# UCL WIKI

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• First Law of Thermodynamics

# First Law of Thermodynamics

## The first law defined

One important concept in thermodynamics is the relationship between heat and work.However, there is no logical relationship between heat and work and so it had to be found experimentally.  The experiments were carried out by J P Joule between 1840 to 01849.  The important experiment involved a system consisting of a well-lagged cylinder containing water going through a cycle composed of two processes.
In the first process, shear work was done on the system when a paddle turned as a weight was lowered, as shown below.
!1st law work.JPG|align=center!As a result of this process the water temperature was found to rise steadily.  When the temperature reached a certain value, the paddle wheel was stopped and the work done noted (historically in foot pounds-force).
In the second process the system was brought into contact with a cold body so that heat was transferred from the system, as shown below. This process was then cut-off when the system reached its original state and again the amount of heat transfer was noted down (in British Thermal units or calorie).
After making the measurements for a variety of systems and for various amounts of work and heat, it was found that the amounts of work and heat were always proportional.  These observations were then formalised into the first law of TD which is stated as:
"If any system is carried through a cycle (the end state being precisely the same as the initial state) then the net work is proportional to the net heat transfer".

Or: Where J is a constant known as the Mechanical Equivalent of Heat.  In British units J is 778 ft lbf per Btu and in SI units its 1. Nm per Joules i.e. ## The First Law for Non-cycle Processes

We defined first  law for cyclic processes, but what happens when we have a change of state.  For this purpose consider a system that undergoes a cycle changing from state 1 to 2 by process A and returning from state  2 to 1 by process B, as shown in the image below:  Now consider another cycle, the system changing from State 1 to 2 by process A and returning to state 1 by process C, so that Subtracting the equations we get: or on rearranging: ## Energy Derivation

Since B & C represent arbitrary processes between states 1 and 2, we can conclude that the quantity (dQ -dW) is the same for all processes between states 1 and 2.  Therefore, (dQ -dW) depends only on the initial and final states and not on the path followed between the two states.  That is this quantity is a point function, which means that it is a property of the system, called the energy of the system and is given smybol E- When we integrate this equation we get: where:
1Q2 is the heat transferred to the system during process 1 - 2
E2 and E1 are the energy at the initial and final states
1W2 is the work done during process 1 - 2

E represents all the energy in the system i.e. E = P. E + K.E. + energy associated with position and motion of molecules + chemical energy (e.g. storage battery), etc.
In thermodynamics it is convenient to consider the bulk K. E. and P.E. separately and then to consider all the other energy of the system in a single property that we call internal energy, U.
So: E = U + KE + PE
dQ = dU + d (KE) + d(PE) + dW

Expression for KE
Consider a system initially at rest acted on by a horizontal force F that moves the system a distance dx in the direction of the force.
dW = - Fdx = - d KE  *Expression for PE*
Consider a system initially at rest.  Let this system be acted on by a vertical force F that is of such magnitude that it raises (in elevation) the system with constant velocity an amount dz.  If acceleration due to gravity is g
dW = - F d Z = - dPE
But:   F = Ma = mg
Then:
dPE = FdZ = mgdZ  Thus we can write: 