Show that it follows directly from the equation of state, P RT , that the internal energy of an ideal gas is a function of temperature only Solution from ( ) and p RT , we have But, mathematically, Equating the coefficients of dT and d in these two expressions gives Noting that 2 (T) 2 (T) we obtai...

This question seems to revolve around the thermodynamics of an ideal gas and specifically aims to show that the internal energy of an ideal gas is a function of temperature alone To prove this the que ...

Show that it follows directly from the equation of state, P = RT/, that the internal energy

Question:

Show that it follows directly from the equation of state, P = RT/α, that the internal energy of an ideal gas is a function of temperature only.

Solution: from (??) and p = RT/α, we have image text in transcribed

But, mathematically, image text in transcribed

Equating the coefficients of dT and dα in these two expressions gives image text in transcribed

Noting that ∂²η/(∂α∂T) = ∂²η/(∂T∂α) we obtain image text in transcribed

Thus, (∂T/∂α)_T = 0. Because, in general, the internal energy may be considered either a function of temperature and density or temperature and pressure, this proves that for an ideal gas the internal energy is a function only of temperature.

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