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The Gist

As mentioned in the comments, we have to prevent the formation of aqua complexes of $\ce{Fe^{3+}}$ as these tend to hydrolyze:

$$\begin{multline} \ce{[Fe(H2O)_{2n}(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-2n} + H2O <=>} \\ \ce{[Fe(H2O)_{2n-1}(OH)(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-2n} + H3O+} \end{multline}$$

Note: In the above equilibrium, it was assumed that the denticity of $\ce{NO3^-}$ is 2. This may or may not be the case. I have avoided this ambiguity in the similar equation in the Remarks section.

Now, to prevent formation of aqua complexes, there are (at least) two ways to do this:

1. Decrease the pH to shift the above equilibrium to the reactants side in accordance with Le Chatelier's Principle.
2. Increase the concentration of $\ce{NO3^-}$ in the solution to prevent formation of aqua complexes in accordance with Le Chatelier's Principle, as explained in the section Hydration Isomerism of $\ce{Fe(NO3)3}$ below.

Your suggestion to add $\ce{HNO3}$ does both simultaneously, decreasing the pH and increasing the concentration of $\ce{NO3^-}$.

Hydration Isomerism

For a hydrated complex $\ce{[ML_y].xH2O}$, generally, assuming maximum valency of $\ce{M}$ is $y$, $y+1$ hydration isomers are possible given by the formulae:

$$ \ce{[ML_{y-n}(H2O)_n].L_n.($x-n$)H2O} %nasty workaround, possibly mhchem bug $$

Note: This formula is most accurate for crystalline ($\ce{c}$) phase and the behavior is more complex in aqueous ($\ce{aq}$); however, for simplicity, we will consider this formula for our analysis.

Example

Consider the example of $\ce{[MX2].2H2O}$ where $\ce{M}$ has a maximum valency of 2:

$$ \ce{\underset{\text{isomer 1}}{[MX2].2H2O} <=> \underset{\text{isomer 2}}{[MX(H2O)].X^-.H2O} <=> \underset{\text{isomer 3}}{[M(H2O)2].(X^-)2}} $$

Hydration Isomerism of $\ce{Fe(NO3)3}$

$\ce{Fe(NO3)3}$ has several hydration isomers and is commonly represented as $\ce{[Fe(NO3)3].xH2O}$, where $x$ is usually 9, which we will consider.

The usual coordination number and geometry of $\ce{Fe}$ are $6$ and octahedral, respectively, commonly forming $\lambda^6$ complexes.

Thus, in $\ce{aq}$ phase, you are likely to observe (assuming that the denticity of $\ce{NO3^-}$ maybe $\pu{1}$ or $\pu{2}$), among many other isomers, the following complexes.

1. $\ce{[Fe(H2O)_6].(NO3^-)3.3H2O}$,
2. $\ce{[Fe(H2O)_4(NO3)_2].NO3^-.5H_2O}$, and
3. (very) low amounts of $\ce{[Fe(NO3)3].9H2O}$.

Note: These formulae are not quite accurate, as this Wikipedia article states.

This happens because, in the spectrochemical series, $\ce{H2O}$ is a stronger field ligand than $\ce{NO3^-}$, forming more stable complexes with $\ce{Fe^3+}$. We could represent this as:

$$ \ce{\underset{\text{isomer 1}}{[Fe(NO3)3].9H2O} <=> \underset{\text{isomer 2}}{[Fe(H2O)_4(NO3)_2].NO3^-.5H_2O} <=> \underset{\text{isomer 3}}{[Fe(H2O)_6].(NO3^-)3}} $$

where, in aqueous phase, isomer 3 (hydrolyzed and otherwise) is present in abundance, and isomer 1 rarely observed. We don't want aqua complexes such as isomer 2 and isomer 3 as, reasoned in The Gist section, these tend to hydrolyze:

$$\begin{multline} \ce{[Fe(H2O)_x(NO3)_y]^{n+}.(NO3^-)_n.(H2O)_{9-x} + H2O <=> }\\ \ce{[Fe(H2O)_{x-1}(OH)(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-x} + H3O+} \end{multline} $$

