Chapter 17: Problem 88
The molar heat capacity of a certain substance varies with temperature according to the empirical equation $$C = 29.5 J/mol \cdot K + (8.20 \times 10{^-}{^3} J/mol \cdot K{^2})T$$ How much heat is necessary to change the temperature of 3.00 mol of this substance from 27\(^\circ\)C to 227\(^\circ\)C? (Hint: Use Eq. (17.18) in the form d\(Q\) = n\(C\) d\(T\) and integrate.)
Short Answer
Step by step solution
Identify the Variables
Setup the Heat Transfer Equation
Substitute the Molar Heat Capacity into the Equation
Integrate the Equation
Evaluate the Integral
Calculate the Numerical Value
Conclusion
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Molar Heat Capacity
- 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 \(8.20 \times 10^{-3}T\) 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
- The initial temperature, \(T_i\), is where the process starts; for this exercise, it is 27°C or 300 K.
- The final temperature, \(T_f\), is the end point of the process, which in this exercise is 227°C or 500 K.
The temperature change, \(\Delta T\), is the difference \(T_f - T_i = 500 \,K - 300 \,K = 200 \,K\). This change is a driving force in the thermodynamic equations, affecting the amount of heat transfer required to achieve that change.
Heat Transfer
- In our case, the number of moles, \(n = 3.00\), affects how much total heat is required.
- The molar heat capacity, expressed as \(29.5 + (8.20 \times 10^{-3})T\), 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
- This involves integrating the molar heat capacity equation over the temperature range. Here, \(Q = \int_{300}^{500} 3.00(29.5 + (8.20 \times 10^{-3})T) dT\).
- The integral evaluates to \(Q = 3.00 \left[29.5T + \frac{8.20 \times 10^{-3}}{2} T^2 \right]_{300}^{500}\).
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.