entropyideal-gasthermodynamics
In the above, it's mentioned that any quantity other than MOLAR VOLUME can be written that way.
But how can we say generalize it with certainty when molar volume itself doesn't fit in or to say any property which might be having a relationship with Pressure (like entropy) can satisfy the above equation for ideal gas mixture.
I.E
$$V_i'(T,P) \neq V_i(T,P_i)$$
but
$$S_i'(T,P) = S_i(T,P_i)$$
where $P_i$ is partial pressure.
When you learned this material as a beginning physics student, they taught you that $nC_p\Delta T=Q$, and the focus was typically on a solid or liquid (both of which are nearly incompressible). However, this was only introductory, and was not the correct and precise definition necessary to use in thermodynamics. In thermodynamics, it is recognized that heat capacity is really a physical property of the material, and has nothing to do with any specific process (which in thermo is characterized by W and Q). The new more precise definitions of heat capacities employed in thermodynamics involve the internal energy and the enthalpy of the material (and can apply to any material, including an ideal gas): $$nC_v=\left(\frac{\partial U}{\partial T}\right)_V$$ $$nC_p=\left(\frac{\partial H}{\partial T}\right)_p$$ If the first equation is combined with the first law, then, in heating tests at constant volume, the amount of heat added Q to the system can be used to experimentally measure Cv directly, since the amount of work is zero. If the second equation is combined with the first law, then, in heating tests at constant pressure, the amount of heat added to the system can be used to experimentally measure Cp directly, since, in this case, $W = p\Delta V$. So the subscripts v and p are used to refer to the conditions required to measure these heat capacities directly by determining the heat Q added is such special tests. However, once these measurements have established the values of the two heat capacities, they can be used for all differential changes in state to determine the partial derivatives of U and H with respect to temperature.
Remember that ideal gases do not interact. They don't interact with themselves or with other ideal gases, so in a mixture of ideal gases each component of the mixture behaves as if the other gases were not present.
So for every type of gas present the ideal gas equation holds:
$$ P_i = n_i\frac{RT}{V} \tag{1} $$
And the total pressure is just the sum of all these partial pressures:
$$ P_{tot} = \sum_i P_i = \sum_i n_i\frac{RT}{V} \tag{2} $$
This can be rewritten in various ways using:
$$ \rho_i = \frac{M_i n_i}{V} $$
where $M_i$ is the molecular weight of the gas $i$, so $M_i n_i$ is the total mass present of species $i$. For example we can rewrite this as:
$$ \frac{V\rho_i}{M_i} = n_i $$
and susbtitute this into equation (1) to get the equation you give:
$$ P_i = n_i\frac{RT}{V} = \frac{V\rho_i}{M_i}\frac{RT}{V} = \frac{\rho_i}{M_i}RT $$
However it isn't useful to talk about a partial molar volume as all the gas molecules are uniformly distributed. It would only make sense to talk about $V_i$ if the two gases were separated by a movable partition so they were at the same temperature and pressure but had different volumes:
Best Answer
Foundations
Given a state property $P$, the definition of its partial molar value in a mixture at a given temperature $T$ and total pressure $p$ is below.
$$ \check{P}_i \equiv \left( \frac{\partial P}{\partial n_j} \right)_{T, p, n_j \neq n_i} $$
The definition measures the change in the total intensive property of the system when the number of moles of a component are varied holding all other number of moles constant.
An ideal solution is one where the particles have zero volume, no interactions, and collide elastically. An ideal gas, as derived through kinetic theory, is an example of an ideal solution.
The total molar property $\bar{P}$ of an ideal solution always depends on partial molar values and mole fractions $z_i$ identically as below.
$$ \bar{P} = \sum\ z_i \check{P}_i $$
Partial Molar Volume
When a number of moles of only one component $i$ are added to an ideal solution as $\Delta n_i$, the total volume increases in proportion.
$$\Delta V = \frac{\Delta n_i R T}{p} $$
This is a direct consequence of the mechanical equation of state of the ideal mixture (the ideal gas law). Convert this to a partial derivative to obtain the following:
$$ \left(\frac{\partial V}{\partial n_i}\right)_{T, p, n_j \neq n_i} = \frac{\partial n_i}{\partial n_i} \frac{R T}{p} = \bar{V} $$
Recall that, in an ideal solution, the total molar volume $\bar{V}$ is the same as the molar volume of any component $\bar{V}_i$. Therefore, the partial molar volume of an ideal solution is the molar volume of the component at the given conditions of $T, p$ (total pressure).
$$ \check{V}_i(T, p) = \bar{V}_i(T, p)$$
This formulation stands for all cases of ideal solutions. It requires no theorem, rather it is a consequence of ideal behavior as stated by first principles in the ideal mechanical equation of state for the system.
