Intutive understanding of internal energy Our high school teacher told us that once we calculate the formula of internal energy, it is valid everywhere. I cannot understand how? I asked him, he simply answered because it is state function. It made me even more confused.



From first law, $\mathrm dU=\mathrm dq+\mathrm dW$

In constant volume process, volume differntial $\mathrm dV=0$, hence $\mathrm dW=0$, so

$(\mathrm dU)_V=(\mathrm dq)_V$ and $(\mathrm dq)_V=C_V\mathrm dT$


  So, $(\mathrm dU)_V=C_V\mathrm dT$




Why we use this formula everywhere even when volume is changing (since we derived this relation for a constant volume process), at least for ideal gas?
For an arbitrary fluid (not necessarily an ideal gas), a combination of the first- and second laws of thermodynamics tells us that, $$\left(\frac{\partial u}{\partial v}\right)_T=-\left[P-T\left(\frac{\partial P}{\partial T}\right)_v\right]$$where U is the molar internal energy and v is the molar volume.  For a fluid which obeys the ideal gas law Pv=RT, the right hand side of this equation is zero.  So the internal energy of a substance which obeys the ideal gas law is independent of volume.  An ideal gas is a substance which obeys the ideal gas law because the interactive forces between its molecules are negligible.  So, yes, it is because ideal gas molecules do not have attractive forces.