The variation of `(p)/((x//m))` as a function of `p` when Langmuir's isotherm valid is `:`
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To solve the question regarding the variation of \(\frac{p}{\frac{x}{m}}\) as a function of \(p\) when Langmuir's isotherm is valid, we will follow these steps:
### Step 1: Understand Langmuir's Isotherm
Langmuir's adsorption isotherm describes how the amount of gas adsorbed on a solid surface varies with pressure at a constant temperature. The equation is given by:
\[
\frac{x}{m} = \frac{a p}{1 + b p}
\]
where:
- \(x\) = amount of gas adsorbed,
- \(m\) = mass of the adsorbent,
- \(p\) = pressure,
- \(a\) and \(b\) are constants.
### Step 2: Rearrange the Equation
We need to express \(\frac{p}{\frac{x}{m}}\) in terms of \(p\). From the Langmuir equation, we can rearrange it as follows:
\[
\frac{x}{m} = \frac{a p}{1 + b p}
\]
Taking the reciprocal gives:
\[
\frac{1}{\frac{x}{m}} = \frac{1 + b p}{a p}
\]
### Step 3: Express \(p\) in Terms of \(\frac{x}{m}\)
Now, we can express \(\frac{p}{\frac{x}{m}}\):
\[
\frac{p}{\frac{x}{m}} = \frac{p}{\frac{a p}{1 + b p}} = \frac{(1 + b p)}{a}
\]
### Step 4: Simplify the Expression
This simplifies to:
\[
\frac{p}{\frac{x}{m}} = \frac{1}{a} + \frac{b}{a} p
\]
### Step 5: Identify the Graph Characteristics
The equation \(\frac{p}{\frac{x}{m}} = \frac{1}{a} + \frac{b}{a} p\) is in the form of \(y = mx + c\), where:
- \(y\) is \(\frac{p}{\frac{x}{m}}\),
- \(x\) is \(p\),
- \(m\) (slope) is \(\frac{b}{a}\),
- \(c\) (y-intercept) is \(\frac{1}{a}\).
This indicates that the graph is a straight line with a positive slope.
### Step 6: Conclusion
The variation of \(\frac{p}{\frac{x}{m}}\) as a function of \(p\) when Langmuir's isotherm is valid will yield a straight line graph.
The curve drawn below shows the variations of P as a function of 1//V for a fixed mass and temperature of an ideal gas. It follows from the curve that:
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