A geologist removes a sample of water from a lake in Nevada where the water level has been slowly dropping over time. The sample is found to be 0.02M in borax. The lake is circular in shape and is currently 500 meters in diameter and 50 meters deep. Using the given data predict the level of the lake when borax will begin to precipitate on the shore if the mean temperature of the lake water is 15degC. Assume that the lake is equally deep at all points.
ΔGrxn= 14.095 kj/mol
ΔHrxn= 46.510 kj/mol
ΔSrxn= 108.72 j/mol-K
To calculate the level of the lake when borax will begin to precipitate, we need to determine the concentration of borax in equilibrium with the given conditions. This can be done using the Gibbs free energy equation:
ΔG = ΔH - TΔS
Where:
ΔG = Gibbs free energy change
ΔH = enthalpy change
T = temperature in Kelvin
ΔS = entropy change
First, let's convert the temperature from Celsius to Kelvin:
T = 15 + 273.15 = 288.15 K
Now we can plug in the given values:
ΔG = 14.095 kJ/mol
ΔH = 46.510 kJ/mol
ΔS = 108.72 J/mol-K
Convert ΔG and ΔH to J/mol by multiplying each by 1000:
ΔG = 14.095 × 1000 = 14095 J/mol
ΔH = 43.51 × 1000 = 46510 J/mol
Now we can calculate the concentration of borax at equilibrium using the equation:
ΔG = -RTln(K)
Where:
R = gas constant (8.314 J/(mol·K))
T = temperature in Kelvin
ln = natural logarithm
K = equilibrium constant
Rearranging the equation to solve for K:
K = e^(-ΔG/RT)
K = e^(-14095/(8.314 × 288.15))
Calculating K using a scientific calculator:
K ≈ 6.3 × 10^(-11) mol/L
Note: The equilibrium constant K is the product of the concentrations of the products divided by the concentrations of the reactants, each raised to the power of their stoichiometric coefficient. In this case, borax has a stoichiometric coefficient of 1, so K is just the concentration of borax.
To determine when the borax begins to precipitate on the shore, we need to find the concentration of borax that exceeds the equilibrium concentration (K).
The molar concentration of borax is given as 0.02 M. We can compare this concentration with K to determine if precipitation occurs.
If the concentration of borax (0.02 M) is greater than the equilibrium concentration (K ≈ 6.3 × 10^(-11) mol/L), then precipitation will occur.
Since the question states that the lake is equally deep at all points, precipitation will occur once the level of the lake drops to the point where the concentration of borax reaches the threshold concentration.
Assuming the volume of the lake remains constant, we can calculate the level of the lake when borax will begin to precipitate:
Volume of the lake = πr^2h
Given:
Diameter of the lake (D) = 500 meters
Depth of the lake (h) = 50 meters
Radius of the lake (r) = D/2 = 500/2 = 250 meters
Volume of the lake = π × (250)^2 × 50 = 98174734 m^3
Now let's calculate the moles of borax that would be present in the lake at the given concentration:
Moles of borax = concentration × volume of the lake
Moles of borax = 0.02 mol/L × 98174734 m^3
Finally, we compare the moles of borax with the threshold concentration (K) to determine the level of the lake when borax will begin to precipitate:
If (0.02 mol/L × 98174734 m^3) > (6.3 × 10^(-11) mol/L), then precipitation will occur.
Simplifying the equation:
If (0.02 × 98174734) > 6.3 × 10^(-11), then precipitation will occur.
If (1.96 × 10^6) > 6.3 × 10^(-11), then precipitation will occur.
The inequality is true, so we can conclude that precipitation of borax will occur on the shore of the lake.