Determine range of `E^(@)` values for this reaction `X_(aq)^(2+)+2e^(-)toX(s)` for given conditions:
(a). If the metal X dissolve in `HNO_(3)` but not in `HCl` it can displace `Ag^(+)` ion but not `Cu^(2+)` ion.
(b). If te metal X is HCl acid producing `H_(2)(g)` but does not displace either `Zn^(2+)` or `Fe^(2+)`
Given `E_(Ag^(+)//Ag)^(0)=0.8V" "E_(Fe^(2+)//Fe)^(0)=-0.44`
`E_(Cu^(2+)//Cu)^(0)=0.34V" "E_(NO_(3)^(-)//NO)^(0)=0.96V" "E_(zn^(2+)//Zn)^(0)=-0.76V`

Explanation

To determine the range of \( E^\circ \) values for the reaction \( X_{(aq)}^{2+} + 2e^- \rightarrow X(s) \) based on the given conditions, we will analyze each condition step by step.

### Step 1: Analyze the first condition
The first condition states that the metal \( X \) dissolves in \( HNO_3 \) but not in \( HCl \), and it can displace \( Ag^+ \) ions but not \( Cu^{2+} \) ions.

- **Dissolves in \( HNO_3 \)**: This indicates that the metal \( X \) is reactive enough to undergo oxidation in the presence of a strong oxidizing agent like \( HNO_3 \).
- **Does not dissolve in \( HCl \)**: This suggests that \( X \) is less reactive than hydrogen, as metals that can displace hydrogen from acids typically dissolve in \( HCl \).
- **Displaces \( Ag^+ \) but not \( Cu^{2+} \)**: This implies that the standard reduction potential \( E^\circ \) of \( X^{2+}/X \) must be greater than that of \( Ag^+ \) (which is \( 0.8 \, V \)) but less than that of \( Cu^{2+} \) (which is \( 0.34 \, V \)).

Thus, we can conclude:
\[ 0.34 \, V < E^\circ < 0.8 \, V \]

### Step 2: Analyze the second condition
The second condition states that the metal \( X \) reacts with \( HCl \) producing \( H_2(g) \) but does not displace either \( Zn^{2+} \) or \( Fe^{2+} \).

- **Produces \( H_2(g) \)**: This indicates that \( X \) is more reactive than hydrogen, meaning \( E^\circ \) must be greater than \( 0 \, V \).
- **Does not displace \( Zn^{2+} \) or \( Fe^{2+} \)**: This means that \( E^\circ \) must be less than that of both \( Zn^{2+} \) (which is \( -0.76 \, V \)) and \( Fe^{2+} \) (which is \( -0.44 \, V \)). Therefore, \( E^\circ \) must be less than \( -0.44 \, V \).

Thus, we can conclude:
\[ -0.76 \, V < E^\circ < -0.44 \, V \]

### Step 3: Combine the findings
From the two conditions, we have the following ranges for \( E^\circ \):
1. From the first condition: \( 0.34 \, V < E^\circ < 0.8 \, V \)
2. From the second condition: \( -0.76 \, V < E^\circ < -0.44 \, V \)

### Final Conclusion
The ranges of \( E^\circ \) values for the reaction \( X_{(aq)}^{2+} + 2e^- \rightarrow X(s) \) are:
- For the first condition: \( 0.34 \, V < E^\circ < 0.8 \, V \)
- For the second condition: \( -0.76 \, V < E^\circ < -0.44 \, V \)