What mass of water at 22.0 °C must be allowed to come to thermal equilibrium with a 1.66-kg cube of aluminum initially at 1.50 multiplied by 102 °C to lower the temperature of the aluminum to 57.1 °C? Assume any water turned to stream subsequently recondenses.
Recheck the initial temperature of the aluminum. It is probably 1.50*10^2 (or 150) degrees, not 1.50 multiplied by 102.
Set the heat loss from the aluminum equal to the heat gained by the water, and slve fr the mass of water. You will need to know the aluminum specific heat.
To solve this problem, we need to make use of the equation for heat transfer:
Q = mcΔT
Where:
Q = heat transfer (in joules)
m = mass of the substance (in kilograms)
c = specific heat capacity of the substance (in joules per kilogram per degree Celsius)
ΔT = change in temperature
First, we need to find the heat transfer from the aluminum cube as it cools down from 102 °C to 57.1 °C:
Q_aluminum = (1.66 kg) * c_aluminum * ΔT_aluminum
The specific heat capacity of aluminum is approximately 900 J/(kg°C). Therefore:
Q_aluminum = (1.66 kg) * (900 J/(kg°C)) * (57.1 °C - 102 °C)
Next, we need to find the heat transfer to the water as it heats up from 22.0 °C to the final temperature, which is 57.1 °C:
Q_water = (m_water) * c_water * ΔT_water
We need to find the mass of water, so let's rearrange the equation:
m_water = Q_water / (c_water * ΔT_water)
Now, we substitute the known values:
m_water = Q_aluminum / (c_water * ΔT_water)
Finally, we can calculate the mass of water:
m_water = (Q_aluminum) / ((m_water) * c_water * ΔT_water)
To solve this equation, we need to make an assumption about the mass of the water. Let's assume a mass, calculate the values, and iterate if necessary until we find a solution.
Let's assume a mass of water (m_water) and substitute the values into the equation. We'll keep adjusting the assumed mass until we find a solution.
Once we have solved the equation, we will have the mass of water required to reach thermal equilibrium with the aluminum cube.