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Derive the relationship between Delta H and Delta U for an ideal gas. Explain each term involved in the equation. |
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Answer» <P> Solution :From the first law of thermodynamics, `q= Delta U + p DeltaV` If the process carried out at constant volume, `Delta V= 0`Hence, `q_(v) = Delta U` [Here, `q_V,=` Heat absorbed at constant volume, `Delta U =` CHANGE in internal energy] Similarly, `q_p = Delta H` Here, `q_p =` heat absorbed at constant pressure `Delta H =` enthalpy change of the system. Enthalpy change of a system is equal to the heat absorbed or evolved by the system at constant pressure. As we know that at constant pressure, `Delta H = Delta U + p Delta V` where, `Delta V` is the change in volume. This equation can be rewritten as `Delta H = Delta U + p(V_f - V_i) = Delta U + ) p V_f - pV_i)""...(i)` where, `V_i=` INITIAL volume of the system `V_f=` final volume of the system But for the ideal gases, `pV=n RT` So that `pV_1 = n_(1) RT` and `pV_(2) = n_(2) RT` where, `n_(1) =` number of moles of the gaseous reactants `n_(2) =` number of moles of the gaseous products. Substituting these values in Eq. (i), we get `Delta H = Delta U + (n_(2) RT - n_(1) RT)` `Delta H = Delta U + (n_(2) - n_(1) ) RT` OR `Delta H = Delta U + Deltan_(g) RT` where `Delta n_(g) = n_(2) - n_(1)` is the difference between the number of moles of the gaseous products and gaseous reactants. Putting the values of ` Delta H and Delta U` we get `q_p = q_v, + Delta n_(g) "RT"` Note: Conditions under which `q_(p) = q_(v) "or" Delta H = Delta U` (i) When reaction is carried out in a closed vessel the volume remains constant i.e., `Delta V =0` (II) When reaction involves only solids or liquids or solutions but no gaseous reactant or product. This is because the volume CHANGES of the solids and liquids during a chemical reaction are negligible. (iii) When reaction involves gaseous reactants and products but their number of moles are equal (i.e., `n_(p) = n_(r) )` e.g., `H_(2) (g) + CI_(2)(g) to 2HCI(g)` `C(s) + O_(2) (g) to CO_(2) (g)` Since, `q_p` is different from `q_v` only in those reactions which involves gaseous reactants and products and `(n_p)` gaseous `ne (n_r)` gaseous. |
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