Calculate the final temperature of a mixture of 350g of ice cubes at -18 degrees C and 237 g of water at 100 degrees C
To calculate the final temperature of a mixture, we need to use the principle of energy conservation, specifically the equation known as the heat transfer equation. This equation states that the energy gained by one substance is equal to the energy lost by another substance when they come into contact and reach thermal equilibrium. In this case, the energy gained by the ice cubes will be equal to the energy lost by the water.
The equation for heat transfer is given by:
Q = mcΔT
Where:
Q is the heat gained or lost by the substance (in joules)
m is the mass of the substance (in grams)
c is the specific heat capacity of the substance (in J/g°C)
ΔT is the change in temperature (in °C)
For ice, the specific heat capacity (c) is 2.09 J/g°C, and for water, it is 4.18 J/g°C.
Let's calculate the energy gained by the ice cubes and the energy lost by the water.
Energy gained by the ice cubes:
Q_ice = m_ice * c_ice * ΔT_ice
Where:
m_ice = 350 g (mass of ice cubes)
c_ice = 2.09 J/g°C (specific heat capacity of ice)
ΔT_ice = Tf - Ti (final temperature - initial temperature of ice)
Ti = -18°C (initial temperature of ice)
Energy lost by the water:
Q_water = m_water * c_water * ΔT_water
Where:
m_water = 237 g (mass of water)
c_water = 4.18 J/g°C (specific heat capacity of water)
ΔT_water = Tf - Ti (final temperature - initial temperature of water)
Ti = 100°C (initial temperature of water)
Since energy gained = energy lost, we have:
Q_ice = Q_water
m_ice * c_ice * ΔT_ice = m_water * c_water * ΔT_water
Substituting the values:
350 * 2.09 * (Tf - (-18)) = 237 * 4.18 * (Tf - 100)
Now we can solve for the final temperature (Tf).
548.3 * (Tf + 18) = 987.54 * (Tf - 100)
548.3Tf + 9888.6 = 987.54Tf - 98754
987.54Tf - 548.3Tf = 98754 - 9888.6
439.24Tf = 88865.4
Tf = 88865.4 / 439.24
Tf ≈ 202.2°C
Therefore, the final temperature of the mixture is approximately 202.2°C.
add the following heats needed:
ice going to 0 deg: massice*Cice*(18)
melted ice to finaltemp: massice*cwater(Tf-0)
add the heat from the water:
masswater*cwater(Tf-100)
and finally, add all these heats and set it equal to zero, and solve for Tf.
Sum heats gained=0 (remember, if somthing lost heat, it will work our to be a negative heat gained).