What Is the Equilibrium Temperature When Aluminum Meets Water?

[Brit412]

Homework Statement

A 150g aluminum cylinder is removed from a liquid nitrogen bath, where it has been cooled to -196 degrees C. The cylinder is immediately placed in an insulated cup containing 60.0g of water at 15.0 degrees C. What is the equilibrium temperature of the system?

Homework Equations

Q(water) = m(water) c(water) (∆T)
Q(water to ice)= m(water) Lfusion of ice
Q(aluminum) = m(aluminum) c(aluminum)(∆T)

The Attempt at a Solution

My problem is how to find the equilibrium temperature using those equations?

 

[Carid]

My preference would be to work in °Kelvin not °Celcius. In which case...

HeatLost = HeatGained

HeatLost = MassWater*SpecificHeatWater*(InitialWaterTemp-FinalTemp) + MassWater*LatentHeat

HeatGained = MassAluminium*SpecificHeatAluminium*(FinalTemp-InitialAluminiumTemp)

 

[nlightner]

I would approach this problem by first understanding the concept of thermal equilibrium. Thermal equilibrium is the state in which two objects or systems are at the same temperature and there is no net flow of heat between them. In this scenario, we have a system consisting of the aluminum cylinder and the water in the insulated cup. In order to find the equilibrium temperature, we need to understand how heat is transferred between these two objects.

First, we need to consider the initial and final temperatures of the aluminum cylinder and the water. The aluminum cylinder is initially at -196 degrees C and the water is initially at 15.0 degrees C. We also know the masses of both objects, which will be important in our calculations.

Next, we need to consider the heat transfer equations provided in the problem. The first equation, Q(water) = m(water) c(water) (∆T), represents the heat transfer from the water to reach the final temperature. The second equation, Q(water to ice)= m(water) Lfusion of ice, represents the heat transfer from the water to change its state from liquid to solid (since the final temperature will be below the freezing point of water). The third equation, Q(aluminum) = m(aluminum) c(aluminum)(∆T), represents the heat transfer from the aluminum cylinder to reach the final temperature.

To solve for the equilibrium temperature, we need to set these three equations equal to each other and solve for ∆T (the change in temperature). This will give us the final temperature of the system, which is the equilibrium temperature. We also need to take into account the specific heat capacities and latent heat of fusion for water and aluminum in our calculations.

Once we have the equilibrium temperature, we can check if it falls within the range of the initial temperatures for both objects. If it does, then we have found the correct solution. If not, we need to reassess our calculations and make any necessary adjustments.

In summary, to find the equilibrium temperature in this scenario, we need to understand the concept of thermal equilibrium, consider the initial and final temperatures of the objects, and use the provided heat transfer equations to solve for the final temperature. It is important to also check if the final temperature falls within the range of the initial temperatures for both objects.