Specific Heat capacity of gases

Specific heat capacity definition

The heat capacity of a substance is defined as the amount of heat required to raise the temperature by one-degree but per gram of substance is the specific heat. Usually, the rise of temperature measured the centigrade unit but for the heat capacity measured in the calorie unit.

Molar and specific heat capacity

The amount of heat required to raise the temperature of one gram of a substance by 10K called specific heat and for one mole is called molar heat capacity.

Cp = M × cp
Cv = M × cv

where Cp and Cv are the heat capacities at constant pressure and constant volume respectively and cp and cv are their specific heats.

Problem
The specific heat at constant pressure and constant volumes are 0.125 and 0.075 cal gm-1 K-1 respectively. Calculate the molecular weight and the gas formula. Name the gas if possible.

Solution
M = 40 and ⋎ = 1.66(mono-atomic), Ar(Argon).

Definition of Cp and Cv for gases

The magnitude of heat capacity depends on the pressure and volume, especially in the cases of properties of gases.

  1. The temperature of a gm-mole of gas raised by one degree at constant volume.
  2. Again in another operation same raise of temperature allowing the volume to very.

Thus the observed quantity in the two operations would be different. Hence for mentioning this, the condition has to specify.

  1. Cp for gases at constant pressure.
  2. Cv for the gases at constant volume.

Cp and Cv of some gases at 1 atm pressure and 298 K temp are

Gas CP CV γ
Argon 4.97 2.98 1.66
Helium 4.97 2.98 1.66
Mercury 5.00 3.00 1.67
Hydrogen 6.85 4.86 1.40
Nitrogen 6.96 4.97 1.40
Oxygen 7.03 5.03 1.40
CO2 8.83 6.80 1.30
SO2 9.65 7.50 1.29
water 8.67 6.47 1.34
Methane 8.50 6.50 1.31

Thermodynamics derivation of Cp and Cv

Let dq heat required to increase the temperature dT for one mole of the substances. From the thermodynamic definition

∴ C = dq/dT
where dq = path function

Values of dq depend on the actual process followed. But we can place certain restrictions to obtain precise values of C. The usual restrictions are at constant pressure and at constant volume.

    \[ \boxed{C_{V}=\left (\frac{dq}{dT} \right )_{V}} \]

    \[ \boxed{C_{P}=\left (\frac{dq}{dT} \right )_{P}} \]

From the thermodynamic definition of internal energy and enthalpy,

dU = (dq)V and dH = (dq)P

    \[ \therefore C_{V}=\left (\frac{dq}{dT} \right )_{V}=\left (\frac{dU}{dT} \right )_{V} \]

    \[ \therefore C_{P}=\left (\frac{dq}{dT} \right )_{P}=\left (\frac{dH}{dT} \right )_{P} \]

Unit of the heat capacity

From the thermodynamics definition

    \[ C_{V}=\left (\frac{dq}{dT} \right )_{V}=\left (\frac{dU}{dT} \right )_{V} \]

    \[ C_{P}=\left (\frac{dq}{dT} \right )_{P}=\left (\frac{dH}{dT} \right )_{P} \]

    \[ \therefore unit\, of\, heat\, capacity=\frac{unit\, of\, energy}{unit\, of\, temperature} \]

Thus CGS and SI units of the heat capacity

erg K-1 and Joule K-1

But if we maintained molar and specific heat capacity then per mole and per gram or kg used in these units.

Mechanical work or energy and Heat capacity

A gas can be heated at constant pressure and constant volume in a cylinder fitted with a piston. The gas expands against the piston gives the mechanical work.

In order to attain a 10 rise in temperature, the heat supplied should be sufficient to provide the kinetic energy to the molecules and also able to do extra mechanical work.

Specific Heat capacity of gases from mechanical energy
Mechanical energy

Heat capacity Cp is some mechanical work that is required as additional energy to absorb for lifting piston from volume V1 to V2.

