Kai Chek, Jerome, Luke, Dawn: Mathematical Modeling 5


The probe is placed into hot water, then ice cubes are slowly added.


On placing probe into hot water: Graph will increase from room temperature to the temperature of the hot water, becoming linear once the temperature of the probe reaches the temperature of the hot water (ie. 70 degrees C)

On adding ice cubes, the graph will decrease with a downward sloping gradient, when the temperature of the probe decreases due to the ice. The temperature will eventually stabilize, becoming linear when the ice and hot water reaches thermal equilibrium (ie. a state at which there is no active heat transfer and temperature is constant)

(Note, the graph does not reach the x-axis, as the temperature cannot reach 0 due to the existing temperature of the hot water)

Observations and Analysis

Heating curve

The heating curve (exponential) is slightly similar to our prediction, but our prediction had a gradient that was a lot smaller than the graph. Furthermore, the temperature stabilized faster than we expected it to.
There is an asymptote slightly above the actual temperature of the probe.
The y-intercept represents the starting temperature, the temperature of the probe when time is 0.
The steep upward slope is due to the gain in heat by the probe from the hot water.

Cooling curve

The graph is similar to the prediction in that it decreases downward with a sloping gradient. However, the initial gradient is steeper, and the gradient does not change as much as we predicted.

The graph slopes downwards as the water loses heat to the ice cubes, and the temperature decreases. The decreasing gradient is due to the reduction in the rate of heat loss by the hot water to the ice cubes.

The graph is mostly linear as it begins, then becomes non-linear (ie. decreasing exponential) as the gradient decreases.

Sources of Error

There might be impurities in the flasks, affecting the rate of heat transfer.
The heat loss might be to the surroundings, since the glass beaker was not insulated. There should only be heat loss to the ice cubes.

Assumptions made

We assumed that there was no heat lost to the surroundings (this is accounted for in sources of error)
We also assumed that the water and ice used were pure, and the containers were clean.
Another assumption would be that the starting temperature is room temperature (ie. 25-30 degrees celsius).

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