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Chapter 17: Problem 100

The molar heat capacity of a certain substance varies with temperature according to the empirical equation $$C=29.5 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}+\left(8.20 \times 10^{-3} \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}^{2}\right) \mathrm{T}$$ How much heat is necessary to change the temperature of 3.00 mol of this substance from \(27^{\circ} \mathrm{C}\) to \(227^{\circ} \mathrm{C}\) ? (Hint: Use Eq. (17.18) in the form \(d Q=n C d T\) and integrate. \()\)

Short Answer

Expert verified
Integrate the heat equation from \(T_1\) to \(T_2\).

Step by step solution

01

Understand the problem

We need to find the amount of heat required to change the temperature of 3.00 mol of a substance from \(27^{\circ} \mathrm{C}\) to \(227^{\circ} \mathrm{C}\). The molar heat capacity \(C(T)\) is given as a function of temperature: \(C(T) = 29.5 + (8.20 \times 10^{-3})T\). To find \(Q\), we integrate \(dQ = nC(T) dT\) over the temperature range.
02

Set up the integration

The formula to use is \(dQ = nC(T) dT\), where \(n\) is the number of moles, \(C(T) = 29.5 + (8.20 \times 10^{-3})T\), and \(dT\) is the differential temperature change. For 3.00 mol, the expression becomes \(dQ = 3.00 (29.5 + (8.20 \times 10^{-3})T) dT\). We need to integrate this from \(T_1 = 27^{\circ}C\) to \(T_2 = 227^{\circ}C\).

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Key Concepts

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

Thermodynamics
Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. It looks at how energy is transferred and transformed. In the context of this problem, we are interested in energy transfer in the form of heat, which affects the temperature of a substance. Thermodynamics helps us understand how heated systems reach equilibrium, how energy is conserved, and how it flows from one part of a system to another. Some key concepts in thermodynamics include:
  • System and Surroundings: A system is the part of the universe we are focusing on. Everything else is considered the surroundings.
  • State Variables: Properties like temperature, pressure, and volume that describe the state of a system.
  • Processes: Paths taking a system from one state to another, affecting heat, work, and internal energy.
Understanding these concepts helps us apply them to solve problems involving heat changes.
Temperature Change
Temperature change refers to the amount by which the temperature of a substance increases or decreases. This exercise asks us to calculate the heat required for a certain temperature change using the molar heat capacity of the substance involved. When we talk about temperature changes, we often refer to:
  • Initial and Final Temperatures: These define the starting and ending points of our calculations. Here, they are 27掳C and 227掳C.
  • Temperature Difference (螖T): Calculated as the final temperature minus the initial temperature. In this exercise, 螖T is 200掳C.
Recognizing these changes helps us understand how much heat transfer occurs, as the amount of heat required depends on the temperature change, the amount of substance, and its heat capacity.
Heat Transfer
Heat transfer involves the movement of thermal energy from one object or substance to another due to temperature difference. This can occur in various modes: conduction, convection, and radiation.In our exercise, we're concerned with heat transfer needed to change the temperature of a substance. This involves calculating the total heat transferred using the equation:\[ dQ = nC(T) dT \]where:
  • dQ: The infinitesimal amount of heat transferred.
  • n: Number of moles of the substance, here 3.00 moles.
  • C(T): Molar heat capacity, given as a function of temperature.
  • dT: Infinitesimal temperature change.
To find the total heat (Q), we integrate this expression over the temperature range of interest. This tells us how much energy is transferred to the substance as heat, changing its temperature from 27掳C to 227掳C.
Empirical Equation
An empirical equation is a formula derived from experimental data. It is used to describe a relationship observed in nature without necessarily explaining the underlying reasons. In this problem, the molar heat capacity is given as:\[ C(T) = 29.5 + (8.20 \times 10^{-3})T \]This expression shows that the heat capacity increases linearly with temperature. Here, "29.5" is the base capacity, and "8.20 \times 10^{-3}" is the rate at which it changes with temperature.Why use empirical equations?
  • Predictions: They help predict how systems behave when we change variables like temperature.
  • Simplification: Allow us to model complex behaviors without knowing every detail.
  • Practicality: Derived from data, they are often more accessible than theoretical equations.
For this exercise, the empirical equation is essential in integrating the heat capacity to find the total heat required to raise the temperature of the substance.

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Most popular questions from this chapter

A \(25,000-\mathrm{kg}\) subway train initially traveling at 15.5 \(\mathrm{m} / \mathrm{s}\) slows to a stop in a station and then stays there long enough for its brakes to cool. The station's dimensions are 65.0 \(\mathrm{m}\) long by 20.0 \(\mathrm{m}\) wide by 12.0 \(\mathrm{m}\) high. Assuming all the work done by the brakes in stopping the train is transferred as heat uniformly to all the air in the station, by how much does the air temperature in the station rise? Take the density of the air to be 1.20 \(\mathrm{kg} / \mathrm{m}^{3}\) and its specific heat to be 1020 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) .

An electric kitchen range has a total wall area of 1.40 \(\mathrm{m}^{2}\) and is insulated with a layer of fiberglass 4.00 \(\mathrm{cm}\) thick. The inside surface of the fiberglass has a temperature of \(175^{\circ} \mathrm{C}\) , and its outside surface is at \(35.0^{\circ} \mathrm{C}\) . The fiberglass has a thermal conductivity of 0.040 \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) (a) What is the heat current through the insulation, assuming it may be treated as a flat slab with an area of 1.40 \(\mathrm{m}^{2} ?(\mathrm{b})\) What electric- power input to the heating element is required to maintain this temperature?

The tallest building in the world, according to some architectural standards, is the Taipei 101 in Taiwan, at a height of 1671 feet. Assume that this height was measured on a cool spring day when the temperature was \(15.5^{\circ} \mathrm{C}\) . You could use the building as a sort of giant thermometer on a hot summer day by carefully measuring its height. Suppose you do this and discover that the Taipei 101 is 0.471 foot taller than its official height. What is the temperature, assuming that the building is in thermal equilibrium with the air and that its entire frame is made of steel?

In a container of negligible mass, 0.0400 \(\mathrm{kg}\) of steam at \(100^{\circ} \mathrm{C}\) and atmospheric pressure is added to 0.200 \(\mathrm{kg}\) of water at \(50.0^{\circ} \mathrm{C} .\) (a) If no heat is lost to the surroundings, what is the final temperature of the system?(b) At the final temperature, how many kilograms are there of steam and how many of liquid water?

In an effort to stay awake for an all-night study session, a student makes a cup of coffee by first placing a \(200-\mathrm{W}\) electric immersion heater in 0.320 \(\mathrm{kg}\) of water. (a) How much heat must be added to the water to raise its temperature from \(20.0^{\circ} \mathrm{C}\) to \(80.0^{\circ} \mathrm{C}\) ? (b) How much time is required? Assume that all of the heater's power goes into heating the water.
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