Expanding Metals: Copper & Aluminum Equilibrium Temperature

[squib]

A 38.0 g copper ring has a diameter of 2.54000 cm at its temperature of 0°C. An aluminum sphere has a diameter of 2.54508 cm at its temperature of 108°C. The sphere is placed on top of the ring as in the figure, and the two are allowed to come to thermal equilibrium, with no heat lost to the surroundings. The sphere just passes through the ring at the equilibrium temperature. What is the final temperature in kelvins? For copper, α = 1.7×10-5/°C. For aluminum, α = 2.3×10-5/°C.

tried pi(Dc)+(pi(Dc*a*deltaT)) = pi(Da) - (pi(Da*a*deltaT))

this came up with ~49 for change in temp, but this does not seem to be the right answer... any suggestions?

 

[Doc Al]

Your equation assumes that both materials undergo the same temperature change. Not likely! Instead, write the delta T for each substance in terms of Tf and its initial temperature.

Also, why is everything multiplied by pi? (Since they cancel, it doesn't effect your answer. But why?)

 

[thepatientmental]

It looks like you are on the right track with your calculations, but there may be a few errors in your equation. Here is a step-by-step solution to help you find the correct answer:

1. First, we need to convert the given temperatures to Kelvin. The initial temperature of the copper ring is 0°C, which is equivalent to 273.15 K. The initial temperature of the aluminum sphere is 108°C, which is equivalent to 381.15 K.

2. Next, we need to find the change in temperature (deltaT) that will result in the sphere just passing through the ring at equilibrium. This can be calculated using the following equation:

deltaT = (Dc - Da)/(a*Da)

Where Dc and Da are the initial diameters of the copper ring and aluminum sphere, respectively, and a is the thermal expansion coefficient.

For the copper ring, Dc = 2.54000 cm and a = 1.7×10^-5/°C, so the change in temperature for copper is:

deltaT = (2.54508 cm - 2.54000 cm)/(1.7×10^-5/°C * 2.54000 cm) = 14.7 °C

For the aluminum sphere, Da = 2.54508 cm and a = 2.3×10^-5/°C, so the change in temperature for aluminum is:

deltaT = (2.54508 cm - 2.54000 cm)/(2.3×10^-5/°C * 2.54508 cm) = 10.5 °C

3. Now, we can set up an equation to find the final temperature at equilibrium:

pi(Dc) + pi(Dc*a*deltaT) = pi(Da) - pi(Da*a*deltaT)

Substituting in the values we have calculated, we get:

pi(2.54000 cm) + pi(2.54000 cm * 1.7×10^-5/°C * 14.7 °C) = pi(2.54508 cm) - pi(2.54508 cm * 2.3×10^-5/°C * 10.5 °C)

Solving for pi (which represents the final temperature in Kelvin), we get:

pi = 273.15 K + 14.7 °C