First of all, make clear three basic independent variables, Pvt. This thermodynamic thing should be in your book.
The following involves the knowledge of advanced mathematics.
Du = (? U/？ T)pdT +(？ U/？ Equation p)TdP is a completely differential form of internal energy. P and t after brackets should be angle marks under brackets, indicating that pressure and temperature are constant changes.
The formula U = f(T, p, n) means that the internal energy is only related to t, p, n, and n is the quantity of matter, so it can be said that the internal energy per unit mass is only related to t, p.
You should have studied derivatives. The physical meaning of derivative is the rate of change of one physical quantity relative to another, but you have only learned one-dimensional derivative, that is, one physical quantity is only related to another physical quantity, and derivative is the rate of change (it can be approximately thought that derivative and differential are the same thing). Because the internal energy per unit mass is related to two physical quantities, P and T, we can do 2 yuan differential, that is, total differential, which is the following formula:
Du = (? U/？ T)pdT +(？ U/？ p)TdP
That is, the sum of (let P find the change rate of U to T) and (let T find the change rate of U to P).
Similarly, it can also be written as U = f(T, p, n).
Also write the total differential of unit mass u.
Du = (? U/？ T)vdT +(？ U/？ 5) TdV
Compare the upper and lower expressions because (? U/？ p)TdP≦(？ U/？ V)TdV (the physical meaning can be simply described as: the rate of change of internal energy with pressure and the rate of change of internal energy with specific volume are different at constant temperature, which is obvious at this time)
So (? U/？ t)v≦(？ U/？ T)p (physical meaning: the change rate of internal energy with temperature is different between constant volume process and constant pressure process)
It may be difficult. These formulas involve the differential relationship between the multivariate differential of high numbers and thermodynamics. Senior two is really difficult to understand, but it will be understood slowly.