The first law can be expressed in many ways. One of the more useful expressions is that the change in internal energy, DU, of a system in any process is equal to the heat, q, added to the system, plus the work, w, done on the system by its surroundings:

For a general process: dU = dq - P_exdV

For a reversible process: P_ex = P and dq = TdS so dU = TdS - PdV

****When thermodynamics is being discussed, a reversible process is a process that occurs through a succession of equilibrium states.

****dU = TdS - PdV is true for any process. But generally speaking, TdS is not heat and -PdV is not work. However, when a reversible process is in question, TdS is heat and -PdV is work.

For some dU:

dq - P_exdV = TdS - PdV ® TdS = dq - PdV_ex + PdV

Why can you do what’s just been done? Every variable involved in the equation dU = TdS - PdV is a state function. The value of a state function depends only on that given state and on no other possible state of the system. So the once reversible process, dU = TdS - PdV, becomes a general process and can be set equal to the general process, dU = dq - P_exdV.

TdS = dq - P_exdV + PdV

so

TdS = dq + (P - P_ex )dV

• Case 1: P_ex > P then (spontaneous) dV is negative so (P - P_ex)dV is positive.
• Case 2: P > P_ex then (spontaneous) dV is positive so (P - P_ex)dV is positive.
• Case 3: P = P_ex then (spontaneous) dV is zero so (P - P_ex)dV is zero.

So for any spontaneous process, the mathematical statement of the second law of thermodynamics is

This law says that there is a direction to the way events occur in nature. When a process occurs spontaneously in one direction, it is nonspontaneous in the reverse direction. It is possible to state the second law in many different forms, but they all relate back to the same idea about spontaneity. One of the most common statements found in chemical contexts is that in any spontaneous process the entropy of the universe increases.

We know

now let T1 ® 0

So the mathematical statement of the third law is

The third law of thermodynamics says that the entropy of a pure, crystalline solid at absolute zero temperature is zero. An alternative statement of the third law is that absolute zero is unattainable. In order to realize this, one must consider the heat capacity data near T ® 0. For S₀ to have importance C_p/T must be finite as T ® 0. Thus C_p ® 0. But C_p = dq/dT ® 0 implies dT/dq ® ¥ . In other words, an extremely tiny amount of heat causes a vast change in temperature.