If a system composed of an ideal gas absorbs heat at a constant volume, its temperature will increase. Since the volume is held constant, the system can do no work on its environment in the process, therefore
Differentiating the equation for the internal energy of an ideal gas (3.9) we obtain
[specific heat at constant volume and R]. (3.14)
Note that R as used in this equation is for a particular gas such as dry air. The heat capacity at constant volume, cv, is proportional to the number of degrees of freedom.
Another important process is the heating of the gas at constant pressure. In this case
QA^B = McpAT [heating at constant pressure]. (3.15) We also have
This leads to
If U is a function of state as the First Law claims,5 then it is given by Mcv AT, and we have an identity:
which is a very important relation for ideal gases, holding independently of the value of f. This last tells us that for an ideal gas cp =( f + 1 ) R
That cp is always greater than cv has an easy interpretation: some of the heat absorbed in the isobaric case is "wasted" by the expansion (work done by the system on the environment) rather than being devoted to raising the temperature.
Example 3.8 Dry air: what are cv, cp, using ideal gas rules? Answer: We have cv = |Rd = 717.5 J kg-1 K-1, and cp = cv + Rd = (717 + 287) = 1004 J kg-1 K-1.
Example 3.9 How much heat is required to raise the temperature of a 1kg parcel of air at constant pressure (constant altitude in the atmosphere) by 1°C? Answer: 1004 J. □
Example 3.10 A mass of 2 kg of dry air is heated isobarically from a temperature of 300 K to 310 K. How much heat is required?
Q = McpAT = (2kg)(1004Jkg-1K-1)(10K) = 20080 J = 20.1 kJ. (3.22)
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