The specific heats do not vary greatly over wide ranges in temperature, as shown in VwB&s Figure 511. It is thus often useful to treat them as constant. If so (T2-7) h=cp(T2-T1) These equations are useful in calculating internal energy or enthalpy differences, but it should be remembered that they hold only for an ideal gas with constant specific heats In summary, the specific heats are thermodynamic properties and can be used even if the processes are not constant pressure or constant volume. The simple relations between changes in energy(or enthalpy)and temperature are a consequence of the behavior of an ideal gas, specificall the dependence of the energy and enthalpy on temperature only, and are not true for more complex substances Adapted from"Engineering Thermodynamics", Reynolds, w. C and Perkins, H C McGraw-Hill Publishers 1. All ideal gases: (a) The specific heat at constant volume(Cy for a unit mass or Cv for one kmol)is a function of T (b) The specific heat at constant pressure(cp for a unit mass or Cp for one kmol) is a function of T only (c) A relation that connects the specific heats cp Cv, and the gas constant Cy= R where the units depend on the mass considered. For a unit mass of gas, e. g, a kilogram, and cy would be the specific heats for one kilogram of gas and r is as defined above. Forone kmol of gas, the expression takes the form: CP-Cv=R, where Cp and Cv have been used to denote the specific heats for one kmol of gas and r is the universal gas constant (d) The specific heat ratio, y, =Cp/cy(or Cp/Cv), is a function of Tonly and is greater than unity 2. Monatomic gases, such as He, Ne, Ar, and most metallic vapors (a)Cy(or Cv) is constant over a wide temperature range and is very nearly equal to (3/2R [ (3/2R, for one kmol (b) Cp(or Cp)is constant over a wide temperature range and is very nearly equal to(5/2)R [or (5/2)R, for one kmol (c) y is constant over a wide temperature range and is very nearly equal to 5/3 [y= 1.67 0-80-8 The specific heats do not vary greatly over wide ranges in temperature, as shown in VWB&S Figure 5.11. It is thus often useful to treat them as constant. If so u2 - u1 = cv (T2 - T1) h2 - h1 = cp (T2 - T1) These equations are useful in calculating internal energy or enthalpy differences, but it should be remembered that they hold only for an ideal gas with constant specific heats. In summary, the specific heats are thermodynamic properties and can be used even if the processes are not constant pressure or constant volume. The simple relations between changes in energy (or enthalpy) and temperature are a consequence of the behavior of an ideal gas, specifically the dependence of the energy and enthalpy on temperature only, and are not true for more complex substances. Adapted from "Engineering Thermodynamics", Reynolds, W. C and Perkins, H. C, McGraw-Hill Publishers 15) Specific Heats of an Ideal Gas 1. All ideal gases: (a) The specific heat at constant volume (cv for a unit mass or CV for one kmol) is a function of T only. (b) The specific heat at constant pressure (cp for a unit mass or CP for one kmol) is a function of T only. (c) A relation that connects the specific heats cp, cv, and the gas constant is cp - cv = R where the units depend on the mass considered. For a unit mass of gas, e. g., a kilogram, cp and cv would be the specific heats for one kilogram of gas and R is as defined above. For one kmol of gas, the expression takes the form: CP - CV = R, where CP and CV have been used to denote the specific heats for one kmol of gas and R is the universal gas constant. (d) The specific heat ratio, γ, = cp/cv (or CP/CV), is a function of T only and is greater than unity. 2. Monatomic gases, such as He, Ne, Ar, and most metallic vapors: (a) cv (or CV) is constant over a wide temperature range and is very nearly equal to (3/2)R [or (3/2)R , for one kmol]. (b) cp (or CP) is constant over a wide temperature range and is very nearly equal to (5/2)R [or (5/2)R , for one kmol]. (c) γ is constant over a wide temperature range and is very nearly equal to 5/3 [γ = 1.67]