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Chapter 4: Problem 16

The molar heat capacity at constant pressure \(C_{P} / n\) of a gas varies with the temperature according to the equation $$ \frac{C_{P}}{n}=a+b T-\frac{c}{T^{2}} $$ where \(a, b\), and \(c\) are constants. How much heat is transferred during an isobaric process in which \(n\) moles of gas experience a temperature rise from \(T_{i}\) to \(T_{f} ?\)

Short Answer

Expert verified
The heat transferred is: \(Q = n \left[ a(T_{f} - T_{i}) + \frac{b}{2} (T_{f}^{2} - T_{i}^{2}) + c \left( \frac{1}{T_{f}} - \frac{1}{T_{i}} \right) \right]\).

Step by step solution

01

- Understanding the Equation

The problem gives the molar heat capacity at constant pressure as \( \frac{C_{P}}{n} = a + bT - \frac{c}{T^{2}} \). Here, \(a\), \(b\), and \(c\) are constants, and \(T\) is the temperature.
02

- Write Heat Transfer Equation

Heat transferred \(Q\) in an isobaric process can be given by \(Q = n \int_{T_{i}}^{T_{f}} \frac{C_{P}}{n} dT\).
03

- Substitute \( \frac{C_{P}}{n} \) in the Heat Transfer Equation

Substitute \(\frac{C_{P}}{n} = a + bT - \frac{c}{T^{2}}\) into the integral: \(Q = n \int_{T_{i}}^{T_{f}} (a + bT - \frac{c}{T^{2}}) dT\).
04

- Integrate Each Term Separately

Integrate each term separately: \( \int_{T_{i}}^{T_{f}} a\, dT = a(T_{f} - T_{i}) \), \( \int_{T_{i}}^{T_{f}} bT\, dT = \frac{b}{2} (T_{f}^{2} - T_{i}^{2}) \), \( \int_{T_{i}}^{T_{f}} -\frac{c}{T^{2}} \, dT = c \left( \frac{1}{T_{f}} - \frac{1}{T_{i}} \right) \).
05

- Combine the Results

Combine the results from the integrals: \(Q = n \left[ a(T_{f} - T_{i}) + \frac{b}{2} (T_{f}^{2} - T_{i}^{2}) + c \left( \frac{1}{T_{f}} - \frac{1}{T_{i}} \right) \right]\).
06

- Simplify the Expression

The final expression for the heat transferred is: \(Q = n \left[ a(T_{f} - T_{i}) + \frac{b}{2} (T_{f}^{2} - T_{i}^{2}) + c \left( \frac{1}{T_{f}} - \frac{1}{T_{i}} \right) \right]\).

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Key Concepts

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

headline of the respective core concept
An isobaric process is a process in which the pressure remains constant. This term comes from the Greek words 'iso-' meaning equal and 'baros' meaning pressure.
This simplification means that instead of dealing with varying pressures, we can work with only one constant pressure value, making calculations easier.
Examples of isobaric processes include the heating of a gas inside a piston where the piston can move to maintain constant pressure.
In such processes, any heat added or removed from the system does work by changing the system's volume while ensuring the pressure stays constant.
headline of the respective core concept
Temperature rise in this context means increasing the temperature of a gas from an initial value, denoted as \(T_i\), to a final value, denoted as \(T_f\).
During this change, the gas either absorbs heat (if temperature rises) or releases heat (if temperature drops).
The amount of heat needed to raise the temperature depends on the properties of the gas, such as its molar heat capacity at constant pressure.
In our problem, the temperature rise occurs in an isobaric process, so the heat energy affects the gas in a specific way related to its specific heat capacities at constant pressure.
headline of the respective core concept
The heat transfer equation is a fundamental tool in thermodynamics. For an isobaric process, the heat (\(Q\)) transferred to the gas can be calculated from the molar heat capacity and the temperature change.
The general form of this equation is:
\[ Q = n \int_{T_i}^{T_f} \tfrac{C_P}{n} \dT \],
where: \
  • \(Q\) is the heat transferred
  • \(n\) is the number of moles of gas
  • \(C_P\) is the molar heat capacity at constant pressure
  • \(T_i\) and \(T_f\) are the initial and final temperatures
This equation integrates the heat capacities over the temperature range to find the total heat transferred.
When we substitute the given heat capacity equation, \( \frac{C_P}{n} = a + bT - \frac{c}{T^2} \) into the integral, we can solve each term separately and combine them to find the total heat:
\[Q = n \[ a(T_f - T_i) + \frac{b}{2} (T_f^2 - T_i^2) + c \left( \frac{1}{T_f} - \frac{1}{T_i} \right) \] \].
This final expression gives us a comprehensive way to calculate the heat transferred in the given isobaric process.

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Most popular questions from this chapter

A copper wire of length \(1.317 \mathrm{~m}\) and diameter \(3.26 \times 10^{-4} \mathrm{~m}\) is blackened and placed along the axis of an evacuated glass tube. The wire is connected to a battery, a rheostat, an ammeter, and a voltmeter, and the current is increased until, at the moment the wire is about to melt, the ammeter reads \(12.8 \mathrm{~A}\) and the voltmeter reads \(20.2 \mathrm{~V}\). Assuming that all the energy supplied was radiated and that the radiation from the glass tube is negligible, calculate the melting temperature of copper.

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