresponds to $\log K$. It also provides a crucial hint at the end: $\log 3 = 0.4771$. By comparing the two, it is clear that $K$ must be exactly $3$.

Q: What if the pressure was given in different units like bar or Pascal?

The constant $K$ is empirical and its value depends on the units of pressure used. In this problem, we assume the given $K$ (derived from the intercept) is already consistent with the pressure unit of atmospheres, so no unit conversion is necessary.

Question

For Freundlich adsorption isotherm, a plot of $\log\left(\frac{x}{m}ight)$ (y-axis) and $\log p$ (x-axis) gives a straight line. The intercept and slope for the line is $0.4771$ and $2$, respectively. The mass of gas, adsorbed per gram of adsorbent if the initial pressure is $0.04$ atm, is ......... $	imes 10^{-4}$ g. ($\log 3 = 0.4771$)

Sigma Percentile · Accepted Answer

### The Geometry of Adsorption

When we study the accumulation of a gas on a solid surface, the Freundlich adsorption isotherm provides a beautiful empirical relationship. It states that the mass of gas adsorbed per unit mass of adsorbent, denoted as $\frac{x}{m}$, is proportional to a fractional power of the pressure $p$. Mathematically, this is written as:

$$ \frac{x}{m} = K p^{1/n} $$

While this exponential curve is useful, scientists love straight lines because they make it incredibly easy to extract constants. By taking the logarithm on both sides of the equation, we transform the curve into a linear masterpiece:

$$ \log\left(\frac{x}{m}ight) = \log K + \frac{1}{n} \log p $$

If you look closely, this is nothing but the classic equation of a straight line, $y = mx + c$. Here, our y-axis is $\log\left(\frac{x}{m}ight)$, our x-axis is $\log p$, the y-intercept $c$ is $\log K$, and the slope $m$ is $\frac{1}{n}$.

### Decoding the Constants

The problem hands us the geometric properties of this line on a silver platter. We are told that the intercept is $0.4771$ and the slope is $2$. Let's decode what these numbers mean physically.

First, the intercept:

$$ \log K = 0.4771 $$

The question generously provides a hint: $\log 3 = 0.4771$. Therefore, without breaking a sweat, we can conclude that our constant **$K = 3$**.

Next, the slope:

$$ \frac{1}{n} = 2 $$

We don't even need to find $n$ itself, because the formula directly uses the power $\frac{1}{n}$. We now have all the tools required to solve the mystery.

### The Final Calculation

We are asked to find the mass of gas adsorbed per gram of adsorbent (which is exactly the definition of $\frac{x}{m}$) when the initial pressure $p$ is $0.04$ atm. Let's bring back our original Freundlich equation and substitute our newly found constants:

$$ \frac{x}{m} = K p^{1/n} $$

$$ \frac{x}{m} = 3 	imes (0.04)^2 $$

Now, it's just a matter of careful arithmetic. Squaring $0.04$ gives $0.0016$. Multiplying this by $3$ yields:

$$ \frac{x}{m} = 0.0048 	ext{ g} $$

The question specifically requests the answer in the format of a number multiplied by $10^{-4}$. To comply with this, we shift the decimal point four places to the right:

$$ 0.0048 = 48 	imes 10^{-4} $$

Thus, the integer value that perfectly fills the blank is **$48$**.

Enter Numerical Value:

Visualized Solution

$Visualizing the Isotherm$

$The Freundlich Equation$

$Extracting Constants$

$Finding K$

$Setting up the Calculation$

$Executing the Math$

$Final Formatting$

The Sigma Insight: Adsorption

The Geometry of Adsorption

Decoding the Constants

The Final Calculation

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Enter Numerical Value:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Visualizing the Isotherm

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Freundlich Equation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Extracting Constants

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Finding K

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Setting up the Calculation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Executing the Math

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Formatting