Which one of the following plots correctly describes the variation of osmotic pressure (H) of a fixed amount of a solute against the volume (V) of the solution at a fixed temperature?
Show Hint
Whenever two variables \(y\) and \(x\) satisfy \(y \cdot x = \text{constant}\), their graph is always a rectangular hyperbola asymptotic to the axes.
This is a standard mathematical rule applicable across physics and chemistry.
Step 1: Understanding the Question:
The question asks us to identify the graph that correctly represents the relationship between the osmotic pressure (\(\Pi\)) of a solution and its volume (\(V\)) for a fixed amount of solute at a constant temperature. Step 2: Key Formula or Approach:
The osmotic pressure of a dilute solution is given by the van 't Hoff equation:
\[ \Pi = CRT \]
where:
- \(C\) is the molar concentration of the solute (\(C = \frac{n}{V}\)).
- \(n\) is the number of moles of solute (fixed amount).
- \(R\) is the gas constant.
- \(T\) is the absolute temperature (fixed temperature). Step 3: Detailed Explanation:
Let's substitute the concentration formula into the osmotic pressure equation:
\[ \Pi = \frac{n}{V} RT \]
Since \(n\), \(R\), and \(T\) are all constant, we can write:
\[ \Pi \cdot V = nRT = \text{constant} \]
This equation is mathematically analogous to Boyle's law for ideal gases (\(P \cdot V = \text{constant}\)).
- The relationship between \(\Pi\) and \(V\) is inversely proportional:
\[ \Pi \propto \frac{1}{V} \]
- A plot of \(\Pi\) against \(V\) will yield a rectangular hyperbola.
- In a rectangular hyperbola:
- As \(V \rightarrow 0\), \(\Pi \rightarrow \infty\).
- As \(V \rightarrow \infty\), \(\Pi \rightarrow 0\).
- The curve approaches but never actually touches either the vertical (\(\Pi\)) or horizontal (\(V\)) axis (it is asymptotic to the axes).
- Let's evaluate the given plots:
- Plot (a) shows a perfect rectangular hyperbola that is asymptotic to both axes.
- Plot (b) shows a curve that erroneously intersects or touches the axes.
- Plot (c) and (d) show straight lines, which do not represent an inverse relationship. Step 4: Final Answer:
The correct graph is represented by Plot (a), which corresponds to option (A).
