I struggled with answering this question. The main problem is that the
speed at which the temperatures change depends sensitively on things like
how heat diffuses from the centers of the eggs into the water and the size
of the pieces of the "handful of ice"; crushed ice will absorb energy much
faster than a big block of ice. So I cannot answer the question, which
presumes that only the amount of water determines the speed to come to
equilibrium—there are many more factors which play. I think it likely
that it would take longer times for large amounts of water to come to
equilibrium, so I could glibly suggest that the answer is the smaller
amount of water is faster; but the problem has too many variables to know
for sure. What I have decided to do is demonstrate the physics of how the
final temperature T is achieved and work that out for one example:
I will find the minimum amount of ice we must add to 1 liter of water so
that the equilibrium of the ice/water/eggs system will be T=0°C.
Heat is just the flow of energy. If an object has a mass M and a
temperature T1 and an amount of heat Q is
added its temperature increases to T2. Then the change
of temperature is proportional to Q, Q=MC(T2-T1),
where C is a proportionality constant called the specific heat of
the material absorbing energy. If the object is instead having energy taken
from it, T2-T1<0 so Q<0.
If a solid is melting it is absorbing energy but not getting warmer; in
that case one needs a constant called latent heat of fusion, L which is the
energy absorbed per kilogram of material melted, Q=ML. So, the
following is the computation.
I will calculate the amount of ice which is
needed for the final temperature to be T=0°C
starting with 1 kg (a liter) of water.
I struggled with answering this question. The main problem is that the
speed at which the temperatures change depends sensitively on things like
how heat diffuses from the centers of the eggs into the water and the size
of the pieces of the "handful of ice"; crushed ice will absorb energy much
faster than a big block of ice. So I cannot answer the question, which
presumes that only the amount of water determines the speed to come to
equilibrium—there are many more factors which play. I think it likely
that it would take longer times for large amounts of water to come to
equilibrium, so I could glibly suggest that the answer is the smaller
amount of water is faster; but the problem has too many variables to know
for sure. What I have decided to do is demonstrate the physics of how the
final temperature T is achieved and work that out for one example:
I will find the minimum amount of ice we must add to 1 liter of water so
that the equilibrium of the ice/water/eggs system will be T=0°C.
Heat is just the flow of energy. If an object has a mass M and a
temperature T1 and an amount of heat Q is
added its temperature increases to T2. Then the change
of temperature is proportional to Q, Q=MC(T2-T1),
where C is a proportionality constant called the specific heat of
the material absorbing energy. If the object is instead having energy taken
from it, T2-T1<0 so Q<0.
If a solid is melting it is absorbing energy but not getting warmer; in
that case one needs a constant called latent heat of fusion, L which is the
energy absorbed per kilogram of material melted, Q=ML. So, the
following is the computation.
I will calculate the amount of ice which is
needed for the final temperature to be T=0°C
starting with 1 kg (a liter) of water.
