You are having a garden party and are preparing a punch bowl. You add one pound of ice at -20°C to 5.00 gallons of punch at 25°C. Since the punch is mostly water, you can assume it has the same specific heat as water. What is the final temperature of the punch?
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25.1 degrees celsius
Is it 5 degrees Celsius?
To determine the final temperature of the punch, we can use the principle of heat transfer. The equation we will use is:
Q(m1c1ΔT1) = Q(m2c2ΔT2)
Where:
Q represents the heat transferred
m represents the mass
c represents the specific heat capacity
ΔT represents the change in temperature
Subscripts 1 and 2 represent the initial and final states, respectively.
In this scenario, the ice will transfer heat to the punch until thermal equilibrium is reached. The ice will warm up (ΔT1) from -20°C to the final temperature, while the punch will cool down (ΔT2) from 25°C to the final temperature. Since the punch is mostly water, we can approximate its specific heat capacity as the specific heat capacity of water, which is about 4.184 J/g°C.
Given:
m1 (mass of ice) = 1 lb ≈ 454 g
c1 (specific heat capacity of ice) = 2.09 J/g°C
ΔT1 (change in temperature of ice) = Tf (final temperature of punch) - (-20°C) = Tf + 20°C
m2 (mass of punch) = 5.00 gallons ≈ 18.93 L ≈ 18930 g (since 1 L ≈ 1000 g for water)
c2 (specific heat capacity of punch) = 4.184 J/g°C
ΔT2 (change in temperature of punch) = 25°C - Tf
Next, we can rewrite the equation and solve for Tf:
Q(m1c1ΔT1) = Q(m2c2ΔT2)
(m1c1ΔT1) = (m2c2ΔT2)
(m1c1(Tf + 20°C)) = (m2c2(25°C - Tf))
(454 g)(2.09 J/g°C)(Tf + 20°C) = (18930 g)(4.184 J/g°C)(25°C - Tf)
Simplifying the equation:
(945.86 J/°C)(Tf + 20°C) = (79183.52 J/°C)(25°C - Tf)
945.86 Tf + 18917.2 J + 1895.74°C = 1979597.4 J - 79183.52 Tf
Rearranging the equation:
945.86 Tf + 79183.52 Tf = 1979597.4 J - 18917.2 J - 1895.74°C
Combining like terms:
80129.38 Tf = 1953784.46 J - 1895.74°C
Now, let's solve for Tf:
80129.38 Tf = 1823888.8 J - 1895.74°C
Dividing both sides by 80129.38:
Tf ≈ 22.80°C
Therefore, the final temperature of the punch is approximately 22.80°C after adding the ice.