Shifting the Equilibrium

As pointed out in other answer, increasing the amount of $\ce{NO3^-}$ in the solution will, in accordance with Le Chatelier principle, shift the equilibrium towards the right. You are essentially carrying out the following reaction:

$$ \ce{\underset{will hydrolyze}{[Fe(H2O)_6].(NO3^-)3.(H2O)3(aq)} ->[\ce{H+NO3^-}][high concentration] \underset{won't hydrolyze}{[Fe(NO3)3]\cdot 9H2O}(aq)} $$

Remarks

Denticity of Nitrate Anion

Denticity of the nitrate anion maybe 1 or 2. Therefore, in:

$$ \ce{[Fe(H2O)_x(NO3)_y]^{n+}.(NO3^-)_n.(H2O)_{9-x}} $$

$x+y \in [3,6]; x>0, y>0$.

Two-Forked Action of Addition of Nitric Acid

Addition of $\ce{HNO3}$ is a two-forked strategy, decreasing the pH and increasing the concentration of $\ce{NO3^-}$, both avoiding formation of aqua complexes of ferric cation.

The Gist

As mentioned in the comments, we have to prevent the formation of aqua complexes of $\ce{Fe^{3+}}$ as these tend to hydrolyze:

$$\begin{multline} \ce{[Fe(H2O)_{2n}(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-2n} + H2O <=>} \\ \ce{[Fe(H2O)_{2n-1}(OH)(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-2n} + H3O+} \end{multline}$$

Note: In the above equilibrium, it was assumed that the denticity of $\ce{NO3^-}$ is 2. This may or may not be the case. I have avoided this ambiguity in the similar equation in the Remarks section.

Now, to prevent formation of aqua complexes, there are (at least) two ways to do this:

1. Decrease the pH to shift the above equilibrium to the reactants side in accordance with Le Chatelier's Principle.
2. Increase the concentration of $\ce{NO3^-}$ in the solution to prevent formation of aqua complexes in accordance with Le Chatelier's Principle, as explained in the section Hydration Isomerism of $\ce{Fe(NO3)3}$ below.

Your suggestion to add $\ce{HNO3}$ does both simultaneously, decreasing the pH and increasing the concentration of $\ce{NO3^-}$.

Hydration Isomerism

For a hydrated complex $\ce{[ML_y].xH2O}$, generally, assuming maximum valency of $\ce{M}$ is $y$, $y+1$ hydration isomers are possible given by the formulae:

$$ \ce{[ML_{y-n}(H2O)_n].L_n.($x-n$)H2O} %nasty workaround, possibly mhchem bug $$

Note: This formula is most accurate for crystalline ($\ce{c}$) phase and the behavior is more complex in aqueous ($\ce{aq}$); however, for simplicity, we will consider this formula for our analysis.

Example

Consider the example of $\ce{[MX2].2H2O}$ where $\ce{M}$ has a maximum valency of 2:

$$ \ce{\underset{\text{isomer 1}}{[MX2].2H2O} <=> \underset{\text{isomer 2}}{[MX(H2O)].X^-.H2O} <=> \underset{\text{isomer 3}}{[M(H2O)2].(X^-)2}} $$

Hydration Isomerism of $\ce{Fe(NO3)3}$

$\ce{Fe(NO3)3}$ has several hydration isomers and is commonly represented as $\ce{[Fe(NO3)3].xH2O}$, where $x$ is usually 9, which we will consider.

The usual coordination number and geometry of $\ce{Fe}$ are $6$ and octahedral, respectively, commonly forming six-coordinated complexes.

Thus, in $\ce{aq}$ phase, you are likely to observe (assuming that the denticity of $\ce{NO3^-}$ maybe $\pu{1}$ or $\pu{2}$), among many other isomers, the following complexes.

