A graph drawn between `'log.(x)/(m)’` and log p on 'X' and 'Y' axis respectively could be - regarding physical adsorption

Options

A. A hyperbola
B. A parabola
C. A straight line with positive slope
D. A straight line with negative slope

Answer

To solve the question regarding the graph drawn between `log(x/m)` and `log(p)` in the context of physical adsorption, we will use the Freundlich adsorption isotherm. Here is a step-by-step solution:

### Step 1: Understand the Freundlich Adsorption Isotherm
The Freundlich adsorption isotherm is given by the equation:
\[ \frac{x}{m} = k p^{\frac{1}{n}} \]
where:
- \( x \) = amount of adsorbate adsorbed,
- \( m \) = mass of the adsorbent,
- \( p \) = pressure,
- \( k \) = constant,
- \( n \) = constant (greater than 1).

### Step 2: Take the Logarithm of Both Sides
To analyze the relationship between \( \frac{x}{m} \) and \( p \), we take the logarithm of both sides:
\[ \log\left(\frac{x}{m}\right) = \log(k) + \log(p^{\frac{1}{n}}) \]

### Step 3: Simplify the Equation
Using the properties of logarithms, we can simplify the equation:
\[ \log\left(\frac{x}{m}\right) = \log(k) + \frac{1}{n} \log(p) \]
This can be rearranged to:
\[ \log\left(\frac{x}{m}\right) = \frac{1}{n} \log(p) + \log(k) \]

### Step 4: Identify the Form of the Equation
The equation now resembles the linear equation format \( y = mx + c \), where:
- \( y = \log\left(\frac{x}{m}\right) \),
- \( x = \log(p) \),
- \( m = \frac{1}{n} \) (the slope),
- \( c = \log(k) \) (the y-intercept).

### Step 5: Determine the Slope and Intercept
From the equation:
- The slope \( m = \frac{1}{n} \) is positive since \( n > 1 \).
- The y-intercept is \( \log(k) \).

### Conclusion
The graph of \( \log\left(\frac{x}{m}\right) \) versus \( \log(p) \) is a straight line with a positive slope.

### Final Answer
The graph drawn between `log(x/m)` and `log(p)` is a **straight line with a positive slope**.

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