# What is positive work and negative work

## Heat, work and energy

Understanding of heat, work and energy is a prerequisite for understanding the first law of thermodynamics. In this article you will learn about all of these topics. We explain step by step, accompanied by learning videos, how warmth, work and energy are defined.

**Table of Contents**

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### Types of energy

For the first main clause we will again use the piston pump from the example in section “0. Main clause ”at hand. First of all, it makes sense to consider what forms of energy can be present in this pump.

The **inner energy** $ u $ is based on the thermal energy that is present at a certain temperature level and which we can describe using the caloric equation of state.

If we hold the piston pump out of the window and drop it, it falls down and hits the floor. This unsurprising circumstance reveals two other forms of energy that a system like the piston pump can exhibit.

The **potential energy**: If we move a system from a stable state (for example standing on the ground) against the acceleration of gravity upwards, the potential energy in this system increases. It increases proportionally with the geodetic height $ z $ (the distance to the stable state, e.g. the earth's surface)

$$ E_ {pot} = m \ cdot g \ cdot z $$

The **kinetic energy**: The movement of a mass results in an energy that can be transferred when the speed of this mass changes (e.g. through a shock). The kinetic energy is calculated with

$$ E_ {kin} = \ frac {1} {2} \ cdot m \ cdot v ^ 2 $$

The piston pump thus contains internal energy and can also contain kinetic and potential energy. How can these energies change?

### Warmth and work

The heat as a form of energy, which is present at a certain temperature level, is already in "*Again the temperature "* encountered. Heat is the amount of energy that can be present in a system. But it is also a process variable. We can add heat to a system (heat water on the stove) or take it out (put warm beer in the freezer).

Work is a new term. We understand it to be a force that is applied to a system along a certain path in order to change the state of the system. When I inflate my tire, I first have to push the piston of my piston pump down. I do work. We call work on a closed system volume change work because we can change the volume of the system along a certain path through the force exerted. The volume change work is defined as follows

**Volume change work** $ W_V = - \ displaystyle \ int p \ \ text {d} V $

But why is volume change work defined negatively?

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### Selfish rule of signs

The piston pump has a volume $ V_1 $ and is compressed to a volume $ V_2

$$ W_ {V, 12} = - p \ displaystyle \ int_ {V_1} ^ {V_2} \ \ text {d} V = \ underbrace {-p \ cdot \ underbrace {(V_2 - V_1)} _ {<0 }} _ {> 0} $$

$ V_2

**Selfish sign rule:**Heat or work that is added to a system is always greater than zero. Heat and work withdrawn from a system are less than zero.

### Volume change share of various changes of state

For isobaric changes of state, we have calculated the equation for calculating the work of volume change:

$$ W_ {V, 12} = - p \ cdot (V_2 - V_1) $$

If we want to represent this equation specifically, we have to divide it by the mass m of the system.

$$ w_ {v, 12} = - p \ cdot (v_2 - v_1) $$

We will derive all equations in a specific form. If you prefer the other representation, you can simply multiply the equations by the mass.

For the other changes in state we have to calculate a little more, because the pressure cannot simply be drawn as a constant in front of the integral. Let's look at isothermal changes in state. It should be noted that we are looking at the changes of state of ideal gases.

The equation for calculating the work of volume change in specific form is

$$ w_ {v} = - \ displaystyle \ int p \ \ text {d} v $$

We have to reformulate the pressure via the temperature in order to be able to integrate according to the volume. Since we integrate according to the specific volume, we use the ideal gas law formulated according to the pressure

$$ p = \ frac {R \ cdot T} {v} $$

and obtained by inserting it into the integral

$$ W_ {V} = -R \ cdot T \ displaystyle \ int_ {v_1} ^ {v_2} \ frac {1} {v} \ \ text {d} v = -R \ cdot T \ cdot \ ln \ left (\ frac {v_2} {v_1} \ right) = R \ cdot T \ cdot \ ln \ left (\ frac {p_2} {p_1} \ right) $$

as an equation for calculating the work of volume change in the case of isothermal changes in state.

Isochoric changes of state are carried out at constant volume. The volume change work is consequently zero.

