1.The first law

The system is the pot containing water closed by the lip. This system is placed on its surrounding which is the burner of a stove that is a resistive heating element connected to a battery. This element converts electrical energy to heat. The surrounding supplies energy (heat Q) transferred to the system. The temperature of the system is increased from T_i to T_f and the volume of water increases from V_i to V_f. This volume expansion gives a certain work (W = ∫pdV). This work W is responsible of lifting the lip. To recap, we give heat, we receive work.

The first law of Thermodynamics states two things:

1. The given heat Q is not exactly equal to the work received; a part of the heat is used to increase temperature, say to increase the internal energy U of the system of ΔU from U_i to U_f. Forethermore, we have: Q = W + (U_f - U_i), or
   Q - W = ΔU = U_f - U_i
   That is the the conservation of energy
   
2. We can have any difference Q - W that can equal ΔU. So the work W does not depend on the initial or on the final state of the system, it depends on the difference between them. This difference Q - W is equal to the net energy tranferred to the system from the surroundings. This difference is equal to the change in the internal energy of the system ΔU. The quantities Q and W that depend on the details of the process are not variables of state for the system but, in contrast, U_i and U_f are variables of state for the system.

The internal energy of a system is interpreted on the molecular level. Neglecting the potential energy of interaction between molecules for an ideal or a dilute gas, the internal energy is simply the sum of the kinetic energy of each molecule in the system. A change (increasing or decreasing)in the internal energy for an ideal gas corresponds to a change in the kinetic energy, therfore in temperature; for a denser gas, the potential energy between molecules contributes to this change.

Sign conventions:

Heat Q is positive if energy is added to the system from its surroundings and negative if it is extracted from the system.
Work W done by the system is positive if energy is transferred to the surroundings and negative if energy is transferred to the system from the surroundings.

2. Applications

2.1 Isochoric process

Process in wchich the volume remains fixed. In this case, dV = 0, then W = 0, that is no work is done by the system. Therfore Q = ΔU which means the heat added to the system is used to change the internal energy.

2.2 Adiabatic process

Process in wchich no energy is added from the surroundings. In this case Q = ), then ΔU = - W
This kind of process occurs when the system is insulated with adiabatic walls or in the case that of the process is performed rapidly so that the amount of heat transferred is negligible. (adiabatic compression in an automobile engine).

2.3 Isobaric process

Process in wchich the pression remains constant. The infinitesimal stage is written as:
dW = pdV, then W = p_i ΔV. p_i is the constant pressure.

2.4 Isothermal process

Process in wchich the temperature remains constant, that is the internal energy remains constant. Therefore Q = W . All the amout of heat is converted to work. This kind of process occurs during the phase change because at this point no temperature, then no internal energy is changing.

2.5 Cyclic process

Process in wchich the initial internal energy U_ireturns back to the final internal energy U_f; (the system is returned to the same state from which it started). Therfore ΔU = 0; then: Q = W.

2.6 Free expansion

When initially a gas is is present in a half of a container, the other half is vacuum, connecting the two parts lead to the free expansion of the gas in order to occupy the entire region. The system ( gas) changes from the initial sate (P_i, V_i, T_i,U_i) to the final state (P_f, V_f,T_f, U_f)
As the container is insolated and no work is done ( to the surroundings) then Q = W = 0 . Therefore ΔU = 0 or U_i = U _f. For a real gas, the temperature changes slightly; for a dilue gas or ideal gas, the temperature does not change. In this case, so the internal energy depends only in the temperature.

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