If 51.6 mL of ethanol (density = 0.789 g/mL) initially at 6.4°C is mixed with 48.9 mL of water (density = 1.0 g/mL) initially at 29.1°C in an insulated beaker, and assuming that no heat is lost, what is the final temperature of the mixture?
use density to convert volume ethanol and volume H2O to mass in grams. Then
[mass ethanol x specific heat ethanol x (Tfinal-Tinitial)] + [mass H2O x specific heat H2O x (Tfinal-Tinitial)] = 0.
Substitute and solve for Tfinal. You must look up specific heat ethanol and H2O.
To solve this problem, we can use the principle of conservation of energy. The heat gained by the cooler water will be equal to the heat lost by the warmer ethanol. We can use the formula:
Q = m * c * ΔT
Where:
Q = Heat energy
m = Mass
c = Specific heat capacity
ΔT = Change in temperature
First, let's calculate the heat gained by the water. We know that the density of water is 1.0 g/mL, and the volume of water is 48.9 mL. Therefore, the mass of water is:
mass_water = density_water * volume_water
mass_water = 1.0 g/mL * 48.9 mL
mass_water = 48.9 g
Next, we need to calculate the heat lost by the ethanol. The density of ethanol is 0.789 g/mL, and the volume of ethanol is 51.6 mL. Therefore, the mass of ethanol is:
mass_ethanol = density_ethanol * volume_ethanol
mass_ethanol = 0.789 g/mL * 51.6 mL
mass_ethanol = 40.68 g
Now, let's calculate the heat gained by the water using the specific heat capacity of water, which is approximately 4.18 J/g°C:
Q_water = mass_water * specific_heat_water * ΔT_water
We need to find the change in temperature for the water. The initial temperature of water is 29.1°C, and the final temperature of the mixture is unknown. Therefore, the change in temperature for water is:
ΔT_water = final_temperature - initial_temperature_water
Similarly, let's calculate the heat lost by the ethanol using the specific heat capacity of ethanol, which is approximately 2.44 J/g°C:
Q_ethanol = mass_ethanol * specific_heat_ethanol * ΔT_ethanol
We need to find the change in temperature for the ethanol. The initial temperature of the ethanol is 6.4°C, and the final temperature of the mixture is unknown. Therefore, the change in temperature for the ethanol is:
ΔT_ethanol = final_temperature - initial_temperature_ethanol
Since no heat is lost, the heat gained by the water must be equal to the heat lost by the ethanol. Therefore:
Q_water = Q_ethanol
Using the above equations:
mass_water * specific_heat_water * ΔT_water = mass_ethanol * specific_heat_ethanol * ΔT_ethanol
Now, let's substitute the values we have:
48.9 g * 4.18 J/g°C * (final_temperature - 29.1°C) = 40.68 g * 2.44 J/g°C * (final_temperature - 6.4°C)
Simplifying the equation, we can solve for the final_temperature.