Question

A graph plotted between $P$ vs $d$ (where $P$is the osmotic pressure of a solution of a solute of molar mass $m$ and $d$ is its density at temperature$T$). Pick out the correct statements about the plots.
(A) $\left[ {\dfrac{p}{d}} \right]$ is independent of molar mass
(B) The intercept of the plot $\dfrac{{ST}}{m}$
(C) The intercept of the plot is zero
(D) ${\left[ {\dfrac{p}{d}} \right]_{d \to 0}}$ is independent of temperature

Answer 1

According to the process of osmosis, there exists a relation between Osmotic pressure of solute, molar mass, density, and temperature. So, using the equation of osmotic pressure, we can plot a graph between pressure and density. With the help of graphs and equations, we can pick the correct statements. Some basic mathematical terms are used here.

Formula Used: Osmotic pressure:
$\pi = \dfrac{{nRT}}{V}$ where, $\pi$ is the osmotic pressure, $n$ is the number of moles of solute, $V$ is the volume of solution, $R,T$ are the gas constant and temperature respectively.

Complete step-by-step solution: First, we will understand the process of osmosis. Osmosis is defined as “The spontaneous flow of solvent molecules from the region of lower concentration to the region of higher concentration through a semipermeable membrane”. Now we will discuss the term osmotic pressure. Osmotic pressure is a colligative property. The excess of hydrostatic pressure which is developed as a result of osmosis is called osmotic pressure.
The equation is expressed as, $\pi = \dfrac{{nRT}}{V}$ . The question says that the graph is plotted between osmotic pressure and density. So, now we will convert the expression in osmotic pressure and density. So, the expression can be expressed as, $P = \dfrac{{dRT}}{M}$ where, $P$ is the osmotic pressure, $d$ is the density at temperature $T$. Now we can write the equation according to the requirement of the question $\dfrac{P}{d} = \dfrac{{RT}}{M}$.
Using the expression $\dfrac{P}{d} = \dfrac{{RT}}{M}$ we can conclude that $\left[ {\dfrac{p}{d}} \right]$ is dependent on both temperature $T$ and molar mass $M$ of the solute. So options (A) and (D) are not correct. Now to find the intercept of the plot $P$vs $d$. We will use the equation $P = \dfrac{{dRT}}{M}$ which represents the equation of a straight line. Now we will compare the general equation of straight lines $y = mx + c$ where $m,c$ are slope and intercept respectively. So by comparing we get $c = 0$. Therefore, the intercept of the plot is zero.

Therefore, the correct option is (C).

Note: The slope of the plot $P$ vs $d$ can also be obtained using the general equation of the straight line, $y = mx + c$. By comparing the equation of the plot, $P = \dfrac{{dRT}}{M}$ with the general equation of the straight line we get $m = \dfrac{{RT}}{M}$. Therefore, the slope is $\dfrac{{RT}}{M}$. For, the intercept we know that the equation passes with origin has zero intercepts.