The expression for the differential of the internal energy \( dU(T, V) \) for a gas described by the following equations:
\( p(T, V) = aT^{1/2} + bT^3 + cV^{-2} \)
\( C_V(T, V) = dT^{1/2}V + eT^2V + fT^{1/2} \)
where \( a \), \( b \), \( c \), \( d \), \( e \), and \( f \) are constants independent of \( T \) and \( V \).
1
\( dU = (dT^{1/2}V + eT^2V + fT^{1/2}) dV + \left( \frac{1}{2}aT^{1/2} - 2bT^3 - cV^{-2} \right) dT\)
2
\( dU = (dT^{1/2}V + eT^2V + fT^{1/2}) dT + \left( \frac{1}{2}aT^{1/2} - 2bT^3 - cV^{-2} \right) dV \)
3
\( dU = (dT^{1/2}V + eT^2V + fT^{1/2}) dV - \left( \frac{1}{2}aT^{1/2} - 2bT^3 - cV^{-2} \right) dT\)
4
\( dU = (dT^{1/2}V + eT^2V + fT^{1/2}) dT - \left( \frac{1}{2}aT^{1/2} - 2bT^3 - cV^{-2} \right) dV \)