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Chapter 2: Problem 101

\(\mathrm{NaCl}\) is doped with \(2 \times 10^{-3} \mathrm{~mol} \% \mathrm{SrCl}_{2}\), the concen- tration of cation vacancies is (a) \(12.04 \times 10^{18} \mathrm{~mol}^{-1}\) (b) \(10.01 \times 10^{18} \mathrm{~mol}^{-1}\) (c) \(12.04 \times 10^{20} \mathrm{~mol}^{-1}\) (d) \(4.02 \times 10^{18} \mathrm{~mol}^{-1}\)

Short Answer

Expert verified
Option (a) \(12.04 \times 10^{18} \: ext{mol}^{-1}\) is closest to the calculated concentration of cation vacancies.

Step by step solution

01

Understand the Doping Problem

We are given that NaCl is doped with SrCl鈧. In this process, some of the Na鈦 ions in the NaCl lattice are replaced by Sr虏鈦 ions since Sr虏鈦 requires two chloride ions (Cl鈦) for charge neutrality but only replaces one Na鈦 ion, thus creating cation vacancies.
02

Calculate the Number of Moles of SrCl2

Given that the doping level of SrCl鈧 is 0.002% of the total moles of NaCl, we calculate the actual number of moles of SrCl鈧. If we assume the number of moles of NaCl is 1 mole, the moles of SrCl鈧 used are \(2 \times 10^{-3} / 100 = 2 \times 10^{-5}\) moles.
03

Relate SrCl鈧 to Vacancy Formation

Each mole of SrCl鈧 doped into NaCl generates exactly the same number of cation vacancies because each Sr虏鈦 replaces a Na鈦 but leaves behind a vacancy (since 2 Na鈦 ions need to be replaced for charge neutrality when Sr虏鈦 is introduced). The same number of moles of cation vacancies are created as the moles of SrCl鈧.
04

Calculate the Concentration of Cation Vacancies

The concentration of vacancies means the number of vacancies per mole of NaCl. Since there are \(2 \times 10^{-5}\) moles of cation vacancies in 1 mole of NaCl, and Avogadro's number is \(6.022 \times 10^{23} \: ext{mol}^{-1}\), the concentration is calculated as \(2 \times 10^{-5} \times 6.022 \times 10^{23} = 1.2044 \times 10^{19} \: ext{mol}^{-1}\).
05