1. $\ce{[Fe(H2O)_6].(NO3^-)3.3H2O}$,
2. $\ce{[Fe(H2O)_4(NO3)_2].NO3^-.5H_2O}$, and
3. (very) low amounts of $\ce{[Fe(NO3)3].9H2O}$.

Note: These formulae are not quite accurate, as this Wikipedia article states.

This happens because, in the spectrochemical series, $\ce{H2O}$ is a stronger field ligand than $\ce{NO3^-}$, forming more stable complexes with $\ce{Fe^3+}$. We could represent this as:

$$ \ce{\underset{\text{isomer 1}}{[Fe(NO3)3].9H2O} <=> \underset{\text{isomer 2}}{[Fe(H2O)_4(NO3)_2].NO3^-.5H_2O} <=> \underset{\text{isomer 3}}{[Fe(H2O)_6].(NO3^-)3}} $$

where, in aqueous phase, isomer 3 (hydrolyzed and otherwise) is present in abundance, and isomer 1 rarely observed. We don't want aqua complexes such as isomer 2 and isomer 3 as, reasoned in The Gist section, these tend to hydrolyze:

$$\begin{multline} \ce{[Fe(H2O)_x(NO3)_y]^{n+}.(NO3^-)_n.(H2O)_{9-x} + H2O <=> }\\ \ce{[Fe(H2O)_{x-1}(OH)(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-x} + H3O+} \end{multline} $$

Shifting the Equilibrium

As pointed out in other answer, increasing the amount of $\ce{NO3^-}$ in the solution will, in accordance with Le Chatelier principle, shift the equilibrium towards the right. You are essentially carrying out the following reaction:

$$ \ce{\underset{will hydrolyze}{[Fe(H2O)_6].(NO3^-)3.(H2O)3(aq)} ->[\ce{H+NO3^-}][high concentration] \underset{won't hydrolyze}{[Fe(NO3)3]\cdot 9H2O}(aq)} $$

Remarks

Denticity of Nitrate Anion

Denticity of the nitrate anion maybe 1 or 2. Therefore, in:

$$ \ce{[Fe(H2O)_x(NO3)_y]^{n+}.(NO3^-)_n.(H2O)_{9-x}} $$

$x+y \in [3,6]; x>0, y>0$.

Two-Forked Action of Addition of Nitric Acid

Addition of $\ce{HNO3}$ is a two-forked strategy, decreasing the pH and increasing the concentration of $\ce{NO3^-}$, both avoiding formation of aqua complexes of ferric cation.

The Gist

As mentioned in the comments, we have to prevent the formation of aqua complexes of $\ce{Fe^{3+}}$ as these tend to hydrolyze:

$$\begin{multline} \ce{[Fe(H2O)_{2n}(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-2n} + H2O <=>} \\ \ce{[Fe(H2O)_{2n-1}(OH)(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-2n} + H3O+} \end{multline}$$

Note: In the above equilibrium, it was assumed that the denticity of $\ce{NO3^-}$ is 2. This may or may not be the case. I have avoided this ambiguity in the similar equation in the Remarks section.

Now, to prevent formation of aqua complexes, there are (at least) two ways to do this:

1. Decrease the pH to shift the above equilibrium to the reactants side in accordance with Le Chatelier's Principle.
2. Increase the concentration of $\ce{NO3^-}$ in the solution to prevent formation of aqua complexes in accordance with Le Chatelier's Principle, as explained in the section Hydration Isomerism of $\ce{Fe(NO3)3}$ below.

Your suggestion to add $\ce{HNO3}$ does both simultaneously, decreasing the pH and increasing the concentration of $\ce{NO3^-}$.

