A 50-g ice cube at is placed in an equal mass of water. What must the initial water temperature be if the final mixture still contains equal amounts of ice and water?
To solve this problem, we need to understand the concept of heat transfer and the specific heat capacities of water and ice.
The equation for heat transfer is:
Q = mcΔT
Where:
Q is the amount of heat transferred
m is the mass of the substance
c is the specific heat capacity of the substance
ΔT is the change in temperature
Since the final mixture contains equal amounts of ice and water, the heat transferred to the ice will be equal to the heat transferred to the water.
Let's calculate the heat transferred to the ice and water separately:
1. Heat transferred to the ice:
For ice, the specific heat capacity (c) is 2.09 J/g°C.
The mass of the ice is 50 g.
The change in temperature (ΔT) for the ice is the final temperature of the mixture minus the melting point of ice (0°C).
So, for the ice:
Q_ice = m_ice * c_ice * ΔT_ice
Q_ice = 50 g * 2.09 J/g°C * (final temperature - 0°C)
2. Heat transferred to the water:
For water, the specific heat capacity (c) is 4.18 J/g°C.
The mass of the water is also 50 g. The change in temperature (ΔT) for the water is the final temperature of the mixture minus the initial water temperature.
So, for the water:
Q_water = m_water * c_water * ΔT_water
Q_water = 50 g * 4.18 J/g°C * (final temperature - initial water temperature)
Since the heat transferred to the ice is equal to the heat transferred to the water, we can set these two equations equal to each other and solve for the initial water temperature:
50 g * 2.09 J/g°C * (final temperature - 0°C) = 50 g * 4.18 J/g°C * (final temperature - initial water temperature)
Simplifying the equation:
2.09 * (final temperature - 0) = 4.18 * (final temperature - initial water temperature)
Now, you can solve for the initial water temperature by rearranging the equation:
2.09 * final temperature = 4.18 * final temperature - 4.18 * initial water temperature
2.09 * final temperature - 4.18 * final temperature = -4.18 * initial water temperature
-2.09 * final temperature = -4.18 * initial water temperature
Now divide both sides by -4.18 to isolate the initial water temperature:
initial water temperature = (-2.09 * final temperature) / (-4.18)
Simplifying further:
initial water temperature = 0.5 * final temperature
Therefore, if the final mixture contains equal amounts of ice and water, the initial water temperature must be half of the final temperature.