A 3.40 kg block of copper at a temperature of 74 °C is dropped into a bucket containing a mixture of ice and water whose total mass is 1.20 kg. When thermal equilibrium is reached the temperature of the water is 14 °C. How much ice was in the bucket before the copper block was placed in it?(ci = 2000 J/(kg.°C), cw = 4186 J/(kg.°C), Lf=3.35 × 105 J/kg, Lv=2.26 × 106 J/kg, ccopper = 387 J/(kg.°C). Neglect the heat capacity of the bucket.)
To solve this problem, we need to determine the amount of ice that was in the bucket before the copper block was placed in it. We can use the principle of energy conservation to find the solution.
First, let's determine the heat transfer between the copper block and the water/ice mixture.
The heat transfer can be calculated using the equation:
Q = m * c * ΔT
Where:
Q - heat transfer
m - mass
c - specific heat capacity
ΔT - change in temperature
For the copper block, the heat transfer is given by:
Qcopper = mcopper * ccopper * (Tfinal - Tinitial)
Substituting the given values into the equation:
Qcopper = 3.40 kg * 387 J/(kg.°C) * (14 °C - 74 °C)
Simplifying:
Qcopper = 3.40 kg * 387 J/(kg.°C) * (-60 °C)
Qcopper = -74904 J
Note that the negative sign indicates heat loss from the copper block.
Next, let's determine the heat transfer between the water/ice mixture and the copper block.
The heat transfer can be divided into two parts: the heat transfer to melt the ice and the heat transfer to warm up the resulting water.
The heat transfer to melt the ice is given by:
Qice = mice * Lf
Where:
Qice - heat transfer to melt the ice
mice - mass of the ice
Lf - latent heat of fusion for ice
Substituting the given values into the equation:
Qice = mice * 3.35 × 10^5 J/kg
The heat transfer to warm up the resulting water is given by:
Qwater = mwater * cw * ΔT
Where:
Qwater - heat transfer to warm up the water
mwater - mass of the water
cw - specific heat capacity of water
ΔT - change in temperature
Substituting the given values into the equation:
Qwater = mwater * cw * (Tfinal - 0 °C)
Note that we assume the temperature of the ice is 0 °C since it is in the process of melting.
Now, let's apply the principle of energy conservation:
Qcopper = Qice + Qwater
Substituting the calculated values:
-74904 J = mice * 3.35 × 10^5 J/kg + (1.20 kg - mice) * 4186 J/(kg.°C) * (14 °C - 0 °C)
Simplifying:
-74904 J = mice * 3.35 × 10^5 J/kg + (1.20 kg - mice) * 4186 J/(kg.°C) * 14 °C
Solving this equation will give us the mass of the ice in the bucket before the copper block was placed in it.