Hydration Isomerism

For a hydrated complex $\ce{[ML_y].xH2O}$, generally, assuming maximum valency of $\ce{M}$ is $y$, $y+1$ hydration isomers are possible given by the formulae:

$$ \ce{[ML_{y-n}(H2O)_n].L_n.($x-n$)H2O} %nasty workaround, possibly mhchem bug $$

Note: This formula is most accurate for crystalline ($\ce{c}$) phase and the behavior is more complex in aqueous ($\ce{aq}$); however, for simplicity, we will consider this formula for our analysis.

Example

Consider the example of $\ce{[MX2].2H2O}$ where $\ce{M}$ has a maximum valency of 2:

$$ \ce{\underset{\text{isomer 1}}{[MX2].2H2O} <=> \underset{\text{isomer 2}}{[MX(H2O)].X^-.H2O} <=> \underset{\text{isomer 3}}{[M(H2O)2].(X^-)2}} $$

Hydration Isomerism of $\ce{Fe(NO3)3}$

$\ce{Fe(NO3)3}$ has several hydration isomers and is commonly represented as $\ce{[Fe(NO3)3].xH2O}$, where $x$ is usually 9, which we will consider.

The usual valency and geometry of $\ce{Fe}$ are $6$ and octahedral, respectively, commonly forming $\lambda^6$ complexes.

Thus, in $\ce{aq}$ phase, you are likely to observe (assuming that the denticity of $\ce{NO3^-}$ maybe $\pu{1}$ or $\pu{2}$), among many other isomers, the following complexes.

1. $\ce{[Fe(H2O)_6].(NO3^-)3.3H2O}$,
2. $\ce{[Fe(H2O)_4(NO3)_2].NO3^-.5H_2O}$, and
3. (very) low amounts of $\ce{[Fe(NO3)3].9H2O}$.

Note: These formulae are not quite accurate, as this Wikipedia article states.

This happens because, in the spectrochemical series, $\ce{H2O}$ is a stronger field ligand than $\ce{NO3^-}$, forming more stable complexes with $\ce{Fe^3+}$. We could represent this as:

$$ \ce{\underset{\text{isomer 1}}{[Fe(NO3)3].9H2O} <=> \underset{\text{isomer 2}}{[Fe(H2O)_4(NO3)_2].NO3^-.5H_2O} <=> \underset{\text{isomer 3}}{[Fe(H2O)_6].(NO3^-)3}} $$

where, in aqueous phase, isomer 3 (hydrolyzed and otherwise) is present in abundance, and isomer 1 rarely observed. We don't want aqua complexes such as isomer 2 and isomer 3 as, reasoned in The Gist section, these tend to hydrolyze:

$$\begin{multline} \ce{[Fe(H2O)_x(NO3)_y]^{n+}.(NO3^-)_n.(H2O)_{9-x} + H2O <=> }\\ \ce{[Fe(H2O)_{x-1}(OH)(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-x} + H3O+} \end{multline} $$

Shifting the Equilibrium

As pointed out in other answer, increasing the amount of $\ce{NO3^-}$ in the solution will, in accordance with Le Chatelier principle, shift the equilibrium towards the right. You are essentially carrying out the following reaction:

$$ \ce{\underset{will hydrolyze}{[Fe(H2O)_6].(NO3^-)3.(H2O)3(aq)} ->[\ce{H+NO3^-}][high concentration] \underset{won't hydrolyze}{[Fe(NO3)3]\cdot 9H2O}(aq)} $$

Remarks

Denticity of Nitrate Anion

Denticity of the nitrate anion maybe 1 or 2. Therefore, in:

$$ \ce{[Fe(H2O)_x(NO3)_y]^{n+}.(NO3^-)_n.(H2O)_{9-x}} $$

$x+y \in [3,6]; x>0, y>0$.

Two-Forked Action of Addition of Nitric Acid

Addition of $\ce{HNO3}$ is a two-forked strategy, decreasing the pH and increasing the concentration of $\ce{NO3^-}$, both avoiding formation of aqua complexes of ferric cation.

