Establish the expression of the differential of the internal energy dU(S (T, V) , V)as a function of the temperature T and the volume V. In the particular case of a gas that satisfies the relation pV = NR T, show that dU (S (T, V) , V) is proportional to dT.

Question

Establish the expression of the differential of the internal energy [latex]dU \Bigl(S (T, V),V\Bigr)[/latex] as a function of the temperature T and the volume V. In the particular case of a gas that satisfies the relation pV = NR T, show that [latex]dU \Bigl(S (T, V),V\Bigr)[/latex] is proportional to dT.

Accepted Answer

According to the mathematical definition (4.76), the differential [latex] dU \Bigl(S(T,V),V\Bigr) [/latex] is expressed as,

[latex] df \Bigl(x(y,z),y\Bigr) = \Biggl(\frac{\partial f\Bigl(x(y,z),y\Bigr) }{\partial x (y,z)}\frac{\partial x(y,z)}{\partial y} + \frac{\partial f \Bigl(x(y,z),y\Bigr) }{\partial y } \Biggr) dy +\Biggl(\frac{\partial f \Bigl(x(y,z),y\Bigr) }{\partial x (y,z)}\frac{\partial x (y,z)}{\partial z} \Biggr) dz [/latex]. (4.76)

[latex] dU \Bigl(S(T,V),V\Bigr) = \Biggl(\frac{\partial U\Bigl(S(T,V),V\Bigr) }{\partial S (T,V)}\frac{\partial S(T,V)}{\partial T} \Biggr) dT +\Biggl(\frac{\partial U \Bigl(S(T,V),V\Bigr) }{\partial S (S,V)}\frac{\partial S (T,V)}{\partial V} +\frac{\partial U \Bigl(S(T,V),V\Bigr) }{\partial V }\Biggr) dV [/latex].

Using the definitions (2.9), (2.10), (4.77) and the Maxwell relation (4.71), we obtain,

[latex] T (S, V, N_1, . . . , N_r) ≡ \frac {\partial U (S, V, N_1, . . . ,N_r) }{\partial S}[/latex]. (2.9)

[latex] p (S, V, N_1, . . . , N_r) ≡ - \frac {\partial U (S, V, N_1, . . . ,N_r) }{\partial v}[/latex]. (2.10)

[latex] \frac{\partial f}{\partial y}\mid _z ≡ \frac{\partial f \Bigl(x(y,z),y\Bigr) }{\partial x (y,z)} \frac{\partial x(y,z)}{\partial y} + \frac{\partial f \Bigl(x(y,z),y\Bigr) }{\partial y }[/latex].

[latex]\frac{\partial f}{\partial z}\mid _y ≡ \frac{\partial f \Bigl(x(y,z),y\Bigr) }{\partial x (y,z)} \frac{\partial x(y,z)}{\partial z} [/latex]. (4.77)

[latex] \frac{\partial p}{\partial T} = \frac{\partial S}{\partial V}[/latex]. (4.71)

[latex] dU \Bigl(S(T,V),V\Bigr) = \frac{\partial U}{\partial T}\mid _V dT + \Bigl(T \frac{\partial p (T,V)}{\partial T} - p (T,V)\Bigr) dV [/latex].

In the particular case of a gas that satisfies the relation p V = NR T, the terms inside the brackets cancel each other out and the differential reduces to,

[latex] dU \Bigl(S(T,V),V\Bigr) = \frac{\partial U}{\partial T}\mid _V dT [/latex].

which is indeed proportional to dT.

Question 4.3: Establish the expression of the differential of the internal...

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