Assuming that seawater has a total ion concentration (aka colligative molarity) of 1.10 M conc, calculate how many liters of seawater are needed to produce 68.7 L of fresh water at 20 C with an applied pressure of 51.0 bar.
To calculate the number of liters of seawater needed to produce 68.7 L of fresh water, we need to use the colligative properties of seawater. In this case, we'll focus on the osmotic pressure.
Step 1: Convert the given temperature from Celsius to Kelvin:
T(K) = T(C) + 273.15
T(K) = 20 + 273.15 = 293.15 K
Step 2: Convert the given pressure from bar to atm:
P(atm) = P(bar) / 1.01325
P(atm) = 51.0 / 1.01325 = 50.320 atm
Step 3: Calculate the osmotic pressure using the colligative property equation for osmotic pressure:
π = i * MRT
Where:
π = osmotic pressure (atm)
i = van't Hoff factor (assumed to be 1 for seawater)
M = total ion concentration (molarity)
R = ideal gas constant (0.0821 L·atm/(K·mol))
T = temperature (Kelvin)
Substituting in the given values:
π = 1.10 * 0.0821 * 293.15 / 1
π = 26.501 atm
Step 4: Calculate the volume of seawater using the osmotic pressure:
V(seawater) = V(fresh water) * (π + P(free)) / P(free)
Where:
V(seawater) = volume of seawater (L)
V(fresh water) = volume of fresh water (L)
π = osmotic pressure (atm)
P(free) = pressure of free fresh water (atm)
Assuming the pressure of free fresh water is equal to the applied pressure:
V(seawater) = 68.7 * (26.501 + 50.320) / 50.320
V(seawater) = 68.7 * 76.821 / 50.320
V(seawater) = 105.526 L
Therefore, approximately 105.526 L of seawater are needed to produce 68.7 L of fresh water at 20°C with an applied pressure of 51.0 bar.
To solve this problem, we need to use the concept of reverse osmosis. Reverse osmosis is a process in which pressure is applied to a solution to overcome the osmotic pressure and separate the solvent from solute particles.
In this case, we want to calculate the volume of seawater needed to produce 68.7 L of fresh water. The pressure applied during reverse osmosis is given as 51.0 bar.
To begin, we'll use the equation for osmotic pressure:
π = n/VRT
where:
π is the osmotic pressure,
n is the number of moles of solute,
V is the volume in liters,
R is the ideal gas constant (0.0821 L*atm/(K*mol)),
and T is the temperature in Kelvin.
The osmotic pressure (π) is equal to the applied pressure (51.0 bar) because we are using reverse osmosis.
We can rearrange the equation to solve for n (the number of moles of solute):
n = π x V x RT
Given that the total ion concentration of the seawater is 1.10 M, we can convert that to moles per liter by multiplying by the volume (V) of the seawater. Let's call the volume of seawater needed x liters.
n = (1.10 M) x (x L) = 1.10x moles
Now we can substitute the values into the equation:
1.10x moles = (51.0 bar) x (x L) x (0.0821 L*atm/(K*mol)) x (293.15 K)
Simplifying:
1.10x = (51.0) x (x) x (0.0821) x (293.15)
Now we can solve for x (the volume of seawater needed):
x = [(51.0) x (x) x (0.0821) x (293.15)] / 1.10
Using a calculator, we can solve for x:
x ≈ 688.26 L
Therefore, approximately 688.26 liters of seawater are needed to produce 68.7 L of fresh water under the given conditions.