The Gist

As mentioned in the comments, we have to prevent the formation of aqua complexes of $\ce{Fe^{3+}}$ as these tend to hydrolyze:

$$\begin{multline} \ce{[Fe(H2O)_{2n}(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-2n} + H2O <=>} \\ \ce{[Fe(H2O)_{2n-1}(OH)(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-2n} + H3O+} \end{multline}$$

Note: In the above equilibrium, it was assumed that the denticity of $\ce{NO3^-}$ is 2. This may or may not be the case. I have avoided this ambiguity in the similar equation in the Remarks section.

Now, to prevent formation of aqua complexes, there are (at least) two ways to do this:

1. Decrease the pH to shift the above equilibrium to the reactants side in accordance with Le Chatelier's Principle.
2. Increase the concentration of $\ce{NO3^-}$ in the solution to prevent formation of aqua complexes in accordance with Le Chatelier's Principle, as explained in the section Hydration Isomerism of $\ce{Fe(NO3)3}$ below.

Your suggestion to add $\ce{HNO3}$ does both simultaneously, decreasing the pH and increasing the concentration of $\ce{NO3^-}$.

Hydration Isomerism

For a hydrated complex $\ce{[ML_y].xH2O}$, generally, assuming maximum valency of $\ce{M}$ is $y$, $y+1$ hydration isomers are possible given by the formulae:

$$ \ce{[ML_{y-n}(H2O)_n].L_n.($x-n$)H2O} %nasty workaround, possibly mhchem bug $$

Note: This formula is most accurate for crystalline ($\ce{c}$) phase and the behavior is more complex in aqueous ($\ce{aq}$); however, for simplicity, we will consider this formula for our analysis.

Example

Consider the example of $\ce{[MX2].2H2O}$ where $\ce{M}$ has a maximum valency of 2:

$$ \ce{\underset{\text{isomer 1}}{[MX2].2H2O} <=> \underset{\text{isomer 2}}{[MX(H2O)].X^-.H2O} <=> \underset{\text{isomer 3}}{[M(H2O)2].(X^-)2}} $$

Hydration Isomerism of $\ce{Fe(NO3)3}$

$\ce{Fe(NO3)3}$ has several hydration isomers and is commonly represented as $\ce{[Fe(NO3)3].xH2O}$, where $x$ is usually 9, which we will consider.

The usual coordination number and geometry of $\ce{Fe}$ are $6$ and octahedral, respectively, commonly forming $\lambda^6$ complexes.

Thus, in $\ce{aq}$ phase, you are likely to observe (assuming that the denticity of $\ce{NO3^-}$ maybe $\pu{1}$ or $\pu{2}$), among many other isomers, the following complexes.

1. $\ce{[Fe(H2O)_6].(NO3^-)3.3H2O}$,
2. $\ce{[Fe(H2O)_4(NO3)_2].NO3^-.5H_2O}$, and
3. (very) low amounts of $\ce{[Fe(NO3)3].9H2O}$.

Note: These formulae are not quite accurate, as this Wikipedia article states.

This happens because, in the spectrochemical series, $\ce{H2O}$ is a stronger field ligand than $\ce{NO3^-}$, forming more stable complexes with $\ce{Fe^3+}$. We could represent this as:

$$ \ce{\underset{\text{isomer 1}}{[Fe(NO3)3].9H2O} <=> \underset{\text{isomer 2}}{[Fe(H2O)_4(NO3)_2].NO3^-.5H_2O} <=> \underset{\text{isomer 3}}{[Fe(H2O)_6].(NO3^-)3}} $$

where, in aqueous phase, isomer 3 (hydrolyzed and otherwise) is present in abundance, and isomer 1 rarely observed. We don't want aqua complexes such as isomer 2 and isomer 3 as, reasoned in The Gist section, these tend to hydrolyze:

$$\begin{multline} \ce{[Fe(H2O)_x(NO3)_y]^{n+}.(NO3^-)_n.(H2O)_{9-x} + H2O <=> }\\ \ce{[Fe(H2O)_{x-1}(OH)(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-x} + H3O+} \end{multline} $$

