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[tex] dU = C_V\,dT [/tex]

But I don't understand why they use C_V, neither the volume nor the preassure is constant during the adiabtic process.. I know that

[tex] dU = C_V\,dT = C_p\,dT - p\,dV [/tex]

But why can that expression for dU be used for the internal energy. In the book "Physics for Scientists and Engineers", they say that because U is only a function of temperatur, then

[tex] U(T)=\int_0^{T} C_V\,dT [/tex]

Without explaining why.. I hope someone can explain this. Thx in advance.