If you have 290.0 mL of water at 25.00 °C and add 120.0 mL of water at 95.00 °C, what is the final temperature of the mixture? Use 1.00 g/mL as the density of water.
heat gained by cool water + heat lost by warm water = 0
[mass cool water x specific heat x (Tfinal-Tinitial)] + [mass warm water x specific heat x (Tfinal-Tinitial)] = 0
Substitute and solve for Tfinal.
To find the final temperature of the mixture, we can use the principle of conservation of energy. The total energy of the system should remain constant.
To calculate the final temperature, we can use the formula:
m1 * c1 * (Tf - T1) = m2 * c2 * (T2 - Tf)
where:
m1 and m2 are the masses of the substances (water in this case),
c1 and c2 are the specific heat capacities of the substances (water in this case),
Tf is the final temperature,
T1 is the initial temperature of one substance (25.00 °C in this case),
and T2 is the initial temperature of the other substance (95.00 °C in this case).
Step 1: Calculate the masses of the substances.
The mass of water is equal to its density times its volume.
Mass of water at 25.00 °C = density * volume = (1.00 g/mL) * 290.0 mL = 290.0 g
Mass of water at 95.00 °C = (1.00 g/mL) * 120.0 mL = 120.0 g
Step 2: Calculate the specific heat capacities.
The specific heat capacity of water is approximately 4.18 J/(g·°C).
c1 = c2 = 4.18 J/(g·°C)
Step 3: Substitute the known values into the formula.
(290.0 g) * (4.18 J/(g·°C)) * (Tf - 25.00 °C) = (120.0 g) * (4.18 J/(g·°C)) * (95.00 °C - Tf)
Step 4: Solve for Tf.
(120.0 g) * (4.18 J/(g·°C)) * (95.00 °C - Tf) = (290.0 g) * (4.18 J/(g·°C)) * (Tf - 25.00 °C)
Simplify and solve for Tf:
(120.0 g) * (4.18 J/(g·°C)) * 95.00 °C - (120.0 g) * (4.18 J/(g·°C)) * Tf = (290.0 g) * (4.18 J/(g·°C)) * Tf - (290.0 g) * (4.18 J/(g·°C)) * 25.00 °C
(120.0 g) * (4.18 J/(g·°C)) * 95.00 °C + (290.0 g) * (4.18 J/(g·°C)) * 25.00 °C = (120.0 g + 290.0 g) * (4.18 J/(g·°C)) * Tf
[(120.0 g) * (4.18 J/(g·°C)) * 95.00 °C + (290.0 g) * (4.18 J/(g·°C)) * 25.00 °C] / [(120.0 g + 290.0 g) * (4.18 J/(g·°C))]
Tf ≈ 45.5 °C
Therefore, the final temperature of the mixture is approximately 45.5 °C.