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The state of a real gas if changed from pressure p1, temperature T1 to pressure p2 temperature T2. The change in enthalpy, h1 – h2, is given by
1. \(\mathop \smallint \limits_{{T_1}}^{{T_2}} {C_p}dT\)
2. \(\mathop \smallint \limits_{{T_1}}^{{T_2}} {C_p}dT + \mathop \smallint \limits_{{p_1}}^{{p_2}} {\left( {\frac{{\partial V}}{{\partial p}}} \right)_T}dp\)
3. \(\mathop \smallint \limits_{{T_1}}^{{T_2}} {C_p}dT + \mathop \smallint \limits_{{p_1}}^{{p_2}} \left[ {V - T{{\left( {\frac{{\partial V}}{{\partial T}}} \right)}_p}} \right]dp\)
4. \(\mathop \smallint \limits_{{T_1}}^{{T_2}} {C_p}dT + \mathop \smallint \limits_{{p_1}}^{{p_2}} \left[ {V - T{{\left( {\frac{{dV}}{{dP}}} \right)}_T}} \right]dp\)

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Correct Answer - Option 3 : \(\mathop \smallint \limits_{{T_1}}^{{T_2}} {C_p}dT + \mathop \smallint \limits_{{p_1}}^{{p_2}} \left[ {V - T{{\left( {\frac{{\partial V}}{{\partial T}}} \right)}_p}} \right]dp\)

Tds = dh – VdP

\(Tds = {c_p}dT - T{\left( {\frac{{\partial V}}{{\partial T}}} \right)_P}dP\)

From the above two Tds equations

\(dh = {c_p}dT - T{\left( {\frac{{\partial V}}{{\partial T}}} \right)_P}dP + VdP\)

\(dh = \mathop \smallint \limits_{{T_1}}^{{T_2}} {c_p}dT + \mathop \smallint \limits_{{P_1}}^{{P_2}} \left[ {V - T{{\left( {\frac{{\partial V}}{{\partial T}}} \right)}_P}} \right]dP\)

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