In a crazy scenario, the Oylan Vinegar Company has 10,000 gallons of vinegar with 0.622% concentration. This vinegar solution cannot be sold because it's too weak, but the plant manager can't bear to discard it. OVC also has a large batch of vinegar that is 9.50% acetic acid by mass. Calculate how many gallons of the strong vinegar would need to be added to the 10,000 gallons of weak vinegar to obtain a solution that is the appropriate 5.10& acetic acid, by mass.
To solve this problem, we can use the concept of mass and concentration (% mass/volume).
Let's start by calculating the amount of acetic acid present in the weak vinegar solution.
Mass of acetic acid in the weak vinegar = (0.622/100) * (10,000 * 128) lbs
Note: We need to convert gallons to lbs, considering that the density of vinegar is approximately 8.34 lbs/gallon.
Now, let's determine the total mass of the weak vinegar solution.
Mass of weak vinegar = 10,000 * 128 lbs
Next, we need to find the mass of the strong vinegar that needs to be added.
To obtain a final solution with 5.10% acetic acid, by mass, we can set up the following equation:
(Mass of acetic acid in the weak vinegar + Mass of acetic acid in the strong vinegar) / Total mass of final solution = 5.10/100
Solving for the mass of the strong vinegar:
Mass of acetic acid in the strong vinegar = (5.10/100) * (Total mass of final solution)
Now, plug in the values:
(Mass of acetic acid in the weak vinegar + Mass of acetic acid in the strong vinegar) / Total mass of final solution = 5.10/100
(0.622/100) * (10,000 * 128 lbs) + Mass of acetic acid in the strong vinegar / (10,000 * 128 lbs + Mass of acetic acid in the strong vinegar) = 5.10/100
Simplify and solve for Mass of acetic acid in the strong vinegar:
(0.622 * 128 * 10,000 lbs) + (Mass of acetic acid in the strong vinegar) = (5.10/100) * (10,000 * 128 lbs + Mass of acetic acid in the strong vinegar)
Once you get the value for Mass of acetic acid in the strong vinegar, divide it by the concentration of the strong vinegar (9.50%) to determine the total mass of the strong vinegar that needs to be added.
Finally, divide the total mass of the strong vinegar by the density of vinegar (approximately 8.34 lbs/gallon) to obtain the number of gallons needed.
To solve this problem, we need to calculate the number of gallons of strong vinegar (9.50% acetic acid) that need to be added to the weak vinegar (0.622% acetic acid) in order to obtain a solution that is 5.10% acetic acid.
Let's break down the steps involved in solving this problem:
Step 1: Convert the percentages to decimal values.
The weak vinegar has a concentration of 0.622% acetic acid, which can be converted to 0.00622 as a decimal.
The strong vinegar has a concentration of 9.50% acetic acid, which can be converted to 0.095 as a decimal.
The desired concentration is 5.10% acetic acid, which can be converted to 0.051 as a decimal.
Step 2: Determine the amount of acetic acid in the weak vinegar.
Since we have 10,000 gallons of weak vinegar, we can calculate the amount of acetic acid in it by multiplying the volume by the concentration:
Amount of acetic acid in the weak vinegar = 10,000 gallons * 0.00622
Step 3: Determine the amount of acetic acid needed in the final solution.
To calculate the amount of acetic acid needed for the final solution, we use the equation:
Amount of acetic acid needed = total volume of the final solution * desired concentration
Since the total volume is the sum of the weak vinegar (10,000 gallons) and the strong vinegar (unknown amount), we can write the equation as:
Amount of acetic acid needed = (10,000 gallons + volume of strong vinegar) * 0.051
Step 4: Set up an equation to solve for the volume of strong vinegar.
Now we can set up an equation to solve for the volume of strong vinegar needed:
Amount of acetic acid in the weak vinegar + Amount of acetic acid in the strong vinegar = Amount of acetic acid needed in the final solution
10,000 gallons * 0.00622 + volume of strong vinegar * 0.095 = (10,000 gallons + volume of strong vinegar) * 0.051
Step 5: Solve the equation for the volume of strong vinegar.
Simplify the equation and solve for the volume of strong vinegar:
0.0622 + 0.095 * volume of strong vinegar = 0.051 * 10,000 + 0.051 * volume of strong vinegar
0.095 * volume of strong vinegar - 0.051 * volume of strong vinegar = 0.051 * 10,000 - 0.0622
(0.095 - 0.051) * volume of strong vinegar = 0.051 * 10,000 - 0.0622
0.044 * volume of strong vinegar = 510 - 0.0622
volume of strong vinegar = (510 - 0.0622) / 0.044
Step 6: Calculate the volume of strong vinegar needed.
Now we can calculate the volume of strong vinegar needed by substituting the values into the equation:
volume of strong vinegar = (510 - 0.0622) / 0.044
After performing the calculation, the volume of strong vinegar needed to be added to the weak vinegar to obtain the desired concentration of 5.10% acetic acid can be obtained.