Expanding Metals: Copper & Aluminum Equilibrium Temperature

In summary, a 38.0 g copper ring with a diameter of 2.54000 cm at 0°C and an aluminum sphere with a diameter of 2.54508 cm at 108°C are placed in thermal equilibrium with no heat lost to the surroundings. The sphere just passes through the ring at the equilibrium temperature. To find the final temperature in kelvins, the equation pi(Dc)+(pi(Dc*a*deltaT)) = pi(Da) - (pi(Da*a*deltaT)) is used, but it is not accurate as it assumes both materials undergo the same temperature change. Instead, the delta T for each substance should be written in terms of Tf and its initial temperature. Additionally, the
  • #1
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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?
 
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  • #2
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?)
 
  • #3


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
 

Related to Expanding Metals: Copper & Aluminum Equilibrium Temperature

1. What is the purpose of studying the equilibrium temperature of expanding metals, specifically copper and aluminum?

The purpose of studying the equilibrium temperature of expanding metals is to understand how these materials react to changes in temperature and to determine the point at which they reach thermal equilibrium. This information can be useful in various industries such as construction, manufacturing, and electronics.

2. How do copper and aluminum differ in their equilibrium temperature?

Copper and aluminum have different thermal properties, resulting in a difference in their equilibrium temperature. Copper has a lower specific heat capacity compared to aluminum, which means it requires less energy to increase its temperature. Thus, copper has a lower equilibrium temperature compared to aluminum.

3. What factors affect the equilibrium temperature of expanding metals?

The equilibrium temperature of expanding metals is affected by various factors such as the specific heat capacity of the material, its thermal conductivity, and the rate at which heat is applied or removed. Other factors that can influence equilibrium temperature include the initial temperature of the metal and its surroundings, as well as the environment's temperature and humidity.

4. How is the equilibrium temperature of expanding metals determined in experiments?

In experiments, the equilibrium temperature of expanding metals can be determined by measuring the temperature of the metal at different time intervals until it reaches a constant value. This constant value is the equilibrium temperature. The experiment can be repeated at different initial temperatures and with different heating or cooling rates to analyze the effects of different factors on the equilibrium temperature.

5. What are the practical applications of understanding the equilibrium temperature of expanding metals?

The understanding of the equilibrium temperature of expanding metals has several practical applications. For example, it can help in the design and construction of buildings and structures that use copper or aluminum materials. It can also be useful in optimizing manufacturing processes and improving the efficiency of electronic devices. Furthermore, this knowledge can aid in the development of new materials with desired thermal properties.

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