How many milliliters of water at 25.0°C with a density of 0.997 g/mL must be mixed with 165 mL of coffee at 91.0°C so that the resulting combination will have a temperature of 62.1°C? Assume that coffee and water have the same density and the same specific heat (4.18 J/g·°C) across the temperature range.
To solve this problem, we will use the equation:
(q1 = mcΔT1) + (q2 = mcΔT2) = 0
where:
q1 = heat gained by water
q2 = heat lost by coffee
m = mass of water or coffee (in grams)
c = specific heat of water or coffee (in J/g·°C)
ΔT1 = change in temperature for water
ΔT2 = change in temperature for coffee
We need to calculate the mass of water needed to mix with the coffee.
Given:
Density of water = 0.997 g/mL
Volume of coffee = 165 mL
Temperature of water = 25.0°C
Density of coffee = density of water
Step 1: Calculate the mass of coffee.
Mass of coffee = volume x density
Mass of coffee = 165 mL x 0.997 g/mL
Mass of coffee = 164.505 g
Step 2: Calculate q1 for water.
q1 = mcΔT1
q1 = mass of water x specific heat x ΔT1
q1 = (mass of water) x (4.18 J/g·°C) x (62.1°C - 25.0°C)
q1 = (mass of water) x (4.18 J/g·°C) x (37.1°C)
Step 3: Calculate q2 for coffee.
q2 = mcΔT2
q2 = (164.505 g) x (4.18 J/g·°C) x (62.1°C - 91.0°C)
q2 = (164.505 g) x (4.18 J/g·°C) x (-28.9°C)
Step 4: Set up the equation.
q1 + q2 = 0
(mass of water) x (4.18 J/g·°C) x (37.1°C) + (164.505 g) x (4.18 J/g·°C) x (-28.9°C) = 0
Step 5: Solve for the mass of water.
(mass of water) x (4.18 J/g·°C) x (37.1°C) = (164.505 g) x (4.18 J/g·°C) x (-28.9°C)
(mass of water) x (37.1) = (164.505) x (-28.9)
(mass of water) = (164.505) x (-28.9) / (37.1) (in grams)
By substituting the values, we get:
(mass of water) = -4124.3145 g
However, mass cannot be negative, so we need to take the absolute value:
(mass of water) ≈ 4124.3145 g
Therefore, approximately 4124.3145 grams (or mL, as density equals 1 g/mL) of water at 25.0°C with a density of 0.997 g/mL must be mixed with 165 mL of coffee at 91.0°C to achieve a resulting temperature of 62.1°C.
To solve this problem, we need to use the principle of energy conservation, which states that the heat lost by the hot coffee will be equal to the heat gained by the cold water. The formula we can use is:
Qcoffee + Qwater = 0
where Qcoffee is the heat gained by the coffee, and Qwater is the heat gained by the water.
The formula to calculate the heat gained or lost is:
Q = mcΔT
where Q is the heat gained or lost, m is the mass of the substance, c is the specific heat capacity, and ΔT is the change in temperature.
1. Calculate the heat gained by the coffee:
Qcoffee = mcoffee * ccoffee * ΔTcoffee
We are given:
mcoffee = 165 mL (The density of coffee is not given, but since we are assuming coffee and water have the same density, we can assume it is also 0.997 g/mL.)
ccoffee = 4.18 J/g·°C
ΔTcoffee = Tfinal - Tcoffee
= 62.1°C - 91.0°C
Convert mL to grams:
mcoffee = 165 mL * 0.997 g/mL
Calculate the heat gained by the coffee.
2. Calculate the heat gained by the water:
Qwater = mwater * cwater * ΔTwater
We are asked to find the mass of water (mwater) that needs to be added. Let's assume it is x grams.
x grams of water will occupy x mL since the density of water is also 0.997 g/mL.
So, mwater = x grams
The change in temperature of the water is:
ΔTwater = Tfinal - Twater
= 62.1°C - 25.0°C
Calculate the heat gained by the water.
3. Set up the energy conservation equation:
Qcoffee + Qwater = 0
Since energy is conserved, the heat gained by the coffee must be equal to the heat gained by the water.
4. Solve for x:
Qcoffee = -Qwater
Substitute the values we calculated for Qcoffee and Qwater.
5. Solve for x and convert it to milliliters (since mass and density are related by mass = volume * density):
x grams = x mL
x is the mass (in grams) of water needed to be added. Convert it to milliliters.
Thus, the final answer will be the value of x, which represents the number of milliliters of water that needs to be added to the coffee.
heat gained by cold water + heat lost by hot coffee = 0
[mass cold H2O x specific heat H2O x (Tfinal-Tinitial)] + [mass coffee x specific heat coffee x (Tfinal-Tinitial) = 0
Substitute and solve for mass cold H2O, then convert to mL.
mass = volume x density.
You know density and mass, solve for volume in mL.