# Heat Capacity

Need for the function: Heat content or enthalpy function is particularly used to measure heat changes accompanying a process at constant volume. It therefore becomes necessary to introduce a new function called heat capacity which relates the heat changes to the temperature changes at constant pressure or at constant volume.

The heat capacity of a system is defined as the quantity of heat required for increasing the temperature of one mole of a system through 1°C. Heat capacity may be given as follows.

C = δq/dT    ….(1)

(i) Heat capacity at constant volume

By first law of thermodynamics,

We have
δq = dE + PdV …(2)

on substituting the value of δq in equation (2) We get

$\large C = \frac{dE + P dV}{dT}$   ….(3)

If the volume is kept constant then

$\large C_v = (\frac{dE }{dT})_v$    …..(4)

Hence the heat capacity at constant volume of a given system may be defined as the rate of change of internal energy with temperature.

(ii) Heat capacity at constant pressure

If the pressure is held constant, equation (3), becomes as follows.

$\large C_p = \frac{dE + P dV}{dT}$

$\large C_p = (\frac{\delta q}{dT})$

Hence the heat capacity at constant pressure of a system may be defined as the rate of change of enthalpy with temperature

### Limitations of first Law of Thermodynamics

1. This law fails to tell us under what conditions and to what extent it is possible to bring about conversion of one form of energy into the other.

2. The first law fails to contradict the existence of a 100% efficient heat engine or a refrigerator.

### Second law of thermodynamics

It has been stated is several forms as follows.

(i) All the spontaneous process are irreversible in nature.

(ii) It is impossible to obtain work by cooling a body below it lowest temperature

(iii) It is impossible to take heat from a hot reservoir and convert it completely into work by a cyclic process without transferring a part of it to a cold reservoirs.

(iv) Heat cannot of itself pass from a colder body to hotter body without the intervention of external work.

(v) It is impossible to construct a machine functioning in cycle which can convert heat completely into equivalent amount of work without producing change elsewhere.

(vi) The entropy of universe is always increasing in the course of every spontaneous process.

(vii) Spontaneous or natural process are always accompanied with an increase in entropy.

### Efficiency of a Heat Engine

The relationship between W , the net work done by the system and q2, the quantity of heat absorbed at the higher temperature T2, in case of the cyclic process (i.e. the carnot cycle), can be obtained from the following two equations.

W = R (T2 − T1)ln(V2/V1) ; q2 = RT2ln(V2/V1)

$\large W = q_2 \frac{T_2 -T_1}{T_2}$

The fraction of the heat absorbed by an engine which it can convert into work gives the efficiency(η)

$\large \eta = \frac{W}{q_2} = \frac{T_2 -T_1}{T_2}$

The net heat absorbed by the system, q is equal to q2 − q1 and according to the first law of thermodynamics, this must be equivalent to the net work done by the system.

Thus, W = q2 − q1

$\frac{q_2 – q_1}{q_2} = \frac{T_2 -T_1}{T_2} = \eta$

Since (T2 − T1)/T2 is invariably less than l , the efficiency of a heat engine is always less than 1.