CP – CV = Mechanical work = PdV
= P(V2 – V1)
= PV2 – PV1

If we consider the gas to be an ideal gas
PV = RT

Thus CP – CV = R(T+1) – RT

∴ CP – CV = R = 2 calories

Kinetic energy and specific heat

Consider the monoatomic gases like argon or helium. If such gases are heated at constant volume, it utilized for increasing the kinetic energy of the translation.

Since the monoatomic gas molecules can not any absorption in vibrational or rotational motion.

If no heat used to do any mechanical work of expansion when the volume of the gas remains constant. Kinetic energy for one-mole gas at the temperature T denoted by E.

    \[ E=\frac{3}{2}PV=\frac{3}{2}RT \]

Increase of kinetic energy for 1° rise in temperature for monoatomic gas helium or argon

    \[ \Delta E=\frac{3}{2}R\left ( T+1 \right )-\frac{3}{2}RT=\frac{3}{2}R \]

=3 Calories

The heat supplied at a constant volume equal to the rise in kinetic energy per unit degree rise in temperature.

∴ Cv = ΔE = 3 calories

For a mole of monoatomic gas the ratio of Cp/Cv universally expressed by the symbol ɣ.

    \[ \gamma =\frac{C_{P}}{C_{V}}=\frac{\left ( C_{V}+R \right )}{C_{V}} \]

= (3 + 2)/3
= 1.66

Heat capacity of polyatomic gas

For polyatomic molecules, the haet supplied used up not only in increasing kinetic energy but also increasing vibrational or rotational energies.

Let x calories used for increasing vibrational or rotational purposes.

    \[ \therefore C_{V}=\Delta E+x=\frac{3}{2}R+x \]

CP = ΔE + mechanical work + x

    \[ =\frac{3}{2}R+R+x \]

    \[ =\frac{5}{2}R+x \]

CP – CV = 2 calories remain constant for all gases but Cp/Cv for gases is differing gas to gas.

    \[ \gamma =\frac{\frac{3}{2}R+x}{\frac{5}{2}R+x}=\frac{5+x}{3+x} \]

Heat capacity of gases from the energy equation

The molar energy of monoatomic gas

    \[ E=\frac{3}{2}RT \]

Heat supplied to one gm-mole of a gas kept at a constant volume to increase the temperature by one degree has the CV.

    \[ \boxed{\therefore C_{V}=\left (\frac{dE}{dT} \right )_{V}=\frac{3}{2}R} \]

For polyatomic linear gas

    \[ E=\frac{3}{2}RT+RT+\left ( 3N-5 \right )RT \]

    \[ \boxed{\therefore C_{V}=\left (\frac{dE}{dT} \right )_{V}=\frac{3}{2}R+R+(3N-5)R} \]

For polyatomic non-linear gas

    \[ E=\frac{3}{2}RT+\frac{3}{2}RT+\left ( 3N-6 \right )RT \]

    \[ \boxed{C_{V}=\left (\frac{dE}{dT} \right )_{V}=\frac{3}{2}R+\frac{3}{2}R+\left ( 3N-6 \right )R} \]

Theoretical and experimental heat capacity of gases

Theoretical and experimental values revels due to the following facts

Due to the perfect arrangement of the monoatomic gases

CV/R = 1.5

Thus this value is independent of temperature over a wide range.

But for polyatomic gases, two points of disarrangement found for the observed heat capacities

  1. Observed values are always lower than the experimental values.
  2. They noticeably depend on the temperature.

\frac{C_{V}}{R}=\frac{3}{2}+\frac{2}{2}+1=3.5(calculated)

But the observed values of this lies in the range of 2.5 to 3.5.

Classical mechanics does not describe the variation of these molecular properties of polyatomic molecules. Thus for this purpose, we use quantum mechanics.

The principal of equipartition derived from the classical consideration of continuous absorption of energy governed by maxwell distribution.

But vibrational and rotational energy takes place in discrete quantities. Thus the observed value of specific heat capacity of substance explained only on the basis of quantum theory.