CITRIS CONCENTRATE: Great orange juice is made from fresh oranges sliced and squeezed. The alternative is to prepare the juice from a concentrated product. Water and valuable oils are removed from fresh citris at a processing plant then "concentrate" is shipped to consumers with the instructions,
"Mix concentrate with 3 cans of tap water."
Suppose the precise initial temperature of the solid concentrate and tap water is known:
- Can the second temperature, the temperature of the mixed OJ be determined
- Can any precise, second LOWEST temperature of the OJ be determined
- OJ is best served cold. How cold might it be
Calculate the very least possible temperature of OJ mixed.
- The concentrate is "one volume" of solid water @ - 15oC.
Assume the average specific heat of the solid to be ~ cp,solid,avg = 1.36 J/( g oC). The tap water is initially 20oC and water as liquid has an average specific heat ~ cp,solid,avg = 4.16 J/( g oC).
Liquid to solid (or the solid to liquid) phase change involves a heat of fusion (or heat of melting) and occurs at the temperature, 0oC. The melting energy has been measured: hsf = -333.4 J/g.
- The three volumes of water are initially 25oC. The average specific heat of water is 4.16 J/(goC).
Solution:
Mass Equation: The initial state of orange juice is a mix of "one can" of concentrate, with "3 cans" of tap water.
The mass equation states simply that the mass of the system finally will be the mass of the concentrate and water.
masssys, initial=
mass(solid water, -15oC)+
mass(liquid water, 20oC)
masssys, final =
mass[(all solid at ?C) or (all liquid at ?C) or (solid and liquid at 0oC]
So the Mass Equation for this mixing of water event is:
m2 - m1 = 0
[mass
[(all solid at ?C) or (all liquid at ?C) or (solid and liquid at 0oC]
]
- [mass(solid water, -15oC)+
mass(liquid water, 20oC)] = 0
One might ponder how large is the can, does the can size matter? The mass of a "can" of solid concentrate is essentially the same as the mass of that "can" filled with water. (The density of water as ice or water is about 4% less than as liquid.)
The mass ofhere is a mass of "one can" of concentrate
However, we are not mixing by mass, we are mixing by volumes. No matter. For this event the difference in liquid water mass per volume from solid mass per volume is insignificant.
The is event is called a thermal equilibration. Occuring at one atmosphere, the concentrate, initially ice at -15oC are placed together and attain the same temperature. We will assume the second state is liquid and solid at 0oC.
To write the energy equation we will identify three masses. The mass of ice as "icemelt" which melts. Then "icesolid" being that ice at -15oC that is warmed to 0oC but that remains solid and the water that is liquid at room temperature and is cooled to 0oC.
The mass equation is:
m2 - m1=0
But m2 = mwater,1 + mice,melt + mice,solid
And m1 = mice,1 + mwater,1
This Example has not been completed.