Question

A 500mL Soda water is packed under 2.5 atm and CO₂ pressure at 298 K. Find out the quantity of CO₂ present in the bottle if Henry’s law constant for CO₂ in water is 1.67x108 Pa at 298 K.

Answer 1

Step 1: Given Data

${\mathrm{K}}_{\mathrm{H}}=1.67\times {10}^8$

Pa Pressure of${\mathrm{CO}}_2=2.5\mathrm{atm}$

Step 2: Formula used

According to Henry’s law, the amount of gas that is dissolved in a liquid is directly proportional to the partial pressure of that gas above the liquid when the temperature is kept constant.

The constant of proportionality for this relationship is called Henry’s law constant (usually denoted by ‘${\mathrm{k}}_{\mathrm{H}}$). The mathematical formula of Henry’s law is given by:

$$\begin{gathered} \mathrm{P}\propto \mathrm{C} \\ \left(\mathrm{or}\right) \\ \mathrm{P}={\mathrm{k}}_{\mathrm{H}}.\mathrm{C} \\ \mathrm{Or}\mathrm{X}={\mathrm{k}}_{\mathrm{H}}.\mathrm{P} \end{gathered}$$

Where ‘P’ denotes the partial pressure of the gas in the atmosphere above the liquid., ‘C’ denotes the concentration of the dissolved gas., ‘${\mathrm{k}}_{\mathrm{H}}$’ is Henry’s law constant of the gas., X is a mole fraction of gas.

Step 3: Determining pressure of the gas

We know that

$1atm=1.01235x{10}^{5}Pa$

So Pressure of${\mathrm{CO}}_2=2.5\mathrm{x}1.01235\mathrm{x}{10}^5\mathrm{Pa}$

$=2.533125\mathrm{x}{10}^5\mathrm{Pa}$

Step 4: Applying Henry's Law

According to the relationship, $\mathrm{p}={\mathrm{K}}_{\mathrm{H}}\mathrm{x}\mathrm{X}$

$$\begin{gathered} \mathrm{X}=\mathrm{p}/{\mathrm{K}}_{\mathrm{H}} \\ =2.533125\mathrm{x}{10}^5\mathrm{Pa}/1.67\times {10}^8\mathrm{Pa} \\ \mathrm{X}=1.52\mathrm{x}{10}^{-3} \end{gathered}$$

Step 5: Calculating moles of water

Given that the bottle contains$500mL$of soda water

Volume of water = $500mL$

Density of water = $1g/mL$

$$\begin{gathered} Mass=VolumeXDensity \\ Mass=500x1 \\ =500 \end{gathered}$$

Molar mass of water is $18g/mol$

Number of moles of water = Mass/(Molar mass)

$$\begin{gathered} n=500/18 \\ =27.78 \end{gathered}$$

Step 6: Calculating mass of gas

We know that mole fraction$X=\frac{nC{O}_2}{(nC{O}_2+n{H}_{2}O)}$

Value of mole fraction is negligible as compared to one

Hence$X=\frac{nC{O}_2}{(n{H}_{2}O)1.52x{10}^{-3}}$

$=\frac{nC{O}_2}{27.78nC{O}_2}$

Molar mass of$C{O}_2=44g/mol$

Mass = molar mass x Number of moles

$$\begin{gathered} =0.042X44 \\ =1.848g \end{gathered}$$

Hence, the quantity of Carbon dioxide present in the bottle is $1.848g$