Conservation of energy in thermodynamics

The first law of thermodynamics and conservation of energy

The first law of thermodynamics states the conservation of energy of our universe. Thus this law states as 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.

Derivation of the first law of thermodynamics

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

The first law of thermodynamics and conservation of energy
The first law of thermodynamics

At constant pressure, the gas expands from V1 to V2 and the temp changes from T1 to T2. Thus according to the first 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 internal energy 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 equation

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

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 for n mole of a gas

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

The first law of thermodynamics ideal gas

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 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

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

Law of conservation energy examples

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

This is another statement of the first law and known as the law of conservation of energy.  Thus 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 during a cyclic process

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

Hence 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

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

Change in internal energy isochoric process

Volume change in isochoric process
dV = 0

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

Thus heat supplied 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 law of conservation energy
0 = dU + w
or, w = -dU

Thus in an isolated system, internal energy uses for the work done by the system.