Shifting the Equilibrium

As pointed out in other answer, increasing the amount of $\ce{NO3^-}$ in the solution will, in accordance with Le Chatelier principle, shift the equilibrium towards the right. You are essentially carrying out the following reaction:

$$ \ce{\underset{will hydrolyze}{[Fe(H2O)_6].(NO3^-)3.(H2O)3(aq)} ->[\ce{H+NO3^-}][high concentration] \underset{won't hydrolyze}{[Fe(NO3)3]\cdot 9H2O}(aq)} $$

Remarks

Denticity of Nitrate Anion

Denticity of the nitrate anion maybe 1 or 2. Therefore, in:

$$ \ce{[Fe(H2O)_x(NO3)_y]^{n+}.(NO3^-)_n.(H2O)_{9-x}} $$

$x+y \in [3,6]; x>0, y>0$.

Two-Forked Action of Addition of Nitric Acid

Addition of $\ce{HNO3}$ is a two-forked strategy, decreasing the pH and increasing the concentration of $\ce{NO3^-}$, both avoiding formation of aqua complexes of ferric cation.

The Gist

As mentioned in the comments, we have to prevent the formation of aqua complexes of $\ce{Fe^{3+}}$ as these tend to hydrolyze:

$$\begin{multline} \ce{[Fe(H2O)_{2n}(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-2n} + H2O <=>} \\ \ce{[Fe(H2O)_{2n-1}(OH)(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-2n} + H3O+} \end{multline}$$

Note: In the above equilibrium, it was assumed that the denticity of $\ce{NO3^-}$ is 2. This may or may not be the case. I have avoided this ambiguity in the similar equation in the Remarks section.

Now, to prevent formation of aqua complexes, there are (at least) two ways to do this:

1. Decrease the pH to shift the above equilibrium to the reactants side in accordance with Le Chatelier's Principle.
2. Increase the concentration of $\ce{NO3^-}$ in the solution to prevent formation of aqua complexes in accordance with Le Chatelier's Principle, as explained in the section Hydration Isomerism of $\ce{Fe(NO3)3}$ below.

Your suggestion to add $\ce{HNO3}$ does both simultaneously, decreasing the pH and increasing the concentration of $\ce{NO3^-}$.

Hydration Isomerism

For a hydrated complex $\ce{[ML_y].xH2O}$, generally, assuming maximum valency of $\ce{M}$ is $y$, $y+1$ hydration isomers are possible given by the formulae:

$$ \ce{[ML_{y-n}(H2O)_n].L_n.($x-n$)H2O} %nasty workaround, possibly mhchem bug $$

Note: This formula is most accurate for crystalline ($\ce{c}$) phase and the behavior is more complex in aqueous ($\ce{aq}$); however, for simplicity, we will consider this formula for our analysis.

Example

Consider the example of $\ce{[MX2].2H2O}$ where $\ce{M}$ has a maximum valency of 2:

$$ \ce{\underset{\text{isomer 1}}{[MX2].2H2O} <=> \underset{\text{isomer 2}}{[MX(H2O)].X^-.H2O} <=> \underset{\text{isomer 3}}{[M(H2O)2].(X^-)2}} $$

Hydration Isomerism of $\ce{Fe(NO3)3}$

$\ce{Fe(NO3)3}$ has several hydration isomers and is commonly represented as $\ce{[Fe(NO3)3].xH2O}$, where $x$ is usually 9, which we will consider.

The usual valency and geometry of $\ce{Fe}$ are $6$ and octahedral, respectively, commonly forming $\lambda^6$ complexes.

Thus, in $\ce{aq}$ phase, you are likely to observe (assuming that the denticity of $\ce{NO3^-}$ maybe $\pu{1}$ or $\pu{2}$), among many other isomers, the following complexes.

