m58400 Heat Capacities of an Ideal Gas By the end of this section, you will be able to: Define heat capacity of an ideal gas for a specific process Calculate the specific heat of an ideal gas for either an isobaric or isochoric process Explain the difference between the heat capacities of an ideal gas and a real gas Estimate the change in specific heat of a gas over temperature ranges a6b46764-7ec3-48a4-ac87-1361295c95cb We learned about specific heat and molar heat capacity in Temperature and Heat; however, we have not considered a process in which heat is added. We do that in this section. First, we examine a process where the system has a constant volume, then contrast it with a system at constant pressure and show how their specific heats are related. Let’s start with looking at , which shows two vessels A and B, each containing the same type of ideal gas at a temperature T and a volume V. The only difference between the two vessels is that the piston at the top of A is fixed, whereas the one at the top of B is free to move against a constant external pressure p. We now consider what happens when the temperature of the gas in each vessel is slowly increased to T+dT with the addition of heat.

Two vessels are identical except that the piston at the top of A is fixed, whereas that atop B is free to move against a constant external pressure p.

Since the piston of vessel A is fixed, the volume of the enclosed gas does not change. Consequently, the gas does no work, and we have from the first law dEint=dQ−dW=dQ. We represent the fact that the heat is exchanged at constant volume by writing dQ=nCVdT,where CV is the molar heat capacity at constant volume of the gas, and, as before, n is the number of moles of the gas. In addition, since dEint=dQ for this particular process, dEint=nCVdT.We obtained this equation assuming the volume of the gas was fixed. However, internal energy is a state function that depends on only the temperature of an ideal gas. Therefore, dEint=nCVdT gives the change in internal energy of an ideal gas for any process involving a temperature change dT.When the gas in vessel B is heated, it expands against the movable piston and does work dW=pdV. In this case, the heat is added at constant pressure, and we write dQ=nCpdT,where Cp is the molar heat capacity at constant pressure of the gas. Furthermore, since the ideal gas expands against a constant pressure, d(pV)=d(nRT)becomes pdV=nRdT.Finally, inserting the expressions for dQ and pdV into the first law, we obtain dEint=dQ−pdV=(nCp−nR)dT.We have found dEint for both an isochoric and an isobaric process. Because the internal energy of an ideal gas depends only on the temperature, dEint must be the same for both processes. Thus, nCVdT=(nCp−nR)dT,and Cp=CV+R. The derivation of was based only on the ideal gas law. Consequently, this relationship is approximately valid for all dilute gases, whether monatomic like He, diatomic like O2, or polyatomic like CO2or NH3. In the preceding chapter, we found the molar heat capacity of an ideal gas under constant volume to be CV=d2R, where d is the number of degrees of freedom of a molecule in the system. shows the molar heat capacities of some dilute ideal gases at room temperature lists the experimental values for some gases. The heat capacities of real gases are somewhat higher than those predicted by the expressions of CV and Cp given in . This indicates that vibrational motion in polyatomic molecules is significant, even at room temperature. Nevertheless, the difference in the molar heat capacities, Cp−CV, is very close to R, even for the polyatomic gases. Molar Heat Capacities of Dilute Ideal Gases at Room Temperature Type of Molecule Cp(J/mol K) CV(J/mol K) Monatomic 52R=20.79 32R=12.47 Diatomic 72R=29.10 52R=20.79 Polyatomic 4R=33.26 3R=24.94

For an ideal gas, the molar capacity at constant pressure Cp is given by Cp=CV+R=dR/2+R, where d is the number of degrees of freedom of each molecule/entity in the system. A real gas has a specific heat close to but a little bit higher than that of the corresponding ideal gas with Cp≃CV+R.

How can an object transfer heat if the object does not possess a discrete quantity of heat? Most materials expand when heated. One notable exception is water between 0°C and 4°C, which actually decreases in volume with the increase in temperature. Which is greater for water in this temperature region, Cp or CV ? Typically Cp is greater than CV because when expansion occurs under constant pressure, it does work on the surroundings. Therefore, heat can go into internal energy and work. Under constant volume, all heat goes into internal energy. In this example, water contracts upon heating, so if we add heat at constant pressure, work is done on the water by surroundings and therefore, Cp is less than CV. Why are there two specific heats for gases Cp and CV, yet only one given for solid?

The temperature of an ideal monatomic gas rises by 8.0 K. What is the change in the internal energy of 1 mol of the gas at constant volume? 100 J For a temperature increase of 10°C at constant volume, what is the heat absorbed by (a) 3.0 mol of a dilute monatomic gas; (b) 0.50 mol of a dilute diatomic gas; and (c) 15 mol of a dilute polyatomic gas? If the gases of the preceding problem are initially at 300 K, what are their internal energies after they absorb the heat? a. 11,600 J; b. 3220 J; c. 116,000 J Consider 0.40 mol of dilute carbon dioxide at a pressure of 0.50 atm and a volume of 50 L. What is the internal energy of the gas? When 400 J of heat are slowly added to 10 mol of an ideal monatomic gas, its temperature rises by 10°C. What is the work done on the gas? 850 J One mole of a dilute diatomic gas occupying a volume of 10.00 L expands against a constant pressure of 2.000 atm when it is slowly heated. If 400.0 J of heat are added in the process, what is its final volume?

molar heat capacity at constant pressure quantifies the ratio of the amount of heat added removed to the temperature while measuring at constant pressure molar heat capacity at constant volume quantifies the ratio of the amount of heat added removed to the temperature while measuring at constant volume