Four ice cubes at exactly 0 Celsius having a total mass of 55.0g are combined with 120g of water at 77 Celsius in an insulated container. If no heat is lost to the surroundings, what will be the final temperature of the mixture?
You have three sources of heat in and heat out.
1. heat to melt ice +
2. heat to raise T of melted ice to final T -
3. heat lost from 120g H2O
#1 + #2 + #3 = 0
(mass ice x heat fusion) + [mass melted ice x specific heat melted ice (H2O) x (Tfinal-Tinitial)] + [mass water already there x specific heat H2O x (Tfinal-Tinitial)] = 0
Substitute and solve for Tfinal.
To find the final temperature of the mixture, we can use the principle of conservation of energy. The heat lost by the hot water is equal to the heat gained by the ice cubes.
To solve this problem, we can use the equation:
(m1 x c1 x ΔT1) + (m2 x c2 x ΔT2) = 0
Where:
m1 = mass of the hot water (in grams)
c1 = specific heat capacity of water (4.18 J/g°C)
ΔT1 = change in temperature of the hot water (final temperature - initial temperature)
m2 = mass of the ice cubes (in grams)
c2 = specific heat capacity of ice (2.09 J/g°C)
ΔT2 = change in temperature of the ice cubes (final temperature - initial temperature)
In this case:
m1 = 120g (mass of the hot water)
c1 = 4.18 J/g°C (specific heat capacity of water)
ΔT1 = final temperature - initial temperature (Tf - 77°C)
m2 = 55.0g (mass of the ice cubes)
c2 = 2.09 J/g°C (specific heat capacity of ice)
ΔT2 = final temperature - initial temperature (Tf - 0°C)
Since the ice cubes are initially at 0°C, we can assume that their initial temperature is the same as the final temperature of the mixture (Tf).
Substituting the given values into the equation, we have:
(120g x 4.18 J/g°C x (Tf - 77°C)) + (55.0g x 2.09 J/g°C x (Tf - 0°C)) = 0
Now, we can solve for Tf. Let's simplify the equation:
(120 x 4.18 x Tf) - (120 x 4.18 x 77) + (55.0 x 2.09 x Tf) - (55.0 x 2.09 x 0) = 0
(120 x 4.18 Tf) - (120 x 4.18 x 77) + (55.0 x 2.09 Tf) = 0
Now, we can collect like terms:
(120 x 4.18 Tf) + (55.0 x 2.09 Tf) = (120 x 4.18 x 77)
496.8 Tf + 114.95 Tf = 366,288
611.75 Tf = 366,288
Now, we can solve for Tf:
Tf = 366,288 / 611.75
Tf ≈ 598.59°C
So, the final temperature of the mixture will be approximately 598.59°C.