The first law of thermodynamics states that energy is conserved. Energy is never created or destroyed. Energy merely changes forms. So if a system undergoes a change that results in an increase (or decrease) in the amount of energy it has, that energy has to have come from (or gone to) somewhere else. We keep track of energy flow in two ways:
Heat, which we give the symbol \(q\)
Work, which we give the symbol \(w\)
The internal energy for a system is the total energy for that system (potential + kinetic). We are interested in tracking the internal energy as it allows us to know if energy is coming into or out of a system. If there is a change in the internal energy of a system, then energy must have been exchanged between the system and the surroundings. This energy flow is in the form of either heat or work. Therefore, we equate any change in the internal energy of a system with the sum of the heat and the work.
\[\Delta U = q + w\]
Where the change in the internal energy is \(\Delta U\) (sometimes \(\Delta E\) is used for changes in internal energy), the heat is \(q\) and the work is \(w\). The change in internal energy can be positive or negative (as can the heat and the work). The change is defined as the final internal energy minus the initial internal energy
\[\Delta U = U_f - U_i\]
So a negative change means the final energy is lower than the initial energy. This results in energy "out of the system." In words this might be stated as energy flow "out of the system" or "released by the system." A positive change indicates the system has "absorbed energy" or "increased in energy" or "taken in energy."
\(\Delta U < 0 \) Energy goes from system to surroundings
\(\Delta U > 0 \) Energy goes from surroundings to system
The same is true for the sign convention of heat.
\(q < 0 \) Heat is flowing out of the system into the surroundings. Heat is released.
\(q > 0\) Heat is flowing from the surrounding into the system. Heat is absorbed.
\(w < 0\) is work out of the system. We say the system does work on the surroundings.
\(w > 0\) is work into the system. The surroundings do work on the system.
Heat is energy flow from high temperature to low temperature. Heat flow can result in temperature change or heat can flow into or out of the system without any change in temperature. To deal with this, it is best to break processes up into these two situations.
For the first, heat flows and the temperature changes. The amount of temperature change depends on the amount of heat and the heat capacity of the system. Heat capacity relates the temperature change to the amount of heat. Heat capacity (\(C\)) is an extensive property. The more material the higher the heat capacity. Therefore, when calculating an actual temperature change, it is important to know the heat capacity of the object you are working with.
\[q = C\Delta T\]
Here \(C\) is the heat capacity. The heat capacity has units of energy per temperature. Typically we will use J K-1. As it is an extensive quantity it is often tabulated in an intensive form. For example, you might want the heat capacity per mole. This is the molar heat capacity (\(C_{\rm m}\)). This will have units of J mol-1 K-1. Alternatively, the heat capacity might be the "specific heat capacity" (\(C_{\rm s}\)). This is the heat capacity per mass. This could have units of J g-1 K-1. As a result, some calculations require you to multiply the molar or specific heat capacity by the number of moles (\(n\)) or by the mass (\(m\)) of a substance. Pay attention to the units and you'll keep it straight. Here are the formulas for heat using molar heat capacity and specific heat capacity:
\[{\rm molar:} \hskip20pt q = n \;C_{\rm m}\; \Delta T \]
\[{\rm specific:} \hskip20pt q = m \;C_{\rm s}\; \Delta T \]
There can also be heat flow without temperature change. For this case, the energy that flows in or out is not changing the thermal kinetic energy, but instead it is changing the potential energy of the system. The potential energy can change if there is either a chemical change or a physical change such as a phase transition. We will address this further in the thermochemistry section.
Work is energy related to a force acting over a distance. There are many different forms of work, but in chemistry we are interested in only two of them. Pressure volume work (or PV work), which is related to compression or expansion of materials. And electrical work, which is related to current flow and electrical potential. For now, we will focus all of our attention on PV work and we will address electrical work later on in the context of electrochemistry.
Pressure volume work is important when a system is changing volume. This is because in order to change its volume it will have to push against the force of the pressure that is exerted on the system. Since the atmosphere is always exerting pressure down upon us, any expansion is always acting against this force.
PV work is best visualized as a volume change of a gas in a cylinder with a movable piston. The pressure pushing down on the piston has some force per area. As the gas expands the piston is pushed upward and this requires some energy (a force acting across a distance). The work is equal to the force x distance. For a gas in a cylinder this works out to be the pressure x the change in volume.
Specifically, we will address the case of expansion or compression against a constant external pressure since this is a case that is typically important in chemistry. For this case
\[w = -P_{ex} \Delta V\]
where w is the work, \(P_{ex}\) is the external pressure and \(\Delta V = V_f - V_i\) is the change in the volume of the gas. The negative sign is to keep our sign convention for energy going into or out of the system. Expansion will lower the energy of the system (energy out). Since this process increases the volume, \(\Delta V\) will be positive. To make the work negative we add a negative sign. With a constant external pressure, if there is no change in volume then there is no work.
It should be noted that the work for expansion can depend on how the expansion is done (constant pressure, changing the pressure, one step, 20 steps...). We will be dealing almost exclusively with the case of the constant external pressure. However, you should realize that while the magnitude will change depending on the process, the sign will not. Expansion is always work out, so w<0. Finally, if there is not change in volume, then w = 0
It is important to be careful with your units when combining energies calculated for work and heat. Work almost always ends up with units for pressure times volume (an energy). For example, L-atm. You need to covert this to Joules in most cases. 101.325 J = 1 L-atm (also 100 J = 1 L-bar).