Question

Obtain expressions for dU, dH, dA, and dG for an ideal gas and for a van der Waals gas.

Answer 1

The first law

The first law of thermodynamics is a statement of the conservation of energy. Specifically, the first law addresses the interconvertability of work and heat as forms of energy.

In words the law is

Internal energy = heat + work

This can be expressed symbolically as

dU = dq + dw

or in integral form as

DU = q + w

The Delta indicates that U is a state function. This means that the change in internal energy depends only on variables of state T, V, etc. and not on the path taken from one state to another. Heat and work, on the other hand, do depend on the path taken.

Calculating the work is possible starting with the definition

work = force x distance

dw = -F dr

This can be recast by dividing the work by an area and by multiplying the displacement by an area

dw = -(F/A) d(rA) or dw = -P dV

We can see immediately that pressure volume work follows from the standard definition of a force operating through a distance. Note the thermodynamic sign convention that states that the work done by the system is negative and the work done on the system is positive.

The heat can be calculated using the constant volume heat capacity.

heat = heat capacity x temperature change

dq = C_v dT

The second law

The second law of thermodynamics tells us that the entropy of system is

dS = dq/T

for a reversible process. Note that this definition only applies to a reversible process, but that is consistent because the entropy of the system will always be calculated along a reversible path. In general, if we consider the system and the surroundings the second law states that the entropy of both (i.e. the universe) tends toward a maximum. If a change is reversible the heat exchanged between the system and surrounds dq_rev will have a magnitude such that dq_rev/T exactly is equal to the entropy change in the system dS. For an irreversible process the heat exhanged between the system and surroundings, dq_irr will be less than TdS such that

dS > dq_irr/T

For a reversible process we can write a combined expression for the first and second laws of thermodynamics

dU