Is the change ΔG for an ideal gas unknown?

When one is introduced to the first law of thermodynamics, one learns that enthalpy is only a function of temperature, and calculating its changes is straightforward
\begin{equation}
 \Delta H^\pu{ig} = H^\pu{ig}(T\_2) - H^\pu{ig}(T\_1) =
 \int\_{T\_1}^{T\_2} C\_\mathrm{p}^\pu{ig}(T) \; dT \tag{1}
\end{equation}
When the second law is introduced, one also learns that the entropy is a function of pressure and temperature, and calculating its changes is also straightforward
\begin{equation}
 \Delta S^\pu{ig} = S^\pu{ig}(p\_2,T\_2) - S^\pu{ig}(p\_1,T\_1) =
 \int\_{T\_1}^{T\_2} \frac{C\_\mathrm{p}^\pu{ig}(T)}{T} \; dT -R \ln\left(\frac{p\_2}{p\_1}\right) \tag{2}
\end{equation}
Finally, when the Gibbs energy is introduced, one is presented with the fundamental equation for constant composition systems
$$ \mathrm{d}G = V\mathrm{d}p - S\mathrm{d}T \tag{3} $$
And Eq. (3) for an ideal gas, at constant temperature $T$, gives also a famous relationship to calculate the changes in pressure. We replace $V^\pu{ig} = RT/p$, integrate, and get
\begin{equation}
 \Delta G^\pu{ig} = G^\pu{ig}(p\_2,T) - G^\pu{ig}(p\_1,T) = RT \ln\left(\frac{p\_2}{p\_1}\right) 
 \tag{4}
\end{equation}


Now, I am interested in calculating the changes in Gibbs energy when the **temperature** changes and the pressure is constant. However, one faces the funny situation of being unable to obtain an expression in this case. For a constant pressure process, integrating Eq. (3) is not particularly attractive since one gets the entropy in the right-hand side
\begin{equation}
 \left(\frac{\partial G}{\partial T}\right)\_p = -S \tag{5}
\end{equation}
It is possible to manipulate Eq. (5) and obtain the Gibbs-Helmholtz equation to eliminate $S$ for $H$
\begin{equation}
 \left[\frac{\partial (G^\pu{ig}/T)}{\partial T}\right]\_p = -\frac{H^\pu{ig}}{T^2} \tag{6}
\end{equation}
and, for example, try to establish a relationship in a simpler case, of a constant heat capacity gas. Eq. (1) integrated for a reference temperature $T\_0$ automatically leads to
\begin{equation}
 \left[\frac{\partial (G^\pu{ig}/T)}{\partial T}\right]\_p = -\frac{H^\pu{ig}(T\_0) +
 C\_\mathrm{p}^\pu{ig}(T - T\_0)}{T^2} \tag{7}
\end{equation}
And no matter what the effort, I can never obtain a simple expression that *only* depends on the fundamental variables $p$ and $T$
$$ \Delta G^\pu{ig} = G^\pu{ig}(p,T\_2) - G^\pu{ig}(p,T\_2) = f(p,T\_2) - f(p,T\_1) $$
And thus, the changes of $\Delta G^\pu{ig}$ for a non-isothermal process in an ideal gas are impossible to calculate. A similar argument can be raised for calculating the changes in the Helmholtz energy $\Delta A^\pu{ig}$ when the temperature changes, and again, one cannot find an equation.


Is this correct? And if it is, why is so?


**Edit** Doing by definition of $\Delta G$ won't work because one faces an issue in the entropy term, impossible to resolve
\begin{align}
 \Delta G^\pu{ig} &=
 [H^\pu{ig}(T\_2) - T\_2S^\pu{ig}(p\_2,T\_2)] -
 [H^\pu{ig}(T\_1) - T\_1S^\pu{ig}(p\_1,T\_1)] \\ &=
 [H^\pu{ig}(T\_2) - H^\pu{ig}(T\_1)] - \color{red}{[T\_2S^\pu{ig}(p\_2,T\_2) -
 T\_1S^\pu{ig}(p\_1,T\_1)]}
\end{align}


