Molar heat capacity of gases

Heat capacity in chemistry

The heat capacity of a substance is defined as the amount of heat required to raise the temperature by one degree.

Usually, the rise in temperature measured in centigrade and heat capacity expressed in calories and heat capacity depends upon the amount of material in the system.

Heat capacity per gram of substance is called specific heat and per mole called molar heat capacity or simply heat capacity of the substances.

Heat capacity at constant volume and constant pressure

The magnitude of heat capacity depends on the pressure and volume, especially in the cases of properties of gases. The temperature of a gm -mole of gas raised by one degree at constant volume. Again in another operation same raise of temperature allowing the volume to very.

The quantity of heat in the two operations would be different. Hence, in mentioning the heat capacity of gas, the condition has to specify.

  1. Heat capacity at constant volume denoted by Cv.
  2. Heat capacity at constant pressure denoted by Cp.

Definition of the molar heat capacity

The amount of heat required to raise the temperature of one gram of a substance by 10K called specific heat.

The heat input required to rise by 10K the temperature of one mole of the substances is called its molar heat capacity or simply heat capacity. The heat capacity denoted by C.

Molar heat capacity = molar mass × specific heat.

Cp = M × cp

Cv = M × cv

where Cp and Cv are the molar heat capacities at constant pressure and constant volume respectively. 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 atomicity of the gas. Name the gas if possible.

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

Specific heat capacity formula

Let dq heat required to increase the temperature dT for one mole of the substances and its heat capacity = C.

\therefore C=\frac{dq}{dT}

dq is a path function and values of dq depend on the actual process followed. We can place certain restrictions to obtain precise values of heat capacity. The usual restrictions are at constant pressure and at constant volume.

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

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

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}

Mechanical work and heat

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.

Molar heat capacity from mechanical work
Heat capacity from mechanical work

CP is some mechanical work 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

CP – CV = R(T+1) – RT

∴ CP – CV = R = 2 calories.

Determining of Cv from kinetic energy

Consider the monoatomic gases like argon or helium. If such gases heated at constant volume, the heat supplied will be entirely utilized for increasing the kinetic energy of the translation of the molecules.

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

No energy used to do any mechanical work of expansion when 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 two heat capacities 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 of heat 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

Heat capacity difference CP – CV = 2 calories remains constant for all gases but heat capacity ratio

\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 heat capacity, CV.

\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

\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

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