Question: Four ice cubes are placed in different environments. All the ice cubes are the same size. Your goal ...

Question

Four ice cubes are placed in different environments. All the ice cubes are the same size. Your goal is to determine which of ice cube 1 or 2 fully turns to liquid first, and which of ice cube 3 or 4 fully turns to vapor first.

Ice cube 1 is at room temperature in a slightly more humid atmosphere.
Ice cube 2 is at room temperature in a slightly less humid atmosphere.
Ice cube 3 is on a hot plate at 100°C in a slightly more humid atmosphere.
Ice cube 4 is on a hot plate at 100°C in a slightly less humid atmosphere.

You may neglect heat conduction and heat capacity of air, and assume that the difference in humidity does not significantly change the boiling point. The room is much larger than the ice cube and the ice cubes are of similar size to the hot plate.

Submit the solution as a 2 digit number as follows:

The tens digit is 1 if ice cube 1 melts first, 2 if ice cube 2 melts first and 9 if they take the same time.
The ones digit is 3 if ice cube 3 boils first, 4 if ice cube 4 boils first and 9 if they take the same time.

For example, if ice cube 2 melts first and ice cube 4 boils first, submit the answer 24. Only one attempt will be accepted for this problem.

Answer 1

Solution

The problem asks us to determine which ice cube melts first between 1 and 2, and which ice cube turns to vapor first between 3 and 4. We need to combine these findings into a two-digit number.

Let's analyze each pair:

Part 1: Melting of Ice Cubes 1 and 2

Ice cube 1: Room temperature, slightly more humid atmosphere.
Ice cube 2: Room temperature, slightly less humid atmosphere.

Melting is the phase transition from solid ice to liquid water. This process requires latent heat of fusion. The heat for melting comes from the surrounding room air, which is at a higher temperature than the ice (0°C).

At the surface of the ice cube (or the thin film of meltwater that forms), water molecules continuously evaporate into the surrounding air. Evaporation is a cooling process because it removes latent heat of vaporization from the surface.

Effect of Humidity on Evaporation: A less humid atmosphere (Ice cube 2) means there is a lower partial pressure of water vapor in the air. This creates a steeper concentration gradient for water vapor, leading to a higher net rate of evaporation from the ice/water surface compared to a more humid atmosphere (Ice cube 1).
Effect of Evaporation on Surface Temperature: Since evaporation is a cooling process, a higher rate of evaporation (for Ice cube 2) will lead to more cooling of the ice cube's surface (or the meltwater layer). Consequently, the surface temperature of Ice cube 2 will be slightly lower than that of Ice cube 1.
Effect on Heat Transfer and Melting Rate: The rate of heat transfer from the warmer room air to the ice cube is proportional to the temperature difference between the air and the ice cube's surface ( $\Delta T = T_{room} - T_{surface}$ ). Since Ice cube 2's surface is cooler ( $T_{surface,2} < T_{surface,1}$ ), the temperature difference for Ice cube 2 will be larger ( $T_{room} - T_{surface,2} > T_{room} - T_{surface,1}$ ). A larger temperature difference means a higher rate of heat transfer from the room air to the ice cube. This increased heat transfer directly provides the energy needed for melting.

Therefore, Ice cube 2, in the less humid atmosphere, will absorb heat faster and thus melt faster.

Conclusion for Melting: Ice cube 2 melts first. The tens digit is 2.

Part 2: Vaporization of Ice Cubes 3 and 4

Ice cube 3: On a hot plate at 100°C in a slightly more humid atmosphere.
Ice cube 4: On a hot plate at 100°C in a slightly less humid atmosphere.

The process involves three stages:

Melting the ice to water (at 0°C).
Heating the water from 0°C to 100°C.
Vaporizing the water at 100°C (boiling and evaporation).

The hot plate at 100°C is the primary heat source. The rate of heat transfer from the hot plate to the ice/water is the dominant factor for the first two stages. The humidity of the air above the ice/water does not significantly affect the heat transfer from the hot plate below. Thus, the time taken for both ice cubes to melt and for the water to reach 100°C will be approximately the same.

Once the water reaches 100°C, it starts to vaporize. This vaporization occurs through two mechanisms:

Boiling: A bulk phenomenon where water turns into vapor throughout the liquid volume. The rate of boiling is primarily determined by the rate of heat supplied by the hot plate. Since the hot plate is at 100°C for both, the boiling rate will be similar for both ice cubes.
Evaporation: A surface phenomenon where water molecules escape from the liquid surface into the air. The rate of evaporation depends on the humidity of the surrounding air.

As established in Part 1, a less humid atmosphere (Ice cube 4) leads to a higher rate of evaporation from the water surface compared to a more humid atmosphere (Ice cube 3).

The total rate at which water turns into vapor is the sum of the boiling rate and the evaporation rate. $Rate_{vaporization} = Rate_{boiling} + Rate_{evaporation}$

Since the boiling rate is similar for both, and the evaporation rate is higher for Ice cube 4 (due to less humidity), the total rate of vaporization will be higher for Ice cube 4.

Therefore, Ice cube 4 will fully turn to vapor first.

Conclusion for Vaporization: Ice cube 4 boils/vaporizes first. The ones digit is 4.

Final Answer: Combining the results: Tens digit (for melting): 2 (Ice cube 2 melts first) Ones digit (for vaporization): 4 (Ice cube 4 boils/vaporizes first)

The two-digit number is 24.

The final answer is $\boxed{24}$

Explanation of the solution: For melting at room temperature, lower humidity leads to faster evaporation from the ice surface, causing more cooling. This increased cooling enhances the temperature difference between the room air and the ice, increasing heat transfer to the ice and thus accelerating melting. So, ice cube 2 melts first. For vaporization on a 100°C hot plate, the primary heat source is the hot plate, driving boiling. However, evaporation from the water surface also contributes to vaporization. Lower humidity increases the rate of evaporation. Since both boiling and evaporation contribute to the total vaporization, the ice cube in the less humid atmosphere (ice cube 4) will turn to vapor faster.