If k is constant, can cp and cv change?

Question

From thermodynamics, the ratio of specific heat is k = cp/cv. Also, cp = kR/(k-1) and cv = R/(k-1), where R is R_bar (the universal gas constant) divided by the molecular weight of the substance in question. So my question is, if k is constant, can cp and cv change? For cp or cv to change, it seems that the molecular weight would have to change. Could this happen and under what circumstances?

Answer 1

No. k is also a function of temperature. So when the temperature of the gas changes, the cp and cv values of the gas change which in effect changes the k value. In fact, the k value of an ideal gas decreases when the temperature increases. Let me explain further:



Note that k is also equal to the enthalpy divided by internal energy.

k = H/U = m*cp*T/m*cv*T = cp/cv

And the cp is equal to cv plus the specific gas constant.

H = U + PV

m*cp*T = m*cv*T + m*R*T

cp = cv + R



For k to be constant when there is a change in gas temperature, the ratio of gas enthalpy to the gas internal energy has to remain the same. This is improbable because the internal energy of a gas is actually more sensitive to temperature change than the gas enthalpy.



Take for example superheated steam @ 10 barg & 400 deg C. You'll find that the cp = 2.12882 kJ/kg-K and Cv = 1.63738 kJ/kg-K, so k = 1.306 (Note that we are assuming that the superheated steam is an ideal gas. The actual k value is different (a bit smaller) as the steam compressibility factor has to be considered.) Now we increase the temperature to say 410 deg C, the new cp you will note is 2.14060 and the new cv is 1.64148 yeilding a new k value of 1.304.



Let's see refrigerant r-134a gas @ 150 psia & 150 deg F, the cp is 0.2518 Btu/lb-R & k is 1.1997. Now if the tem is reduced to say 110, the cp becomes 0.2697 & k becomes 1.2787.



Therefore to answer your question, if k is constant then the cp and cv values must be the same.

Answer 2

For a particular species, k is not a constant, but molecular weight is. cp and cv are both functions of temperature. As temperature rises or falls, their ratio will change as R is a constant. Thus, for a given species, the same k value will not occur at two different temperatures. That said, the sensitivity of cp, cv and k to temperature is not so extreme that gas species cannot be approximated as having constant properties over certain ranges of temperatures.

Answer 3

yes they can change as cp and cv is dependent on the type of gas whereas k is constant for all gases.