# Heat, work and law of conservation of energy

Heat

Heat is another mode by which a system can exchange with the surroundings. Wherever a temperature difference exists between the system and surroundings heat either flows in or out of the system. It is not a state function because the quantity of heat involved in a process depends upon the path. Heat like work is an extensive property.

It is not a property of system or surroundings. Heat as such cannot be measured at all, but the effects which it produces are measured. It was formerly measured by the increase in temperature of water. The amount of heat required to raise the temperature of 1 g of water by 1°C becomes a unit of heat, the Calorie.

In modern practice, it is defined in terms of Joule. Because Joule, in 1850, showed that there is a definite relationship between mechanical work done (w) and heat produced (H).

I.e.

$w \propto H \\ w = JH$

‘J’ is known as mechanical equivalent of heat. Its numerical value is taken as $4.185 \times 10^{-7}$  ergs = 4.185 Joules.

Heat is an algebraic quantity and the convention used for heat is q. A + q shows that heat is added to the system and a – q means that the system has lost the heat.

Work

Mechanical work is done whenever there is a change (increase or decrease) in the volume of the system i.e. expansion or compression of a gas.

pressure volume work

This is known as Pressure volume work or PV work or Expansion work.

Consider a gas enclosed in a cyclinder provided with a piston.

If P is the pressure of the gas, it exerts a force F on the piston given by:

F = PA . .. (i)

where A is area of cross-section of the piston. This force can be balanced by an equal force, Fem acting on the piston, (fig). If there is an infinitesimally small movement of the piston (dl) outward, the small amount of work done (dw) by the gas (system) on the surroundings will be given by,

$dw = F \times dl$

$= P \times A \times dl$

$P \times dV ( \therefore dV = A \times dl )$ ………….(ii)

where dV is the small increase in volume of gas that has taken place in the process. This process of expansion may be carried out in infinitesimally slowly (i.e. in a thermodynamic reversible) manner in a series of steps. The work done in each step will be given by PdV.

If, ultimately, the volume of the system changes by a finite quantity, say, from $V_1 \text{to} V_2$ then the total work (w) done by the system on the surroundings will be obtained by the integration of the factor PdV.

I.e.

$w = \int^{V_2}_{V_1} P d V$ ………(iii)

where P is a variable factor.

On the other hand, if there is infinitesimal contraction of the gas resulting from infinitesimal movement of the piston inward, then the small amount of work done by the surroundings on the system will be given by:

Dw = PdV

where ‘dV’ is the decrease in volume of the gas that has taken place in the process. If the work of contraction is carried out in the above manner in a series of steps, the work done (w) by the surroundings on the system is given by integration of the factor PdV.

i.e.

$\int ^{V_1}_{V_2}P dV$

When the volume of the system decreases from $V_1 \text{to} V_2$.

Work done at constant pressure

If the-pressure P remains constant throughout the process, the above integration gives

$w = P(V_2- V_1)$

i.e.

$w = P \Delta V$ ………….(iv)

where $V_1$ is the volume of the system in the initial state,. Then $\Delta V$, evidently, is the change in volume of the system.

Thus,

If $\Delta V$ is positive i.e. the gas expands in the process, w will have a positive value. The work, in this case, is done by the system on the surroundings.

If $\Delta V$ is negative i.e. the gas undergoes contraction, w will have a negative value. The work, in this case, is done by the surroundings on the system.

Work done at constant volume

If the volume is constant,

dV  = 0

then w = 0 …..(v)

Maximum work

In the above example, suppose the pressure applied on the piston is negligibly small in comparison with the pressure of the gas inside the cylinder. The gas will then expand rapidly i.e. irreversibly. In this case, the work done by the system will be negligibly small since the opposing force has been negligibly small.

If the opposing pressure on the piston is zero, the work done by the system will be zero.

PdV = 0 ……(vi)

Hence, it follows that when a gas expands freely (free expansion ) i.e. when it expands against vacuum such that P = 0, no work is done by the system.

It also follows from the above discussion that the magnitude of work done by a system on expansion depends upon the magnitude of the opposing (external) pressure. The close is the opposing pressure to the gaseous system in the cylinder, the greater will be the work performed by it on expansion.

In other words, maximum work is obtained when the two opposing pressures differ only by an infinitesimally small amount from one another.

This condition, evidently, is demanded for an ideal reversible process. Hence the condition for maximum work coincides with that for thermodynamic reversibility.

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