you are working in your kitchen preparing lunch for your family. you have decided to make egg salad sandwiches and are boiling six eggs, each of mass 60.2g, in 0.85L of water at 100c. you wish to take all the eggs out of the boiling water and immediately place them in 22.9c water to cool them down to a comfortable tempreature to hold them and peel them. you decide that you wish the mixture of the water and the eggs to reach an equilibrium tempreature of 40.2c. after explaining this to a family member, she challenges you to determine exactly how much water (kg) you need at 22.9c to achieve your desired equilibrium tempreature. take the average specific heat of an egg over the expected tempreature range to be 3.27x10^3 J/kgc
let M equal the mass of H2O
(mass of eggs) * (specific heat of eggs) * (100 - 40.2) =
... M * (specific heat of H2O) * (40.2 - 22.9)
Well, it seems like you've got yourself into quite a boiling situation there! But don't worry, Clown Bot is here to help you out with a chuckle and a solution.
To determine how much water you need at 22.9°C to achieve your desired equilibrium temperature, we'll have to consider the heat transfer between the eggs and the water.
Now, based on the average specific heat of an egg (3.27x10^3 J/kg°C), we can calculate the amount of heat energy required to cool down each egg from 100°C to 40.2°C.
Heat Energy = mass × specific heat × temperature change
Heat Energy = 60.2g × (3.27x10^3 J/kg°C) × (100°C - 40.2°C)
But hang on, we need to convert grams to kilograms. So that's 60.2g ÷ 1000 = 0.0602kg.
Now, Heat Energy = 0.0602kg × (3.27x10^3 J/kg°C) × (100°C - 40.2°C)
Calculating that will give us the total heat energy that needs to be transferred from all six eggs to the water.
Now, we can use the equation for heat transfer:
Heat Energy (eggs) = Heat Energy (water)
0.0602kg × (3.27x10^3 J/kg°C) × (100°C - 40.2°C) = mass (water) × specific heat (water) × temperature change
We know the specific heat of water (4.18x10^3 J/kg°C) and the initial and final temperatures:
mass (water) × (4.18x10^3 J/kg°C) × (40.2°C - 22.9°C) = 0.0602kg × (3.27x10^3 J/kg°C) × (100°C - 40.2°C)
Now, let's solve for the mass of the water:
mass (water) = (0.0602kg × (3.27x10^3 J/kg°C) × (100°C - 40.2°C)) ÷ ((4.18x10^3 J/kg°C) × (40.2°C - 22.9°C))
*Calculations are important, but don't forget to add a pinch of humor to your kitchen adventures!*
And voilà! After you've worked out the math and nailed the calculations, you'll have the exact amount of water (in kilograms) you need at 22.9°C to achieve that desired equilibrium temperature. Good luck, my culinary friend!
To determine the exact amount of water needed at 22.9°C to achieve the desired equilibrium temperature, we can use the principle of conservation of energy.
First, let's calculate the initial energy of the eggs and boiling water. We'll assume the specific heat capacity of water to be 4.18 J/g°C.
1. Calculate the heat gained by the boiling water:
Q1 = mass of boiling water × specific heat capacity of water × (final temperature - initial temperature)
Q1 = 0.85 kg × 4.18 J/g°C × (40.2°C - 100°C)
2. Calculate the heat lost by the eggs:
Q2 = mass of eggs × specific heat capacity of eggs × (final temperature - initial temperature)
Q2 = 6 eggs × 60.2 g/egg × (40.2°C - 100°C) × (3.27 × 10^3 J/kg°C)
Since energy conservation implies that Q1 = Q2, we can set these two equations equal to each other and solve for the mass of water needed.
0.85 kg × 4.18 J/g°C × (40.2°C - 100°C) = 6 eggs × 60.2 g/egg × (40.2°C - 22.9°C) × (3.27 × 10^3 J/kg°C)
Now we can solve for the mass of water (in kg):
0.85 × 4.18 × (40.2 - 100) = 6 × 60.2 × (40.2 - 22.9) × 3.27 x 10^3
Let's calculate this:
0.85 × 4.18 × (40.2 - 100) = 6 × 60.2 × (40.2 - 22.9) × 3.27 x 10^3
-67.546 = 6 x 60.2 x 17.3 x 3.27 x 10^3
Dividing both sides by (6 x 60.2 x 17.3 x 3.27 x 10^3):
-67.546 / (6 x 60.2 x 17.3 x 3.27 x 10^3) = 1 kg
Therefore, to achieve the desired equilibrium temperature of 40.2°C, approximately 1 kg of water at 22.9°C is needed.
To determine how much water you need at 22.9°C to achieve the desired equilibrium temperature of 40.2°C, you can use the principle of energy conservation.
First, let's calculate the energy required to heat up the eggs and water to the final equilibrium temperature:
1. Calculate the energy required to heat up the eggs:
Energy_egg = mass_egg * specific_heat_egg * (equilibrium_temp - initial_temp_egg)
Here, mass_egg = 6 eggs * 60.2g/egg = 361.2g = 0.3612kg (converting grams to kilograms)
specific_heat_egg = 3.27x10^3 J/kg°C (given in the question)
initial_temp_egg = 100°C (boiling point)
Therefore,
Energy_egg = 0.3612kg * 3.27x10^3 J/kg°C * (40.2°C - 100°C)
2. Calculate the energy released by the eggs when they cool down:
Energy_released = mass_egg * specific_heat_egg * (equilibrium_temp - final_temp_egg)
Here, final_temp_egg = 22.9°C (temperature of the water in which the eggs will be placed to cool down)
Therefore,
Energy_released = 0.3612kg * 3.27x10^3 J/kg°C * (40.2°C - 22.9°C)
3. Calculate the energy required to heat up the water:
Energy_water = mass_water * specific_heat_water * (equilibrium_temp - initial_temp_water)
Here, mass_water = volume_water * density_water
volume_water = 0.85L (given in the question)
density_water = 1kg/L (density of water)
specific_heat_water = 4.18x10^3 J/kg°C (specific heat of water)
Therefore,
mass_water = 0.85L * 1kg/L = 0.85kg
Energy_water = 0.85kg * 4.18x10^3 J/kg°C * (40.2°C - 100°C)
Now, we can set up the energy conservation equation:
Energy_egg released + Energy_released + Energy_water = 0
0.3612kg * 3.27x10^3 J/kg°C * (40.2°C - 100°C) + 0.3612kg * 3.27x10^3 J/kg°C * (40.2°C - 22.9°C) + 0.85kg * 4.18x10^3 J/kg°C * (40.2°C - 100°C) = 0
Simplifying the equation will give you the answer for the mass of water required at 22.9°C.