Derive the relationship between \Delta H and \Delta U for an ideal gas. Explain each term involved in the equation.

Answers (1)

Now, the heat absorbed at constant volume is equal to change in the internal Energy.

\Delta U = q_{p} - p\; \Delta V

qp is the heat absorbed.

\Delta V is the expansion work done by the system

the equation can be represented as U_2 - U_1 = p (V_2 - V_1)

so, we get,

q_{p} = (U_2 + Pv_2) - (U_1 + Pv_1) ----------------- (1)

now we take into consideration, the enthalpy

H = U + pV -------------------------- (2)

Hence, we can rewrite the first equation as,

q_{p} = H_2 - H_1 = \Delta H

now, in case of the state functions.

H is a state function because it depends on U, p and V, all of which are state functions.

So, \Delta H, as well as qp, is path independent.

We can rewrite equation 2 in case of finite changes as,

\Delta H = \Delta U + p\Delta V --------------------- (3)

We measure changes in enthalpy, when heat is being absorbed at a constant pressure.

ΔV = 0, for constant volume. So, equation 3 becomes ΔH = ΔU = qv   

 Let Va be the total volume of gaseous reactants and Vb be the total volume of gaseous products.  Let nA be the number of moles of gas reactants and nB be the number of moles for the gaseous products. The ideal gas law is:

pVa= nA R T

pVb= nB R T

now,

pVb - pVa = (nB - nA) RT

hence,

p\Delta V = \Delta n_g RT ------------------- (4)

now, by using equation 3 and equation 4,

we can write

ΔH = ΔU + Δng RT

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