The Gibbs Theorem
The Gibbs theorem is a hypothetical framework. It is used to express the partial molar values of the thermodynamic state properties $U, H, S, A$, and $G$ for ideal mixtures. While we can obtain the total intensive values of a mixture using a mechanical equation of state, we have no rigorous way to define the partial molar properties without making an additional assumption. In other words, unlike the partial molar volume, the behavior of the partial molar value of any thermodynamic state function cannot be derived by first principles using only a mechanical equation of state for the total mixture. The Gibbs theorem is based on the assumptions that all particles in an ideal mixture are self-contained and that the intensive thermodynamic state property of any component $i$ in the mixture is defined solely by the system temperature $T$ and the partial pressure of the component $p_i$ in the mixture.
Conclusion
The partial molar volume of a component $i$ in an ideal mixture is identically defined and its expression is directly derived when we make the statement that the system behaves ideally. No other assumptions or theorems are required. By comparison, the partial molar value of any thermodynamic state function for a component $i$ in a mixture cannot be defined solely by the mechanical equation of state. An additional statement is required, and that statement is the Gibbs theorem.
Further Discussions
The remaining discussion here is left as a supplement. It is not needed to finalize the conclusive answer given above.
Two of the fundamental assumptions in the kinetic theory used to derive ideal gas behavior are: particles have zero volume and particles do not interact. The second statement is the more important one to determine ideal versus non-ideal behavior in thermodynamic functions. The first statement is the more important one to determine ideal versus non-ideal mixing behavior of volumes. Both must be true to apply the Gibbs theorem.
When particles interact, their thermodynamic energies in a mixture will depend on the composition of the mixture. When they do not interact, their thermodynamic energies in a mixture are the same as though they are at the $T, p$ that they would have to occupy the entire volume of the container entirely by themselves. By this statement, the particles in the ideal mixture have no identifiable partial volume. Instead they behave in energy functions as though they occupy the total volume of the container. That is permitted because all other particles have zero volume.
Note that we do not set the conditions at the total pressure. Even though the ideal particles occupy the total container, they contribute only their part of the collisions in the container. So, in a 50/50 mixture of an ideal gas, when we conceptually remove half of the gas particles, the remaining half still hit the walls of the container with the same frequency as they did before. They do not get to make up the loss of collisions on the wall from the missing particles. By kinetic theory, even when half the particles are removed, the remaining half still contribute only half to the pressure.
In a different view, consider that in any closed system, only two of the three parameters $T, p, V$ are independent. Allow that we choose $T$ as a constant. Subsequently we can pick constant $p$ and allow $V$ to be defined for us or we can pick constant $V$ and allow $p$ to be defined for us.
In summary, the Gibbs theorem is constructed on the basis of a defined $T, p$ state. It is a theorem only about thermodynamic (energy) functions in a mixture. At a defined $T, p$ state in a closed system, the total molar energy functions $\bar{U}, \bar{H} \ldots$ for the mixture are defined (because they are state functions). The Gibbs theorem establishes how these total molar energy functions are distributed among the components. For an ideal mixture, they are distributed in accord with the mole fraction of the components. The theorem is not about molar volume because we cannot independently divide molar volume AND partial pressures in a closed system at a given $T, p$. We can divide one or the other.
We could construct a hypothetical framework using partial volume at total pressure as the basis for a new theorem of partial molar energy functions. This choice of $T, V$ as our basis would not be the Gibbs theorem.
Perhaps a direct answer will serve as well. The Gibbs theorem does not state for partial molar volume that is does not equal molar volume.
$$\check{V}_j(T, p) \neq \bar{V}_j(T, p)$$
This is false on two accounts. First, for an ideal gas, the absolute truth is
$$\check{V}_i(T, p) \equiv \bar{V}_i(T, p)$$
Secondly, this is absolute truth not because we need a theorem. This is absolute truth because in a closed system only two of the three parameters $T, p, V$ are independent.