1. $\ce{[Fe(H2O)_6].(NO3^-)3.3H2O}$,
2. $\ce{[Fe(H2O)_4(NO3)_2].NO3^-.5H_2O}$, and
3. (very) low amounts of $\ce{[Fe(NO3)3].9H2O}$.

Note: These formulae are not quite accurate, as this Wikipedia article states.

This happens because, in the spectrochemical series, $\ce{H2O}$ is a stronger field ligand than $\ce{NO3^-}$, forming more stable complexes with $\ce{Fe^3+}$. We could represent this as:

$$ \ce{\underset{\text{isomer 1}}{[Fe(NO3)3].9H2O} <=> \underset{\text{isomer 2}}{[Fe(H2O)_4(NO3)_2].NO3^-.5H_2O} <=> \underset{\text{isomer 3}}{[Fe(H2O)_6].(NO3^-)3}} $$

where, in aqueous phase, isomer 3 (hydrolyzed and otherwise) is present in abundance, and isomer 1 rarely observed. We don't want aqua complexes such as isomer 2 and isomer 3 as, reasoned in The Gist section, these tend to hydrolyze:

$$\begin{multline} \ce{[Fe(H2O)_x(NO3)_y]^{n+}.(NO3^-)_n.(H2O)_{9-x} + H2O <=> }\\ \ce{[Fe(H2O)_{x-1}(OH)(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-x} + H3O+} \end{multline} $$

Shifting the Equilibrium

As pointed out in other answer, increasing the amount of $\ce{NO3^-}$ in the solution will, in accordance with Le Chatelier principle, shift the equilibrium towards the right. You are essentially carrying out the following reaction:

Remarks

Denticity of Nitrate Anion

Denticity of the nitrate anion maybe 1 or 2. Therefore, in:

$$ \ce{[Fe(H2O)_x(NO3)_y]^{n+}.(NO3^-)_n.(H2O)_{9-x}} $$

$x+y \in [3,6]; x>0, y>0$.

Two-Forked Action of Addition of Nitric Acid

Addition of $\ce{HNO3}$ is a two-forked strategy, decreasing the pH and increasing the concentration of $\ce{NO3^-}$, both avoiding formation of aqua complexes of ferric cation.

$$ \ce{[Fe(H2O)_6].(NO3^-)3.(H2O)3 ->[\ce{H+NO3^-}][high concentration] [Fe(NO3)3]\cdot 9H2O} $$

The Gist

As mentioned in the comments, we have to prevent the formation of aqua complexes of $\ce{Fe^{3+}}$ as these tend to hydrolyze:

$$\begin{multline} \ce{[Fe(H2O)_{2n}(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-2n} + H2O <=>} \\ \ce{[Fe(H2O)_{2n-1}(OH)(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-2n} + H3O+} \end{multline}$$

Note: In the above equilibrium, it was assumed that the denticity of $\ce{NO3^-}$ is 2. This may or may not be the case. I have avoided this ambiguity in the similar equation in the Remarks section.

Now, to prevent formation of aqua complexes, there are (at least) two ways to do this:

1. Decrease the pH to shift the above equilibrium to the reactants side in accordance with Le Chatelier's Principle.
2. Increase the concentration of $\ce{NO3^-}$ in the solution to prevent formation of aqua complexes in accordance with Le Chatelier's Principle, as explained in the section Hydration Isomerism of $\ce{Fe(NO3)3}$ below.

Your suggestion to add $\ce{HNO3}$ does both simultaneously, decreasing the pH and increasing the concentration of $\ce{NO3^-}$.