Select the Closest Answer

The calculated concentration of cation vacancies \(1.2044 \times 10^{19} \: ext{mol}^{-1}\) is closest to the given option (a) \(12.04 \times 10^{18} \: ext{mol}^{-1}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cation Vacancies
In the world of chemistry, a vacancy refers to a missing atom or ion in a crystal lattice. When a cation is missing, it is termed a cation vacancy. Cation vacancies are crucial because they affect the properties of the material, such as conductivity and strength.
In an ionic crystal like NaCl, doping with a compound like SrCl鈧 can create these vacancies. The replacement of some Na鈦 ions with Sr虏鈦 ions leads to vacancies due to the difference in their charges. This concept helps us understand why such modifications alter the properties of the substance.
SrCl鈧 Doping
Doping is a process of intentionally adding impurities to a material to change its properties. In our problem, NaCl, a simple ionic compound, is doped with SrCl鈧.
  • SrCl鈧 has a cation, Sr虏鈦, which has a higher charge than Na鈦.
  • When introduced into NaCl, Sr虏鈦 replaces some of the Na鈦 ions in the lattice.
  • Because Sr虏鈦 has a +2 charge, it pairs with two Cl鈦 ions, unlike Na鈦 which pairs with just one Cl鈦.
  • As a result, for every Sr虏鈦 introduced, one Na鈦 position remains vacant in the crystal lattice.
This incorporation leads to cation vacancies, ultimately affecting the crystal's physical properties.
Charge Neutrality
A critical concept in doping is maintaining charge neutrality. Although additional charges are introduced by doping, the entire system must remain electrically neutral to avoid accumulations of charge.
In the case of NaCl being doped with SrCl鈧:
  • Each Sr虏鈦 ion replaces one Na鈦 ion in the lattice.
  • This leaves a vacant spot since two Na鈦 are required to balance the charge of one Sr虏鈦 with its two associated Cl鈦 ions.
  • The vacancy is a way to maintain the overall charge neutrality of the crystal structure by balancing the charge discrepancy between Sr虏鈦 and Na鈦.
When designing materials with specific properties, maintaining charge neutrality is essential to ensure stability and functionality of the lattice.
NaCl Crystal Lattice
The crystal lattice of NaCl is a well-ordered, repeating pattern of ions. It features alternating Na鈦 and Cl鈦 ions in a cubic arrangement that is characteristic of many crystalline solids.
Upon doping with SrCl鈧:
  • The regularity of the NaCl lattice is altered.
  • Sr虏鈦 ions step in the place of some Na鈦 ions, causing the formation of vacant positions or vacancies.
  • Though these changes disturb the regularity, they also enhance certain properties like electrical conductivity.
Understanding the NaCl lattice helps in comprehending how modifications through doping can lead to significant property changes in the material.
Concentration Calculation
Calculating the concentration of cation vacancies is a methodical process:
  • First, we determine how much SrCl鈧 is added to the NaCl.
  • In this exercise, the doping level is given as 0.002%, which is calculated as a very small fraction of the total moles of NaCl.
  • For simplicity, assume we're dealing with 1 mole of NaCl. Thus, the SrCl鈧 moles are \(2 \times 10^{-5}\) moles.
  • Given that each Sr虏鈦 results in one cation vacancy, the number of moles of cation vacancies equals the moles of SrCl鈧 doped into the lattice.
  • To find the concentration per mole of NaCl, we multiply the number of moles of vacancies by Avogadro's number, \(6.022 \times 10^{23}\).
  • This results in a concentration of vacancies of \(1.2044 \times 10^{19} \, \text{mol}^{-1}\), matching the closest provided \(a\) option.
Accurate concentration calculations are not only necessary for solving textbook problems but also for real-world applications in materials science.

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Most popular questions from this chapter

Which of the following statements is correct? (1) The coordination number of each type of ion in \(\mathrm{CsCl}\) crystal is 8 (2) A metal that crystallizes in BCC structure has a coordination number of 12 (3) A unit cell of an ionic crystal shares some of its ion with other units cells (4) the length of unit cell in \(\mathrm{NaCl}\) is \(552 \mathrm{pm}\) (r \(\mathrm{Na}^{+}=\) \(\left.95 \mathrm{pm}, \mathrm{r} \mathrm{Cl}^{-}=181 \mathrm{pm}\right)\) (a) 1,2 (b) 1,3 (c) 2,3 (d) 2,4

Which of the following liquid has the highest vapour pressure or is most volatile? (a) HF (1) (b) \(\mathrm{NH}_{3}\) (l) (c) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\) (1) (d) \(\mathrm{H}_{2} \mathrm{O}\)

Among the following statement(s) is /are correct? (a) The number of nearest the neighbours in three consecutive layers in ABABAB...... Type solid is \(12 .\) (b) Effective number of molecules per unit cell in \(\mathrm{MgAl}_{2} \mathrm{O}_{4}\) is 4 (c) The distance between a tetrahedral and octahedral hole in MgO lattice is \(\frac{\sqrt{3}}{4}\) a where a is the edge length of the cube. (d) C-atoms are present in tetrahedral and octahedral holes in diamond.

In which of the following structures, the cation has maximum coordination number? (a) \(\mathrm{NaCl}\) (b) \(\mathrm{Na}_{2} \mathrm{O}\) (c) zinc blende (d) \(\mathrm{CaF}_{2}^{2}\)

Which defect causes decrease in the density of a crystal? (a) Schottky (b) F-centre (c) Frenkel (d) Interstitial
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