JEE MAIN - Chemistry (2023 - 6th April Evening Shift - No. 23)
Consider the following pairs of solution which will be isotonic at the same temperature. The number of pairs of solutions is / are ___________.
A. $$1 ~\mathrm{M}$$ aq. $$\mathrm{NaCl}$$ and $$2 ~\mathrm{M}$$ aq. urea
B. $$1 ~\mathrm{M}$$ aq. $$\mathrm{CaCl}_{2}$$ and $$1.5 ~\mathrm{M}$$ aq. $$\mathrm{KCl}$$
C. $$1.5 ~\mathrm{M}$$ aq. $$\mathrm{AlCl}_{3}$$ and $$2 ~\mathrm{M}$$ aq. $$\mathrm{Na}_{2} \mathrm{SO}_{4}$$
D. $$2.5 ~\mathrm{M}$$ aq. $$\mathrm{KCl}$$ and $$1 ~\mathrm{M}$$ aq. $$\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}$$
Explanation
We say that two solutions are isotonic (at the same temperature) if they have the same osmotic pressure, $\pi$. For dilute solutions at the same temperature $T$,
$ \pi = i\, M\, R\, T, $
where
$M$ = molarity of the solution,
$i$ = van 't Hoff factor (number of particles the solute dissociates into),
$R$ = universal gas constant,
$T$ = absolute temperature.
Since $R$ and $T$ are common for both solutions (same temperature), two solutions will be isotonic if
$ i_1 \, M_1 \;=\; i_2 \, M_2. $
Let us check each pair:
1. Pair A: $1\,\mathrm{M}$ NaCl and $2\,\mathrm{M}$ urea
NaCl dissociates into $ \mathrm{Na}^+ $ and $ \mathrm{Cl}^- $.
Hence $i(\mathrm{NaCl}) = 2$.
So $i \, M = 2 \times 1 = 2.$
Urea ($\mathrm{(NH_2)_2CO}$) does not dissociate in water.
Hence $i(\mathrm{urea}) = 1$.
So $i \, M = 1 \times 2 = 2.$
Both solutions have $i M = 2.$
$\therefore$ They are isotonic.
2. Pair B: $1\,\mathrm{M}$ $\mathrm{CaCl_2}$ and $1.5\,\mathrm{M}$ $\mathrm{KCl}$
Calcium chloride ($\mathrm{CaCl_2}$) dissociates into $ \mathrm{Ca}^{2+} $ and $ 2\,\mathrm{Cl}^- $.
Hence $i(\mathrm{CaCl_2}) = 3.$
So $i \, M = 3 \times 1 = 3.$
Potassium chloride ($\mathrm{KCl}$) dissociates into $ \mathrm{K}^+ $ and $ \mathrm{Cl}^- $.
Hence $i(\mathrm{KCl}) = 2.$
So $i \, M = 2 \times 1.5 = 3.$
Both solutions have $i M = 3.$
$\therefore$ They are isotonic.
3. Pair C: $1.5\,\mathrm{M}$ $\mathrm{AlCl_3}$ and $2\,\mathrm{M}$ $\mathrm{Na_2SO_4}$
Aluminum chloride ($\mathrm{AlCl_3}$) dissociates into $ \mathrm{Al}^{3+} $ and $ 3\,\mathrm{Cl}^- $.
Hence $i(\mathrm{AlCl_3}) = 4.$
So $i \, M = 4 \times 1.5 = 6.$
Sodium sulfate ($\mathrm{Na_2SO_4}$) dissociates into $ 2\,\mathrm{Na}^+ $ and $ \mathrm{SO_4}^{2-} $.
Hence $i(\mathrm{Na_2SO_4}) = 3.$
So $i \, M = 3 \times 2 = 6.$
Both solutions have $i M = 6.$
$\therefore$ They are isotonic.
4. Pair D: $2.5\,\mathrm{M}$ $\mathrm{KCl}$ and $1\,\mathrm{M}$ $\mathrm{Al_2(SO_4)_3}$
Potassium chloride ($\mathrm{KCl}$) has $i(\mathrm{KCl}) = 2.$
So $i \, M = 2 \times 2.5 = 5.$
Aluminum sulfate $\mathrm{Al_2(SO_4)_3}$ dissociates into
$ 2\,\mathrm{Al}^{3+} \;+\; 3\,\mathrm{SO_4}^{2-} \quad \Longrightarrow i(\mathrm{Al_2(SO_4)_3}) = 2 + 3 = 5. $
So $i \, M = 5 \times 1 = 5.$
Both solutions have $i M = 5.$
$\therefore$ They are isotonic.
Conclusion
All four given pairs $(A, B, C, D)$ satisfy $i_1 M_1 = i_2 M_2$. Therefore, all of them are isotonic pairs.
Hence, the number of isotonic pairs is:
$ \boxed{4}. $
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