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Chapter 14: Problem 86

A copper pot has a mass of \(1.0 \mathrm{~kg}\) and is at \(100^{\circ} \mathrm{C}\). How much heat must be removed from it to decrease its temperature to precisely \(0^{\circ} \mathrm{C}\) ? The specific heat of copper is \(387 \mathrm{~J} /(\mathrm{kg} \cdot \mathrm{K})\).

Short Answer

Expert verified
To decrease its temperature to precisely \(0^{\circ} \mathrm{C}\), \(38700 \mathrm{~J}\) of heat must be removed from the copper pot.

Step by step solution

01

Identify Given Values and Unknown

The mass \(m\) of the copper pot is \(1.0 \mathrm{~kg}\), the initial temperature \(T_{1}\) is \(100^{\circ}\mathrm{C}\), the final temperature \(T_{2}\) is \(0^{\circ} \mathrm{C}\), and the specific heat capacity \(c\) of copper is \(387 \mathrm{~J/(kg \cdot K)}\). The unknown is the amount of heat \(Q\) that needs to be removed.
02

Calculate the Change in Temperature

The temperature change \(\Delta T\) is given by \(T_{2} - T_{1}\). Substituting the given values, \(\Delta T = 0^{\circ}\mathrm{C} - 100^{\circ}\mathrm{C} = -100^{\circ}\mathrm{C}\). The negative sign indicates a decrease in temperature.
03

Use the Heat Transfer Formula

According to the formula \(Q = mc\Delta T\), the amount of heat removed from or added to a substance is equal to the product of its mass, its specific heat capacity, and the change in its temperature. Substituting the given values, \(Q = (1.0 \mathrm{~kg})(387 \mathrm{~J/(kg \cdot K)})(-100^{\circ}\mathrm{C}) = -38700~\mathrm{J}\). The negative sign confirms that heat is being removed from the pot.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Specific Heat Capacity
The concept of specific heat capacity is fundamental when discussing heat transfer. It refers to the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). This intrinsic property varies from one material to another.

For copper, the specific heat capacity is 387 J/(kg路K).
  • It reflects how much energy is needed to raise the temperature of 1 kg of copper by 1掳C.
  • The lower the specific heat capacity, the less heat is needed to change its temperature.
In our case, the specific heat capacity allows us to calculate how much heat must be removed to cool the copper pot by using the formula: \[Q = mc\Delta T\]Here, "m" is mass, "c" is specific heat capacity, and "\Delta T" is temperature change.
Temperature Change
Understanding the concept of temperature change is crucial when calculating heat transfer. Temperature change is simply the difference between the final and initial temperatures of a substance.
  • In our problem, the initial temperature ( \(T_{1}\)) of the copper pot is 100掳C, and the final temperature ( \(T_{2}\)) is 0掳C.
  • The temperature change ( \(\Delta T\)) is thus \(T_{2} - T_{1} = 0^{\circ} \mathrm{C} - 100^{\circ} \mathrm{C} = -100^{\circ} \mathrm{C}\).

The negative sign indicates that this is a cooling process, where heat is being removed from the substance. This change is vital for the calculation of the heat transferred, as it signifies the direction and magnitude of energy movement.
Copper Properties
Copper is a popular material in industrial and household applications, primarily due to its excellent thermal and electrical conductivities. Below are some important properties of copper that enhance its effectiveness:
  • High thermal conductivity: Copper can effectively transfer heat, making it an ideal material for pots and pans.
  • Durability: It is a resilient metal that can withstand high temperatures without degrading.
  • Specific heat capacity: As previously discussed, this is 387 J/(kg路K), connecting directly to its ability to store heat.
When you heat or cool copper, these properties ensure that it reacts predictably, absorbing or releasing heat according to its specific heat capacity.

In exercises like the one regarding the copper pot, these properties of copper allow us to determine exactly how much energy will change the pot鈥檚 temperature, using well-known formulas and conversion principles.

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Most popular questions from this chapter

Explain why there must exist a numerical value that is the same on the Celsius scale as on the Fahrenheit scale. Show the calculation that yields the special value. 5SM

Two objects that have different sizes, masses, and temperatures come into close contact wirh each other. Thermal energy is transferred A. from the larger object to the smaller object. B. from the object that has more mass to the one that has less mass. C. from the object that has the higher temperature to the object that has the lower temperature. D. from the object that has the lower temperature to the object that has the higher temperature. E. back and forth between the rwo objects until they come to equilibrium.

The ideal gas law can be written as \(P V=N k T\) or \(P V=n R T\). (a) Explain the different contexts in which you might use one or the other, and (b) define all the variables and constants.

You wish to heat \(250 \mathrm{~g}\) of water to make a hor cup of coffee. If the water starts at \(20^{\circ} \mathrm{C}\) and you want your coffee to be \(95^{\circ} \mathrm{C}\), calculate the minimum amount of heat required. SSM

A careful physics student is reading about the concept of temperature and has what she thinks is a bright idea. While reading the rext, she discovers that the Celsius temperature scale and the Kelvin temperature scale both have the same increments for equal temperature differences. However, the Fahrenheit scale was described as having larger increments for the same temperature differences. From this information, the student determines that Celsius (or kelvin) thermometers will always be shorter than Fahrenheit thermometers. Explain the parts of her idea that are valid and the parts that are not.
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