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.