A 5.00 g sample of aluminum pellets (specific heat capacity = 0.89 J/°C·g) and a 10.00 g sample of iron pellets (specific heat capacity = 0.45 J/°C·g) are heated to 100.0°C. The mixture of hot iron and aluminum is then dropped into 94.9 g of water at 24.2°C. Calculate the final temperature of the metal and water mixture, assuming no heat loss to the surroundings.
sum of heats gained is zero.
5*.89(Tf-100)+10*.45(Tf-100)+94.0(4.18)(Tf-24.2)=0
solve for Tf
Okay got it thank you!
To find the final temperature of the metal and water mixture, we can use the principle of conservation of energy. The heat lost by the metal pellets will be equal to the heat gained by the water.
The heat lost by the metal can be calculated using the formula:
Q = m × c × ΔT
Where:
Q = heat lost by the metal
m = mass of the metal pellets
c = specific heat capacity of the metal
ΔT = change in temperature of the metal
For the aluminum pellets:
m(aluminum) = 5.00 g
c(aluminum) = 0.89 J/°C·g
ΔT(aluminum) = ???
For the iron pellets:
m(iron) = 10.00 g
c(iron) = 0.45 J/°C·g
ΔT(iron) = ???
The heat gained by the water can be calculated using the same formula, but with the specific heat capacity of water:
Q = m × c × ΔT
Where:
Q = heat gained by the water
m = mass of the water
c = specific heat capacity of water (4.18 J/°C·g)
ΔT = change in temperature of the water
m(water) = 94.9 g
c(water) = 4.18 J/°C·g
ΔT(water) = ???
Since the heat lost by the metal is equal to the heat gained by the water, we can set up the equation:
Q(metal) = Q(water)
Using the formula for each component:
(m(aluminum) × c(aluminum) × ΔT(aluminum)) + (m(iron) × c(iron) × ΔT(iron)) = (m(water) × c(water) × ΔT(water))
Substituting the given values:
(5.00 g × 0.89 J/°C·g × ΔT(aluminum)) + (10.00 g × 0.45 J/°C·g × ΔT(iron)) = (94.9 g × 4.18 J/°C·g × ΔT(water))
Simplifying the equation:
4.45 ΔT(aluminum) + 4.50 ΔT(iron) = 396.682 ΔT(water)
Now, we can substitute the known values:
4.45 ΔT(aluminum) + 4.50 ΔT(iron) = 396.682 × (T(final) - 24.2)
Where T(final) is the final temperature of the metal and water mixture.
Since the metal and water will reach the same final temperature, ΔT(aluminum) = ΔT(iron) = T(final) - 100.0, we can further simplify the equation:
(4.45 + 4.50) (T(final) - 100) = 396.682 × (T(final) - 24.2)
8.95 T(final) - 895 = 396.682 T(final) - 9726.638
Rearranging the equation:
8.95 T(final) - 396.682 T(final) = -9726.638 + 895
-387.732 T(final) = -8831.638
Dividing both sides by -387.732:
T(final) ≈ 22.756 °C
Therefore, the final temperature of the metal and water mixture is approximately 22.756 °C.
To calculate the final temperature of the metal and water mixture, you can use the principle of conservation of energy. The energy gained by the metal pellets when they cool down will be equal to the energy lost by the water when it heats up.
First, let's calculate the energy gained by the metal pellets when they cool down. The formula to calculate heat energy is:
Q = m * c * ΔT
Where:
Q is the heat energy (in Joules)
m is the mass of the substance (in grams)
c is the specific heat capacity (in J/°C·g)
ΔT is the change in temperature (in °C)
For the aluminum pellets:
Q_aluminum = (5.00 g) * (0.89 J/°C·g) * (100.0°C - T_final)
For the iron pellets:
Q_iron = (10.00 g) * (0.45 J/°C·g) * (100.0°C - T_final)
Since the aluminum and iron pellets are in the same mixture, their energies will be the same:
Q_aluminum = Q_iron
Now, let's calculate the energy lost by the water when it heats up. The formula is the same:
Q_water = (94.9 g) * (4.18 J/°C·g) * (T_final - 24.2°C)
Since the energy gained by the metal pellets is equal to the energy lost by the water, we can set up the equation:
Q_aluminum = Q_water
(5.00 g) * (0.89 J/°C·g) * (100.0°C - T_final) = (94.9 g) * (4.18 J/°C·g) * (T_final - 24.2°C)
Now, you can solve this equation to find the final temperature of the metal and water mixture by isolating T_final:
(5.00 g) * (0.89 J/°C·g) * (100.0°C - T_final) = (94.9 g) * (4.18 J/°C·g) * (T_final - 24.2°C)
447.5°C - 4.45*T_final = 398.382*T_final - 9679.468
4.45*T_final + 398.382*T_final = 447.5°C + 9679.468
402.832*T_final = 10126.968
T_final = 25.13°C
Therefore, the final temperature of the metal and water mixture is approximately 25.13°C.