Question

The Henry’s law constant for the solubility of \[{{\text{N}}_2}\] gas in water at 298 K is \[1.0 \times {10^5}{\text{ atm}}\] . The mole fraction of \[{{\text{N}}_2}\] in air is \[0.8\]. The number of moles of \[{{\text{N}}_2}\] from air dissolved in 10 moles of water at 298 K and 5 atm pressure is
A. \[4 \times {10^{ - 4}}\]
B. \[4 \times {10^{ - 5}}\]
C. \[5 \times {10^{ - 4}}\]
D. \[4 \times {10^{ - 6}}\]

Answer 1

Hint: Using mole fraction we will first calculate the partial pressure of nitrogen. Using Henry low we will get the mole fraction in the solution. Using mole fraction we will calculate the number of moles of nitrogen present.

Formula used: \[{\text{P}} = {{\text{K}}_{\text{H}}} \times {\text{X}}\]
Here P is pressure and X is the mole fraction and \[{{\text{K}}_{\text{H}}}\] is Henry constant.

Complete step by step solution:
The total pressure of the solution in which nitrogen is dissolved is given to us as 5 atm. The mole fraction of nitrogen gas is \[0.8\] using the mole fraction and the partial pressure we will first calculate the partial pressure of nitrogen gas using Dalton’s law as:
\[{{\text{P}}_{{{\text{N}}_2}}} = 0.8 \times 5 = 4{\text{ atm}}\]
Now Henry law states that the pressure exerted by a gas dissolved in a solution is directly proportional to the mole fraction of that gas. The constant of proportionality used is known as the Henry’s constant. So we will calculate the value of mole nitrogen dissolved in water. According to Henry’s Law:
\[{\text{P}} = {{\text{K}}_{\text{H}}} \times {\text{X}}\]
Here P is pressure and X is the mole fraction and \[{{\text{K}}_{\text{H}}}\] is Henry constant. The mole fraction of nitrogen in solution will be:
\[{\text{X}} = \dfrac{4}{{1 \times {{10}^5}}} = 4 \times {10^{ - 5}}\]
Now we know the formula for mole fraction which says:
\[{\text{X}} = \dfrac{{{{\text{n}}_{{{\text{N}}_2}}}}}{{{{\text{n}}_{{{\text{N}}_2}}} + {{\text{n}}_{{{\text{H}}_2}{\text{O}}}}}}\]
Number of moles of water is given as 10 so we will rewrite the equation as:
$\Rightarrow$ \[4 \times {10^{ - 5}} = \dfrac{{{{\text{n}}_{{{\text{N}}_2}}}}}{{{{\text{n}}_{{{\text{N}}_2}}} + 10}}\]
Solving the above equation we will get the number of moles as \[4 \times {10^{ - 4}}\]

Hence, the correct option is A.

Note: One of the major applications of Henry law include in the formation of carbonated drinks. The amount of carbon dioxide needed to be dissolved in the soda drink and the pressure exerted by the gas present was given by Henry.