Physics Topics are often described using mathematical equations, making them precise and quantifiable.
What is the Importance of Ratio of Specific Heats?
Let us consider n mol of an ideal gas. Its volume V, pressure p and absolute temperature T are related by the equation of state
pV = nRT ……. (1)
where, R = universal gas constant or molar gas constant
= 8.31 × 107 erg ᐧ mol-1 ᐧ °C-1
= 8.31 J ᐧ mol-1 ᐧ K-1 ≈ 2 cal ᐧ mol-1 ᐧ K-1
From the first law of thermodynamics,
dQ = dU + dW = dU + pdV ………. (2)
Now, suppose the temperature of the gas is raised by dT at constant volume. So, the heat taken is dQv = nCvdT, where Cv = molar specific heat of the gas at constant volume. Also, as V = constant, dV = 0. So, from equation (2),
dU = nCvdT …… (3)
For an ideal gas, U is a function of T only. So, for the same change dT of temperature, dU will remain the same for all processes. Then equation (2) can be written as,
dQ = nCvdT + pdV ….. (4)
This equation is applicable to all processes.
Now, Suppose the temperature of the gas is raised by the same amount dT at constant pressure. The heat taken is dQp = ncvT. where = molar specific heat at constant
pressure. So, from equation (4),
nCpdT = nCvdT + pdV or, n(Cp – Cv)dT = pdV
or, n(Cp – Cv) = p\(\frac{d V}{d T}\)
From (1), V = \(\frac{n R T}{p}\)
So at constant pressure, \(\frac{d V}{d T}\) = \(\frac{n k}{p}\) or, p\(\frac{d V}{d T}\) = nR
Then we have,
Cp – Cv = R ……. (5)
So, the difference between the two molar specific heats of an ideal gas is equal to the universal gas constant. As R is a constant, Cp – Cv, has the same value for all gases.
The specific heats per unit mass of an ideal gas are,
cv = \(\frac{C_v}{M}\) and cp = \(\frac{C_p}{M}\)
where M = molecular weight of the gas
Then, cp – cv = \(\frac{1}{M}\)(Cp – Cv) = \(\frac{R}{M}\) = r.
where, r = gas constant for unit mass of the gas.
As the molecular weight M is different for different gases, the value of (cp – cv) is also different. Because of this, 1 mol of a gas is more useful in thermodynamic discussions than unit mass of a gas.
It should be noted that, in the application of equation (5), all the three quantities are to be converted into the same unit. If Cp and Cv, are expressed in unit of heat (cal ᐧ mol-1 ᐧ °C-1) and R in unit of work (erg ᐧ mol-1 ᐧ °C-1), then R is to be divided by J to express it in unit of heat. In this case, equation (5) can be written as,
Cp – Cv = \(\frac{R}{J}\) …… (6)
Importance of the Ratio of the Two Molar Specific Heats
The ratio between Cp and Cv is usually denoted by the Greek letter γ:
γ = \(\frac{C_p}{C_v}\) \(=\frac{\text { molar specific heat at constant pressure }}{\text { molar specific heat at constant volume }}\)
As Cp > Cv, we have γ > 1. The ratio between the specific heats as per unit mass is also the same because
\(\frac{c_p}{c_\nu}\) = \(\frac{C_p / M}{C_\nu / M}\) = \(\frac{C_p}{C_v}\) = γ, where M = molecular weight.
The value of γ is important for different applications in physics and chemistry:
i) The value of γ gives an idea of the molecular structure of any gas.
For monatomic gases, γ = \(\frac{5}{3}\) = 1.67 (helium, neon, argon, etc.)
For diatomic gases, γ = \(\frac{7}{5}\) = 1.4 (hydrogen, oxygen, nitrogen, carbon monoxide, etc.)
For triatomic gases, γ = \(\frac{4}{3}\) = 1.33 (water vapour, carbondioxide, ozone, etc.)
For polyatomic gases like ammonia, methane and many organic gases, the value of γ lies between 1.1 and 1.3. So, the knowledge of γ gives an idea about the number of atoms in a molecule of a gas.
ii) The thermodynamic relationships between different properties of a gas in an adiabatic process involve the value of γ.
iii) The expression for the velocity of sound in a gas involves the value of γ.