Definition of Law of Conservation Energy

The first law of thermodynamics principles states the conservation of energy of our universe. Study of this conservation law provides the calculation formula of heat change and transfer and internal energy and work done for the cyclic, isothermal, isochoric process and isolated system in thermodynamics and the definition of this law is,

Energy can neither be created nor be destroyed but it may be transferred one form to another.

In other words, when one form of energy disappears, an exact and same amount of other form appears. Thus the net amount of energy in our universe must be constant.

1st law of thermodynamics and conservation of energy, heat change and internal energy

Law of Conservation of Energy Formula

Let q amount of heat supplied to the system containing one-mole gas in a cylinder fitted with a frictionless weightless moveable piston.

At constant pressure, the gas expands from V1 to V2 and the temp changes from T1 to T2. Thus according to 1st law of thermodynamics

q = dU + w
where dU = change of internal energy

If the work restricted to pressure-volume work than the work performed for this system w = PdV.

∴ q = dU + PdV

But U = ∫(T, V)

    \[ \therefore dU=\left ( \frac{dU}{dT} \right )_{V}dT+\left ( \frac{dU}{dV} \right )_{T}dV \]

Putting this value in the energy law

    \[ q=\left ( \frac{dU}{dT} \right )_{V}dT+\left [ P+\left ( \frac{dU}{dV} \right )_{T} \right ]dV \]

But at constant volume

    \[ \left ( \frac{q}{dT} \right )_{V}=\left ( \frac{dU}{dT} \right )_{V} = C_{V} \]

where Cv = molar heat capacity at constant V.

Thus the generalized mathematical formula of the first law of thermodynamics energy transfer for n mole of a gas

    \[ q=n\, C_{V}\, dT+\left [ P+\left ( \frac{dU}{dV} \right )_{T} \right ]dV \]

Thermodynamics First Law of Ideal Gases

The thermodynamics equation of state

    \[ P+\left ( \frac{dU}{dV} \right )_{T}=T\left ( \frac{dP}{dT} \right )_{V} \]

From ideal gas law

    \[ \left ( \frac{dP}{dT} \right )_{V}= \frac{nR}{V} \]

    \[ \therefore \left ( \frac{dU}{dV} \right )_{T}=\left (\frac{nRT}{V}-P \right )=0 \]

Thus the mathematical form of the first law of thermodynamics for the ideal gas

q = nCVdT + pdV

Thermodynamics First Law of real gases

Van der Waals equation for n mole real gases

    \[ \left ( P+\frac{an^{2}}{V^{2}} \right )\left ( V-nb \right )=nRT \]

    \[ \therefore T\left ( \frac{dP}{dT} \right )_{V}=\frac{nRT}{V-nb}=\frac{an^{2}}{V^{2}} \]

Thus from the first law thermodynamics

    \[ q=n\, C_{V}\, dT+\left (P+\frac{an^{2}}{V^{2}} \right )dV \]

Example of Conservation of Energy

For an isolated system, the value of q and w are zero since no interaction of the system with the surroundings can take place. Hence we can conclude that the value of the energy function of an isolated system is constant.

For example, this is another principles of conservation of energy and this states as

No matter what changes of state of an isolated system, thus the value of energy function always constant.

Change in Internal Energy Formula Thermodynamics

Internal energy changes in the cyclic process
dU = 0
Thus according to the first law
q = w

Therefore, form the conservation law of thermodynamics, heat is completely converted into work for the cyclic process.

Heat change in the isothermal process

Internal energy change in the isothermal process for the ideal gas

dU = nCVdT = 0
∴ q = w

Therefore thermodynamics heat is completely converted into work for the ideal gas in the isothermal process.

Internal Energy in the Isochoric Process

Volume change in isochoric process
dV = 0

The 1st law of thermodynamics when dV = 0
q = dU = nCVdT

Thus heat change to an isochoric process only increases the internal energy or temperature of the system.

Internal energy change in an isolated system

In an isolated system, neither energy nor matter can be transfer to or from it.

∴ q = 0 for isolated system

From the 1st law of thermodynamics
0 = dU + w
or, w = -dU

Therefore, from this conservation law of thermodynamics, an isolated system, internal energy uses for the work done by the system.