25.0 mL of ethanol (density = 0.789 g/mL) initially at 5.5°C is mixed with 32.3 mL of water (density = 1.0 g/mL) initially at 25.3°C in an insulated beaker. Assuming that no heat is lost, what is the final temperature of the mixture?
You will need to look up the specific heat of H2O and the specific heat of ethanol.
heat lost by H2O + heat gained by ethanol = 0
First, use density to convert volumes to grams of each. mass = volume x density
[mass H2O x specific heat H2O x (Tfinal-Tinitial)] + [mass ethanol x specific heat ethanol x (Tfinal-Tinitial)]=0
Substitute and solve for Tfinal.
To find the final temperature of the mixture, we need to use the principle of conservation of energy, which states that the total energy of a closed system remains constant.
First, let's calculate the initial energy of the ethanol and water separately using the formula:
Energy = mass * specific heat * change in temperature
For ethanol:
Mass of ethanol = density * volume = 0.789 g/mL * 25.0 mL = 19.725 g
Change in temperature of ethanol = final temperature - initial temperature = T - 5.5
For water:
Mass of water = density * volume = 1.0 g/mL * 32.3 mL = 32.3 g
Change in temperature of water = final temperature - initial temperature = T - 25.3
Now, since the final temperature of the mixture will be the same for both substances, we can set up the equation:
Energy of ethanol + Energy of water = Energy of mixture
(mass of ethanol) * (specific heat of ethanol) * (T - 5.5) + (mass of water) * (specific heat of water) * (T - 25.3) = (mass of ethanol + mass of water) * (specific heat of mixture) * (T - final temperature)
Substituting the known values:
(19.725 g) * (2.44 J/g°C) * (T - 5.5) + (32.3 g) * (4.18 J/g°C) * (T - 25.3) = (19.725 g + 32.3 g) * (2.071 J/g°C) * (T - final temperature)
Now, we can simplify this equation and solve for the final temperature.
(19.725 * 2.44 * T - 19.725 * 2.44 * 5.5) + (32.3 * 4.18 * T - 32.3 * 4.18 * 25.3) = (19.725 + 32.3) * 2.071 * T - (19.725 + 32.3) * 2.071 * final temperature
(48.02962 * T - 53.87534) + (135.05012 * T - 342.89098) = (52.025 * T) - (52.025 * final temperature)
Now, we can combine like terms:
183.07974 * T - 396.76632 = 52.025 * T - 52.025 * final temperature
Rearranging the equation:
131.05474 * T - 52.025 * T = 52.025 * final temperature - 396.76632
79.02974 * T = 52.025 * final temperature - 396.76632
Now, let's rearrange the equation further to find the final temperature:
52.025 * final temperature = 79.02974 * T + 396.76632
final temperature = (79.02974 * T + 396.76632) / 52.025
Now, we can substitute the given initial temperature of the ethanol, T = 5.5°C, into the equation to find the final temperature:
final temperature = (79.02974 * 5.5 + 396.76632) / 52.025
final temperature ≈ 31.63°C
Therefore, the final temperature of the mixture is approximately 31.63°C.