Chapter 17: Problem 88
The molar heat capacity of a certain substance varies with temperature
according to the empirical equation
Short Answer
Expert verified
The heat necessary is 22107 J.
Step by step solution
01
Identify the Variables
We begin by identifying the given variables:- Initial temperature, - Final temperature, - Number of moles, - Molar heat capacity,
02
Setup the Heat Transfer Equation
Using the hint provided, we know that the differential form of the heat transfer can be expressed as . This tells us how much heat is added when the temperature changes by an infinitesimal amount .
03
Substitute the Molar Heat Capacity into the Equation
Replace with the given function: This modifies in terms of .
04
Integrate the Equation
Calculate the total heat by integrating from to : Solving the integral:
05
Evaluate the Integral
Evaluate the integrated function at the bounds: This simplifies to:
06
Calculate the Numerical Value
Calculate each term inside the brackets:- - - - Substitute these values back into the equation to find :
07
Conclusion
The total heat required to change the temperature of the substance is 22107 J.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Molar Heat Capacity
Molar heat capacity is an important concept in thermodynamics as it measures the amount of heat a substance can hold per mole for a given temperature change. Expressing it as a function of temperature, like in the provided equation, is common for many substances. This equation describes how the heat capacity changes depending on the temperature, combining a constant term and a temperature-dependent term:
In this way, molar heat capacity is crucial for determining how much heat is needed during a temperature change, thus playing a central role when calculating heat transfer in varying thermal conditions.
- The constant term (29.5 J/mol·K) represents the basic heat absorption ability inherent to the substance, regardless of temperature.
- The temperature-dependent term
reflects how the capacity increases or decreases as the substance's temperature changes.
In this way, molar heat capacity is crucial for determining how much heat is needed during a temperature change, thus playing a central role when calculating heat transfer in varying thermal conditions.
Temperature Change
Temperature change is simply the difference between the initial and final temperatures of a substance when it undergoes heating or cooling. In thermodynamic calculations, temperature is typically converted from Celsius to Kelvin for accuracy.
The temperature change, , is the difference . This change is a driving force in the thermodynamic equations, affecting the amount of heat transfer required to achieve that change.
- The initial temperature,
, is where the process starts; for this exercise, it is 27°C or 300 K. - The final temperature,
, is the end point of the process, which in this exercise is 227°C or 500 K.
The temperature change,
Heat Transfer
Heat transfer is the movement of thermal energy from one place to another. In a thermal system, it is often calculated using the relation , where is the differential heat added, is the number of moles, is the molar heat capacity, and is the small change in temperature. This relation tells us how heat added depends on the thermal properties of the substance and the current temperature.
This calculated heat transfer quantifies the energy needed to achieve the desired temperature change, playing a vital role in thermal management and engineering.
- In our case, the number of moles,
, affects how much total heat is required. - The molar heat capacity, expressed as
, is substituted into this formula, creating a differential equation ready to be integrated.
This calculated heat transfer quantifies the energy needed to achieve the desired temperature change, playing a vital role in thermal management and engineering.
Integration in Thermodynamics
Integration in thermodynamics allows for calculating the total heat transfer over a finite range of temperatures. Starting with the expression , we integrate both sides from the initial temperature to the final temperature .
Solving this integral provides the total heat required, using fundamental calculus that bridges changes in state functions across temperature ranges. This method is essential for accurately determining energy exchanges in chemistry and physics.
- This involves integrating the molar heat capacity equation over the temperature range. Here,
. - The integral evaluates to
.
Solving this integral provides the total heat required, using fundamental calculus that bridges changes in state functions across temperature ranges. This method is essential for accurately determining energy exchanges in chemistry and physics.