First law for control mass, cyclic process

Derivation of the first law for a control mass

The first law is based on Joule's experiments: when a closed system undergoes an adiabatic process between two states, the work done depends only on the end states. That defines a property, the energy E:

−W_adiabatic = E₂ − E₁

If heat is also exchanged, the energy of the system changes by the net energy transferred:

Q − W = E₂ − E₁ = ΔE = ΔU + ΔKE + ΔPE

In differential form: δQ − δW = dE; in rate form: Q̇ − Ẇ = dE/dt.

Explanation

• Q is positive when heat is added to the system, W is positive when work is done by the system.
• E is a property (state function) and Q and W are path functions, but the difference Q − W is path independent.
• For a stationary system ΔKE = ΔPE = 0, so Q − W = ΔU = m(u₂ − u₁).

Reduction for a cycle

In a cyclic process the system returns to its initial state, so E₂ = E₁ and ΔE = 0:

∮ δQ − ∮ δW = 0 ⇒ ∮ δQ = ∮ δW

The net heat added in a cycle equals the net work done: Q_net = W_net. Example: for a power cycle, Q_H − Q_L = W_net.