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If you are given Avogadro's number of at...

If you are given Avogadro's number of atoms of a gas X . If half of the atoms are converted into `X_(g)^(+)` by energy `Delta H` . The IE of X is

A

`(2DeltaH)/(N_(A))`

B

`(2N_(A))/(DeltaH)`

C

`(DeltaH)/(2N_(A))`

D

`(N_(A))/(DeltaH)`

Text Solution

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The correct Answer is:
To solve the problem, we need to determine the ionization energy (IE) of gas X based on the information provided. Here’s a step-by-step breakdown of the solution: ### Step 1: Understand the Given Information We are given Avogadro's number of atoms of gas X, which is \( N_A = 6.022 \times 10^{23} \). Half of these atoms are converted into \( X^{+} \) by the energy \( \Delta H \). ### Step 2: Calculate the Number of Atoms Converted Since we have Avogadro's number of atoms, half of this would be: \[ \text{Number of atoms converted} = \frac{N_A}{2} = \frac{6.022 \times 10^{23}}{2} = 3.011 \times 10^{23} \] ### Step 3: Relate Energy to Ionization Energy The energy \( \Delta H \) is used to convert half of the atoms into ions. The ionization energy (IE) is defined as the energy required to remove one mole of electrons from one mole of gaseous atoms. ### Step 4: Calculate the Ionization Energy for One Atom The total energy \( \Delta H \) is used to ionize \( \frac{N_A}{2} \) atoms. Therefore, the energy required to ionize one atom of gas X can be calculated as: \[ \text{Ionization Energy (IE)} = \frac{\Delta H}{\text{Number of atoms converted}} = \frac{\Delta H}{\frac{N_A}{2}} = \frac{2 \Delta H}{N_A} \] ### Step 5: Final Expression for Ionization Energy Thus, the ionization energy of gas X is given by: \[ \text{IE} = \frac{2 \Delta H}{N_A} \] ### Conclusion The ionization energy of gas X is \( \frac{2 \Delta H}{N_A} \).
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