For polytropic and reversibly adiabatic changes in state, the derivation is a bit more complex, since the caloric equation of state must also be used. The corresponding equations can be found in the appendix and are explained in detail in videos at the end of the section.

### Technical work

Without being aware of it, we have so far considered closed systems (such as the piston pump). Closed systems are characterized by the fact that they cannot carry out any material exchange with their environment.

In thermodynamics we sometimes have to deal with circulatory processes. These cycle processes move mass flows $ \ dot {m} _i $ through a system that go through changes of state. The changes of state in a section of the system take place in open (sub-) systems, to which a mass flow $ \ dot {m} _i $ is permanently supplied and discharged.

Such open systems have no defined volume $ V $ that can be changed by volume change work. Nevertheless, work can be carried out in a circular process, which is referred to as technical work.

$$ w_t = \ displaystyle \ int v \ \ text {d} p $$

The technical work is presented specifically, i.e. per mass. If the pressure is changed isochorically in a circulatory system, the following technical work is required for this

$$ w_t = \ displaystyle \ int_ {p_1} ^ {p_2} v \ \ text {d} p = v \ cdot (p_2-p_1) $$

For isothermal changes in state in an open system, the procedure is analogous to volume change work. The specific volume is replaced by an expression of the (constant) temperature based on the ideal gas law.

$$ w_t = \ displaystyle \ int_ {p_1} ^ {p_2} v \ \ text {d} p = R \ cdot T \ cdot \ displaystyle \ int_ {p_1} ^ {p_2} \ frac {1} {p} \ \ text {d} p = R \ cdot T \ cdot \ ln \ left (\ frac {p_2} {p_1} \ right) $$

No technical work can be calculated for isobaric changes of state. The expression $ \ text {d} p $ in the integral is zero for isobaric changes of state, analogous to the work of volume change for isochoric changes of state. The technical work of polytropic and reversible state changes is derived in detail in the videos at the end of the section and is summarized as an equation in the appendix.

So we now know forms of energy in closed systems and can work on them for every change in state. It is time to deepen this new knowledge with an example.

Before we turn to the first law, we have to understand heat as a process variable.

### Heat in closed and open systems

Like work, heat is a process variable that can be added to or removed from systems. It makes sense to be able to calculate heat supply and removal. We already know the isochoric and isobaric heat capacity from working with the caloric equation of state.

**Specific heat capacity**: Amount of heat that has to be added or removed to heat or cool a $ \ si {kg} $ of a substance by one Kelvin.

For isobaric changes of state, the isobaric heat capacity must be multiplied by the temperature changes to be achieved:

$$ q_ {12, p} = c_p \ cdot (T_2-T_1) $$

We are lucky: the equation corresponds to the selfish rule of signs: If we dissipate heat, $ T_2 $ is less than $ T_1 $, the difference becomes negative and thus the dissipated heat is less than zero.

The isochoric heat supply is then also calculated

$$ q_ {12, v} = c_v \ cdot (T_2-T_1) $$

These equations apply to both open and closed systems.

Isothermal heat supplies are something that we have to get into mentally. If we add or remove heat, the temperature changes as a result. However, if the temperature is to remain the same for isothermal heat supply, we have to compensate for the lack or excess of energy by adding or removing work. If we heat air in a pump, the temperature of the air rises. An isothermal change of state occurs, in which air expanding due to the warming is used as volume change work.

Isothermal change of state

This idea is ideal because it neglects the time dependence of heat transfer. Physically, the amount of heat supplied for an isothermal change of state must be the same as the work removed (or vice versa). It therefore applies to a closed system

$$ q_ {T, 12} = -R \ cdot T \ cdot \ ln \ left (\ frac {v_2} {v_1} \ right) $$

and for an open system

$$ q_ {T, 12} = -R \ cdot T \ cdot \ ln \ left (\ frac {p_2} {p_1} \ right) $$

In the case of reversibly adiabatic changes in state, no heat is added or removed. That is why they are adiabatic. As a result, no heat quantities can be calculated for these changes in state.

For polytropic changes in state, the equation for calculating the heat is given in the appendix.

Learn more about heat and work in the following videos ...

**isothermal change of state (closed system)**

**reversible adiabatic change of state (closed system)**

**open system:**

**polytropic change of state: **

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