Energy can be exchanged between the system of interest and its surroundings. However,

the total energy of the system plus the surroundings is constant.

That's the First Law of Thermodynamics. The First Law is also stated as energy is conserved.

This is an empirical law, which means that we know that energy is conserved because of many repeated experiments by scientists. It's been observed that you can't get any more energy out of a system than you put into it . James Prescott Joule did a famous experiment which demonstrated the conservation of energy and showed that heat and work were both of the same nature: energy. His system of interest was water in a thermally insulated container. In this container was also a paddle which was connected to the outside world (surroundings) and connected to weights on a string. Joule measured the work done by the paddle wheel and he also measured the heat created by the wheel turning in the water. Significantly, Joule found that the amount of energy done as work was converted exactly to heat. Energy was changed from one form to another (work to heat); however, no net change of energy in the system plus the surroundings occured. Energy is conserved.

Matter can exist in various states (having a certain density, color, heat capacity, phase, etc.) . Given the values of T, P, V, and n of a sample of a pure substance, we will know it's state. Moreover, we know that whenever the matter is in that state it will have the same properties. Early experiments on the variables of state (such as T, P, V, and n) showed that only two of these variables of state need to be known to know the state of a sample of matter. Once two variables are known, the state of the matter is known and the values of the other variables can be determined.

The variables of state can be divided into two types--extensive variables and intensive variables.
- Extensive variables: depend on the amount of substance present. Examples include the volume, energy, enthalpy, and heat capacity.
- Intensive variables: do not depend on the amount of substance present. Examples include the temperature and pressure.

One thing to note is that any extensive variable can be converted to an intensive variable by dividing it by the moles or the mass (like we did with the heat capacity).

An equation of state is an equation which relates the variables of state (T, P, V, and n). It's particularly useful when you want to know the effect of a change in one of the variables of state. Let's look at some situations where the variables of state change:
- Solids and Liquids: If the pressure on a solid or liquid is increased, the volume does not change much. If the temperature is increased, the volume doesn't change much either. Therefore, an appropriate equation of state describing such systems would be: V(T,P) = constant.
- Gases: In contrast, changing the pressure or temperature of a gas will have an easily observable effect on the volume of that gas. For an ideal gas (no intermolecular interactions and no molecular volume), an appropriate equation of state would be: V(T,P,n) = (nRT)/P. There are many equations of state describing real gases. These equations take into consideration molecular volume and interactions. The most well-known such equations is probably the Van der Waals equation: [P + (an^2)/V^2)] [V - bn] = nRT, where a is an experimentally determined constant for molecular attractions and b is a volume correction for the size of the gas molecules.

The internal energy, E, is just the energy of the system. In thermodynamics, it's useful to define the energy in another way, as the energy plus pressure times volume. This new definition is the enthalpy, H = E + PV. For PV work at constant pressure, the work done is -PV, so you can see that the +PV term in the definition of enthalpy is a correction for the work term in the energy.

We are strictly interested in the changes between the initial and final states of energy and enthalpy because energy and enthalpy are variables of state and depend only on the state of the system. They do not depend on the path used during the change of state. Moreover, you don't need to be concerned with the absolute energy of the two states you're studying; instead it's the difference in the energy between the two states that will be of primary interest. Let's now look at calculating the energy and enthalpy changes that occur when a system changes state, and let's consider calculating the amount of energy needed to cause a desired change in the state of a system.

The energy of a system will change if heat is transferred to or from the system or work is done by the system. If heat and work are the only forms of energy transferred between the systems and surroundings of a closed system, E2 - E1 = q + w. That's the first law.
- Gas expansion at constant temperature. The isothermal expansion of a gas can proceed by two types of paths: reversible and irreversible. Reversible path: In this case, the changes in pressure at any time are very small and the direction of the volume change can be reversed. Pex = Pin - dP. Irreversible path: The direction of the volume change cannot be reversed, and the external and internal pressures will try to reach an equilibrium. The work done, w = - † Pex dV.
- Cyclic paths. In a cyclic path, the initial and final states are the same. Therefore, the change in energy and the change in enthalpy are both equal to zero. All changes in thermodynamic variables of state are equal to zero.