To determine the mass of Argon (Ar) that will dissolve in 500 grams of water at a pressure of 2.5 atm, given the solubility constant [tex]\( k \)[/tex] at [tex]\( 20^{\circ}C \)[/tex] is [tex]\( 1.64 \times 10^{-3} \, M/atm \)[/tex], follow these detailed steps:
1. Given Data and Constants:
- Solubility constant [tex]\( k = 1.64 \times 10^{-3} \, M/atm \)[/tex]
- Molar mass of Argon [tex]\( M_{\text{Ar}} = 39.948 \, g/mol \)[/tex]
- Pressure of Argon [tex]\( P_{\text{Ar}} = 2.5 \, atm \)[/tex]
- Mass of water [tex]\( m_{\text{water}} = 500 \, g \)[/tex]
- Density of water [tex]\( \rho_{\text{water}} = 1.0 \, g/mL \)[/tex]
2. Calculate the volume of water:
Since water has a density of [tex]\( 1.0 \, g/mL \)[/tex], the volume of 500 grams of water can be calculated as:
[tex]\[
\text{Volume of water} = \frac{500 \, g}{1.0 \, g/mL} = 500 \, mL
\][/tex]
Convert this volume to liters:
[tex]\[
\text{Volume of water in liters} = 500 \, mL \times 1 \times 10^{-3} \, L/mL = 0.5 \, L
\][/tex]
3. Use Henry's Law to find the molarity of Argon in solution:
Henry's Law states that the concentration ([tex]\( C \)[/tex]) of a gas in a solution is directly proportional to the pressure ([tex]\( P \)[/tex]) of the gas above the solution:
[tex]\[
C = k \times P
\][/tex]
Plugging in the known values:
[tex]\[
C_{\text{Ar}} = 1.64 \times 10^{-3} \, M/atm \times 2.5 \, atm = 4.1 \times 10^{-3} \, M
\][/tex]
4. Calculate the moles of Argon that will dissolve in the given volume of water:
The molarity ([tex]\( C \)[/tex]) represents the moles of solute per liter of solution, so:
[tex]\[
\text{Moles of Ar} = C_{\text{Ar}} \times \text{Volume of water in liters}
\][/tex]
[tex]\[
\text{Moles of Ar} = 4.1 \times 10^{-3} \, mol/L \times 0.5 \, L = 2.05 \times 10^{-3} \, mol
\][/tex]
5. Calculate the mass of Argon that will dissolve:
The mass of Argon can be determined by multiplying the moles of Argon by its molar mass:
[tex]\[
\text{Mass of Ar} = \text{Moles of Ar} \times M_{\text{Ar}}
\][/tex]
[tex]\[
\text{Mass of Ar} = 2.05 \times 10^{-3} \, mol \times 39.948 \, g/mol = 0.0819 \, g
\][/tex]
Therefore, the mass of Argon that will dissolve in 500 grams of water under the given conditions is approximately 0.0819 grams.
