Question

An aqueous solution of glucose boils at \[{100.02^0}C\]. What is the number of glucose molecules in the solution containing \[100\] g of water? What will be the osmotic pressure of this glucose solution at \[{27^0}C\](Given \[{K_b}\] for water \[ = 0.5K.Kg.{\left( {mol} \right)^{ - 1}},R = 0.082L.atm.{\left( {mol.K} \right)^{ - 1}}\], Avogadro’s constant \[{N_A} = 6.02 \times {10^{23}}mo{l^{ - 1}}\])

Answer 1

Hint: The molality and molarity will be nearly equal for dilute solutions. The molality can be determined from the number of moles of solute and weight of solution in kilograms. The number of molecules can be obtained number of moles multiplied by Avogadro’s number.

Formula used:
\[\pi = CRT\]
Where \[\pi \]is osmotic pressure
C is molar concentration or molality
R is ideal gas constant
T is temperature in kelvin

Complete answer:
Given an aqueous solution of glucose boils at \[{100.02^0}C\]
Temperature = \[27 + 273 = 300K\]
Elevation in boiling point \[{k_{_b}} = 0.5K.Kg.{\left( {mol} \right)^{ - 1}}\]
Universal gas constant \[R = 0.082L.atm.{\left( {mol.K} \right)^{ - 1}}\]
Avogadro’s constant \[{N_A} = 6.02 \times {10^{23}}mo{l^{ - 1}}\])
The boiling point of water is \[{100^0}C\]
But, given an aqueous solution of glucose boils at \[{100.02^0}C\]
Thus, elevation in boiling point \[\Delta {T_b}\]will be
\[\Delta {T_b} = {100.02^0}C - {100^0}C = {0.02^0}C\]
But, \[\Delta {T_b} = {k_b} \times m\]
We know the values of \[\Delta {T_b}\]and \[{k_b}\].
Thus, the molality value will be
\[m = \dfrac{{\Delta {T_b}}}{{{k_b}}} = \dfrac{{0.02}}{{0.5}} = 0.04\]
Thus, molality is \[0.04\].
We know that molality is given by number of moles of solute and weight of solvent
Weight of the solvent is \[100g = 0.1kg\]
Thus, the number of moles of glucose will be
\[n = \dfrac{{0.04}}{{0.1}} = 0.004\]
The number of molecules can be obtained by dividing the number of moles of glucose by Avogadro’s number.
Number of molecules of glucose will be
\[0.004 \times 6.023 \times {10^{23}} = 2.408 \times {10^{21}}molecules\]
Osmotic pressure can be obtained by substituting the values of molality, ideal gas constant and temperature in kelvin.
Thus, osmotic pressure
\[\pi = 0.04 \times 0.0821 \times 300 = 0.9852atm\]

Note:
The molar concentration can also be known as molarity. For dilute solutions like glucose dissolved in water both the terms molality and molarity will be nearly equal. Moles are the units used to express the amount of substance. The number of molecules can be obtained from the number of moles and Avogadro’s number.