What simplifying assumptions do we usually make in working problems involving the ionization of a polyprotic acids? Why are they usually valid? Under what conditions do they fail? Answer by selecting all true statements. Nearly all of the $$ \mathrm{H}^{+} $$comes from the first ionization. Nearly all of the $$ \mathrm{H}^{+} $$comes from the second ionization. If the first two ionization constants are not very different, the approximations fail. If the first two ionization constants are not very different, the approximations are valid. The concentration of $$ \mathrm{H}^{+} $$and the conjugate base of the polyprotic acid are approximately equal. The approximations are usually valid because the first ionization constant for the polyprotic acid is usually much larger than the second ionization constant. The approximations are usually valid because the first ionization constant for the polyprotic acid is usually smaller than the second ionization constant.

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To analyze the ionization of polyprotic acids, we need to consider the following simplifying assumptions and their validity:

1. **Nearly all of the $$ \mathrm{H}^{+} $$ comes from the first ionization.**
   - This statement is **true**. In polyprotic acids, the first ionization typically produces the majority of $$ \mathrm{H}^{+} $$ ions because the first ionization constant ($$ K_{a1} $$) is usually much larger than the subsequent ionization constants.

2. **Nearly all of the $$ \mathrm{H}^{+} $$ comes from the second ionization.**
   - This statement is **false**. The second ionization generally contributes much less $$ \mathrm{H}^{+} $$ compared to the first ionization.

3. **If the first two ionization constants are not very different, the approximations fail.**
   - This statement is **true**. When the first and second ionization constants ($$ K_{a1} $$ and $$ K_{a2} $$) are similar in magnitude, the contributions of both ionizations to the $$ \mathrm{H}^{+} $$ concentration become significant, making the simplifying assumptions less valid.

4. **If the first two ionization constants are not very different, the approximations are valid.**
   - This statement is **false**. As mentioned, if the constants are not very different, the assumptions fail.

5. **The concentration of $$ \mathrm{H}^{+} $$ and the conjugate base of the polyprotic acid are approximately equal.**
   - This statement is **false**. The concentration of $$ \mathrm{H}^{+} $$ is typically much higher than that of the conjugate base after the first ionization.

6. **The approximations are usually valid because the first ionization constant for the polyprotic acid is usually much larger than the second ionization constant.**
   - This statement is **true**. The validity of the approximations relies on the first ionization constant being significantly larger than the second, which ensures that the first ionization dominates the $$ \mathrm{H}^{+} $$ concentration.

7. **The approximations are usually valid because the first ionization constant for the polyprotic acid is usually smaller than the second ionization constant.**
   - This statement is **false**. If the first ionization constant were smaller than the second, the first ionization would not dominate, and the approximations would not hold.

### Summary of True Statements:
- Nearly all of the $$ \mathrm{H}^{+} $$ comes from the first ionization.
- If the first two ionization constants are not very different, the approximations fail.
- The approximations are usually valid because the first ionization constant for the polyprotic acid is usually much larger than the second ionization constant.
- Nearly all $$ \mathrm{H}^{+} $$ comes from the first ionization. - If first and second ionization constants are similar, approximations fail. - Approximations are valid when the first ionization constant is much larger than the second.

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