In an adsorption experiment, a graph of log(x/m) versus log P was found to be linear with a slope of 45°, and the the intercept is found to be 0.3010. The amount of gas adsorbed per gram charcoal under a pressure of 0.8 atm is

Options

A. 1.2
B. 1.4
C. 1.6
D. 1.8

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Freundlich Adsorption Isotherm
The Freundlich adsorption isotherm relates the amount of gas adsorbed per gram of adsorbent (x/m) to the pressure (P) of the gas. The equation can be expressed as:
\[ \frac{x}{m} = k \cdot P^{\frac{1}{n}} \]
where:
- \( x \) = amount of gas adsorbed
- \( m \) = mass of the adsorbent
- \( k \) = Freundlich constant
- \( P \) = pressure
- \( n \) = constant related to the adsorption intensity

### Step 2: Take the Logarithm of Both Sides
Taking the logarithm of both sides gives us:
\[ \log\left(\frac{x}{m}\right) = \log(k) + \frac{1}{n} \log(P) \]

### Step 3: Identify the Slope and Intercept
This equation resembles the equation of a straight line \( y = mx + c \), where:
- \( y = \log\left(\frac{x}{m}\right) \)
- \( x = \log(P) \)
- \( m = \frac{1}{n} \) (slope)
- \( c = \log(k) \) (intercept)

From the problem, we know:
- The slope is 45°, which means \( \tan(45°) = 1 \). Therefore, \( \frac{1}{n} = 1 \) and thus \( n = 1 \).
- The intercept is given as 0.3010, so \( \log(k) = 0.3010 \).

### Step 4: Calculate the Value of k
To find \( k \), we can convert the logarithmic value back to its exponential form:
\[ k = 10^{0.3010} \]
We know that \( 10^{0.3010} \) is equal to 2 (since \( \log_{10}(2) \approx 0.3010 \)). Therefore:
\[ k = 2 \]

### Step 5: Substitute Values into the Freundlich Equation
Now we can substitute \( k \) and the given pressure \( P = 0.8 \) atm into the Freundlich equation:
\[ \frac{x}{m} = k \cdot P^{\frac{1}{n}} \]
Substituting the values:
\[ \frac{x}{m} = 2 \cdot (0.8)^{1} \]
\[ \frac{x}{m} = 2 \cdot 0.8 \]
\[ \frac{x}{m} = 1.6 \]

### Conclusion
The amount of gas adsorbed per gram of charcoal under a pressure of 0.8 atm is:
\[ \frac{x}{m} = 1.6 \]

### Final Answer
Thus, the answer is \( 1.6 \).

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