Hydration Isomerism

For a hydrated complex $\ce{[ML_y].xH2O}$, generally, assuming maximum valency of $\ce{M}$ is $y$, $y+1$ hydration isomers are possible given by the formulae:

$$ \ce{[ML_{y-n}(H2O)_n].L_n.($x-n$)H2O} %nasty workaround, possibly mhchem bug $$

Note: This formula is most accurate for crystalline ($\ce{c}$) phase and the behavior is more complex in aqueous ($\ce{aq}$); however, for simplicity, we will consider this formula for our analysis.

Example

Consider the example of $\ce{[MX2].2H2O}$ where $\ce{M}$ has a maximum valency of 2:

$$ \ce{\underset{\text{isomer 1}}{[MX2].2H2O} <=> \underset{\text{isomer 2}}{[MX(H2O)].X^-.H2O} <=> \underset{\text{isomer 3}}{[M(H2O)2].(X^-)2}} $$

Hydration Isomerism of $\ce{Fe(NO3)3}$

$\ce{Fe(NO3)3}$ has several hydration isomers and is commonly represented as $\ce{[Fe(NO3)3].xH2O}$, where $x$ is usually 9, which we will consider.

The usual valency and geometry of $\ce{Fe}$ are $6$ and octahedral, respectively, commonly forming $\lambda^6$ complexes.

Thus, in $\ce{aq}$ phase, you are likely to observe (assuming that the denticity of $\ce{NO3^-}$ maybe $\pu{1}$ or $\pu{2}$), among many other isomers, the following complexes.

1. $\ce{[Fe(H2O)_6].(NO3^-)3.3H2O}$,
2. $\ce{[Fe(H2O)_4(NO3)_2].NO3^-.5H_2O}$, and
3. (very) low amounts of $\ce{[Fe(NO3)3].9H2O}$.

Note: These formulae are not quite accurate, as this Wikipedia article states.

This happens because, in the spectrochemical series, $\ce{H2O}$ is a stronger field ligand than $\ce{NO3^-}$, forming more stable complexes with $\ce{Fe^3+}$. We could represent this as:

$$ \ce{\underset{\text{isomer 1}}{[Fe(NO3)3].9H2O} <=> \underset{\text{isomer 2}}{[Fe(H2O)_4(NO3)_2].NO3^-.5H_2O} <=> \underset{\text{isomer 3}}{[Fe(H2O)_6].(NO3^-)3}} $$

where, in aqueous phase, isomer 3 (hydrolyzed and otherwise) is present in abundance, and isomer 1 rarely observed. We don't want aqua complexes such as isomer 2 and isomer 3 as, reasoned in The Gist section, these tend to hydrolyze:

$$\begin{multline} \ce{[Fe(H2O)_x(NO3)_y]^{n+}.(NO3^-)_n.(H2O)_{9-x} + H2O <=> }\\ \ce{[Fe(H2O)_{x-1}(OH)(NO3)_{3-n}]^{n+}.(NO3^-)_n.(H2O)_{9-x} + H3O+} \end{multline} $$

Shifting the Equilibrium

As pointed out in other answer, increasing the amount of $\ce{NO3^-}$ in the solution will, in accordance with Le Chatelier principle, shift the equilibrium towards the right. You are essentially carrying out the following reaction:

$$ \ce{\underset{will hydrolyze}{[Fe(H2O)_6].(NO3^-)3.(H2O)3(aq)} ->[\ce{H+NO3^-}][high concentration] \underset{won't hydrolyze}{[Fe(NO3)3]\cdot 9H2O}(aq)} $$

Remarks

Denticity of Nitrate Anion

Denticity of the nitrate anion maybe 1 or 2. Therefore, in:

$$ \ce{[Fe(H2O)_x(NO3)_y]^{n+}.(NO3^-)_n.(H2O)_{9-x}} $$

$x+y \in [3,6]; x>0, y>0$.

Two-Forked Action of Addition of Nitric Acid

Addition of $\ce{HNO3}$ is a two-forked strategy, decreasing the pH and increasing the concentration of $\ce{NO3^-}$, both avoiding formation of aqua complexes of ferric cation.