A 68-g ice cube at 0°C is placed in 796 g of water at 30°C. What is the final temperature of the mixture?
The heat gained by the ice cube in melting and heating up to final temperature T equals the heat lost by the original liquid water.
68*(80 + T) = 796*(30 - T)
80 cal/g is the heat of fusion of ice.
Solve for T.
5440 + 68T = 23,880 -796 T
864 T = 18440
T = 21.3 C
To find the final temperature of the mixture, we can use the principle of conservation of heat energy. The heat gained by the water should be equal to the heat lost by the ice, assuming no heat is lost to the surroundings.
The formula for the heat gained or lost is given by:
Q = mcΔT
Where:
Q is the heat gained or lost
m is the mass of the substance
c is the specific heat capacity of the substance
ΔT is the change in temperature
For the water, the heat gained can be calculated as:
Q_water = m_water * c_water * ΔT_water
For the ice, the heat lost can be calculated as:
Q_ice = m_ice * c_ice * ΔT_ice
Since the heat lost by the ice is equal to the heat gained by the water, we can set up an equation:
Q_water = -Q_ice
m_water * c_water * ΔT_water = -m_ice * c_ice * ΔT_ice
We can rearrange the equation to solve for the final temperature of the mixture:
ΔT_water / ΔT_ice = -m_ice * c_ice / (m_water * c_water)
Now let's substitute the given values into the equation:
m_ice = 68 g
c_ice = 2.09 J/g°C (specific heat capacity of ice)
m_water = 796 g
c_water = 4.18 J/g°C (specific heat capacity of water)
ΔT_water / ΔT_ice = -(68 g * 2.09 J/g°C) / (796 g * 4.18 J/g°C)
Calculating the value, we get:
ΔT_water / ΔT_ice = -0.044
Since ΔT_water is the final temperature minus the initial temperature (30°C), and ΔT_ice is the final temperature minus the melting point of ice (0°C), we can substitute the values back into the equation:
(30°C - T_final) / (T_final - 0°C) = -0.044
Now, we can solve the equation for the final temperature (T_final). Multiplying both sides by (T_final - 0°C):
30°C - T_final = -0.044 * T_final
Simplifying the equation:
30°C = -0.044 * T_final + T_final
Combining like terms:
30°C = 0.956 * T_final
Dividing both sides by 0.956:
T_final = 31.38°C
Therefore, the final temperature of the mixture is approximately 31.38°C.