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Chapter 12: Problem 48

Two bars of identical mass are at \(25^{\circ} \mathrm{C} .\) One is made from glass and the other from another substance. The specific heat capacity of glass is \(840 \mathrm{J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right) .\) When identical amounts of heat are supplied to each, the glass bar reaches a temperature of \(88{ }^{\circ} \mathrm{C},\) while the other bar reaches \(250.0^{\circ} \mathrm{C} .\) What is the specific heat capacity of the other substance?

Short Answer

Expert verified
The specific heat capacity of the other substance is approximately 235.2 J/(kg路C).

Step by step solution

01

Understand the Problem

We need to find the specific heat capacity of the second substance using the given data for the glass and the other substance. We'll use the formula for heat exchange, assuming the same mass and heat energy input for both substances.
02

Recall the Heat Capacity Formula

The formula for heat absorbed or released by a substance is given by \( Q = mc\Delta T \), where \( Q \) is the heat energy, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.
03

Calculate Heat for Glass

For the glass bar, the change in temperature \( \Delta T \) is \( 88^{\circ}C - 25^{\circ}C = 63^{\circ}C \). Using the formula: \[ Q = mc_{\text{glass}}\Delta T = m \times 840 \times 63 \].
04

Set up the Equation for the Other Substance

The other substance reaches \(250^{\circ}C\), so \( \Delta T = 250^{\circ}C - 25^{\circ}C = 225^{\circ}C \). \[ Q = mc_{\text{other}} \times 225 \].
05

Solve for Specific Heat Capacity of the Other Substance

Since the heat added \( Q \) is the same for both substances, set \[ mc_{\text{glass}} \times 63 = mc_{\text{other}} \times 225 \]. Divide both sides by \( m \) to eliminate mass: \( 840 \times 63 = c_{\text{other}} \times 225 \). Solve for \( c_{\text{other}} \): \[ c_{\text{other}} = \frac{840 \times 63}{225} \]. Calculating this gives \( c_{\text{other}} \approx 235.2 \text{ J/(kg路C)} \).

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Key Concepts

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

Heat Energy
When we talk about heat energy, we are referring to the energy that is transferred between systems or objects because of temperature differences.
This concept is paramount in thermal physics, as it explains how and why energy moves from one place to another. In this context, two identical masses were heated by the same amount of energy.
  • Heat energy is often measured in joules (J).
  • It moves from a hotter object to a colder one, seeking thermal equilibrium.
Another important aspect is how this energy interacts with different materials. Some materials absorb heat differently, which relates closely to their specific heat capacity. This is why even though both bars received the same heat, their temperatures changed differently.
Temperature Change
The change in temperature (\(\Delta T\)) of an object when it absorbs or loses heat is directly linked to its specific heat capacity. This is a measure of how much heat energy is required to raise the temperature of one kilogram of a substance by one degree Celsius.
Using the formula \(Q = mc\Delta T\), we can understand how temperature change is calculated. In the case of the two bars, even though they both started at the same temperature (\(25^{\circ}C\)), the glass bar increased to \(88^{\circ}C\), while the other reached \(250^{\circ}C\).
  • Temperature change depends on both the amount of heat and the substance's specific heat capacity.
  • Higher specific heat capacity means more energy needed for the same temperature increase.
This relationship underlines the different behaviors of materials when exposed to identical heat energies.
Thermal Physics
Thermal physics combines the principles of thermodynamics, heat transfer, and statistical mechanics to describe how heat is generated, absorbed, and transferred.
One of the key ideas in this domain is that different materials respond uniquely to heat due to their specific heat capacities. This experiment reflects thermal physics as it shows how the same heat energy can cause varying temperature changes in different materials.
  • Each material鈥檚 atomic or molecular structure affects its heat absorption ability.
  • Thermodynamics principles, like energy conservation, are crucial for calculating heat interactions.
Understanding thermal physics helps us design materials and systems for better heat management, crucial in engineering and everyday applications such as cooking and home heating systems.

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Most popular questions from this chapter

The Eiffel Tower is a steel structure whose height increases by \(19.4 \mathrm{cm}\) when the temperature changes from -9 to \(+41^{\circ} \mathrm{C}\) . What is the approximate height (in meters) at the lower temperature?

Multiple-Concept Example 4 reviews the concepts that are in volved in this problem. A ruler is accurate when the temperature is \(25^{\circ} \mathrm{C}\). When the temperature drops to \(-14^{\circ} \mathrm{C},\) the ruler shrinks and no longer measures distances accurately. However, the ruler can be made to read correctly if a force of magnitude \(1.2 \times 10^{3} \mathrm{N}\) is applied to each end so as to stretch it back to its original length. The ruler has a cross-sectional area of \(1.6 \times 10^{-5} \mathrm{m}^{2},\) and it is made from a material whose coefficient of linear expansion is \(2.5 \times 10^{-5}\left(\mathrm{C}^{\circ}\right)^{-1} .\) What is Young's modulus for the material from which the ruler is made?

A \(0.200-\mathrm{kg}\) piece of aluminum that has a temperature of \(-155^{\circ} \mathrm{C}\) is added to \(1.5 \mathrm{kg}\) of water that has a temperature of \(3.0^{\circ} \mathrm{C}\). At equilibrium the temperature is \(0.0^{\circ} \mathrm{C}\). Ignoring the container and assuming that the heat exchanged with the surroundings is negligible, determine the mass of water that has been frozen into ice.

The latent heat of vaporization of \(\mathrm{H}_{2} \mathrm{O}\) at body temperature \(\left(37.0^{\circ} \mathrm{C}\right)\) is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg} .\) To cool the body of a \(75-\mathrm{kg}\) jogger [average specific heat capacity \(\left.=3500 \mathrm{J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right)\right]\) by \(1.5 \mathrm{C}^{\circ},\) how many kilograms of water in the form of sweat have to be evaporated?

At a fabrication plant, a hot metal forging has a mass of \(75 \mathrm{kg}\) and a specific heat capacity of \(430 \mathrm{J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right) .\) To harden it, the forging is immersed in \(710 \mathrm{kg}\) of oil that has a temperature of \(32^{\circ} \mathrm{C}\) and a specific heat capacity of \(2700 \mathrm{J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right) .\) The final temperature of the oil and forging at thermal equilibrium is \(47^{\circ} \mathrm{C}\). Assuming that heat flows only between the forging and the oil, determine the initial temperature of the forging.
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