## The first law of thermodynamics

Thermal energy is disordered energy.  Work is the conversion of one form of energy into another form.  Heat is the exchange of energy between object because they are in thermal contact with each other.  Can we do work with heat?  Can we convert thermal energy into another form of energy?  Thermodynamics is the study of the way that one does work with heat.

A very simple device that can convert heat into potential energy is a rubber band.  Unlike most other substances, rubber contracts when heated.  We can therefore lift an object by heating a rubber band.  Heat is converted into gravitational potential energy.

A simple rubber band can be the heart of an engine converting heat into electrical energy.  We can use heat to make the rubber band contract.  It then can lift water, converting heat into gravitational potential energy.  This gravitational potential energy of the water can be converted into electrical energy with the help of a turbine generator.  After the load had been lifted, we let the rubber band cool down.  It then has the same internal energy it had before we started the process.  We can repeat this process over and over again.  This certainly is a very inefficient engine.  It wastes a large amount of heat.  Can we make the process of converting heat into other forms of energy more efficient?

### How much work can we possible get out of heat?

Energy conservation limits the amount of work we can get out of a certain amount of heat.  The first law of thermodynamics state that energy is conserved.  We may express it in the following way:

Change in internal energy of a system
= heat put into the system - work done by the system on its surroundings,
or
ΔU = ΔQ - ΔW.

A system can be anything.  It is most convenient if it has well defined boundaries.

• ΔQ is positive if heat flows into the system, negative if it flows out of the system.
• ΔW is positive if the system does work on its surroundings and it is negative if work is done on the system.
• The internal energy U is the sum of the kinetic and potential energy of the atoms and molecules that make up the system.  It is a physical property of the system.

A physical property of a system in a given state can be measured.  It depend only on the state of the system, not on the way the system was put into this state.  For an ideal gas, for example, U = (3/2)NKBT.  The internal energy of the gas depends only on the number N of gas atoms present and on the temperature T of the gas, not on the way the gas has reached that temperature.

#### Problem:

How much heat transfer occurs from a system, if its internal energy decreased by 150 J while it was doing 30.0 J of work?

Solution:

• Reasoning:
Energy conservation <--> the first law of thermodynamics
increase in internal energy of a system
= heat put into the system - work done by the system on its surroundings.
• Details of the calculation:
ΔQ = ΔU + ΔW = -150 J + 30 J = -120 J.
120 J is transferred from the system to its environment.

#### Problem:

A thermodynamic system undergoes a process in which its internal energy decreases by 500 J.  If at the same time 220 J of work is done on the system, find the thermal energy transferred to or from it.

Solution:

• Reasoning:
Energy conservation <--> the first law of thermodynamics
increase in internal energy of a system
= heat put into the system - work done by the system on its surroundings.
• Details of the calculation:
ΔU = ΔQ - ΔW.
-500J = ΔQ + 220 J.
ΔQ = -720 J.

#### Problem:

Suppose a woman does 500 J of work and 9500 J of heat transfer occurs into the environment in the process.  What is the decrease in her internal energy, assuming no consumption of food

Solution:

• Reasoning:
Energy conservation <--> the first law of thermodynamics
increase in internal energy of a system
= heat put into the system - work done by the system on its surroundings.
• Details of the calculation:
ΔU = ΔQ - ΔW = -9500 J - 500 J = -10000 J is the decrease in her internal energy.

#### Problem:

Assume you are running an electric space heater.  Let the space heater be the system under consideration.  It has warmed up and is now running at a constant temperature.  It consumes 500 Watts of electrical power.  How much electrical energy is converted into heat per hour?  (1 Watt = 1 J/s)

Solution:

• Reasoning:
Energy conservation <--> the first law of thermodynamics
increase in internal energy of a system
= heat put into the system - work done by the system on its surroundings.
ΔU = ΔQ - ΔW.
ΔU = 0 since the temperature of the heating element is constant.  Therefore ΔQ = ΔW.
• Details of the calculation:
ΔW is negative, because the work is done on the system by the power supply in providing electrical energy to the system.  ΔQ must be negative.  It is the heat leaving the system.  500 J of heat are leaving the system per second.
500 J/s * 3600 s = 1.8 MJ of electrical energy are converted into heat every hour.  All the electrical energy put into the system is converted into heat.

Any form of ordered energy can be completely converted into thermal energy.

Can we reverse the process?  Can we take the heat generated by the space heater or by some other process and completely convert it back to electrical energy?  Can we design an engine (a device) that accomplices this?  The first law of thermodynamics does not exclude this possibility.  It just says that we will not get more electrical energy out of a system than the energy we put into the system in the form of heat, assuming ΔU = 0, i.e. the internal energy of the system is the same before and after doing the work.