Name: estimating equilibrium with small changes
Uploaded: 2020-10-19T09:47:37.000Z
Duration: 2 min 20 s
Channel: JoVE

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Chemistry

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PS.Chm.5.2

The Small x Assumption

작은 x 가정은 평형식을 간단하게 풀고 근의 공식을 사용하지 않기 위해 특정 조건에서 사용할 수 있는 근사치입니다. 반응의 평형 상수가 작고 반응물의 초기 농도가 충분히 높으면 소량의 반응물만 생성물로 전환될 수 있습니다. 이러한 조건이 충족되면 초기 농도의 변화를 무시할 수 있다는 가정으로부터 평형 농도를 근사화할 수 있습니다.

그러나 가정의 타당성을 보여주기 위해서는 농도 변화 x가 초기 농도의 5 퍼센트 미만이라는 것을 확인해야 합니다. 0.66몰 이염화 카르보닐 기체의 분해는 일산화탄소와 염소기체를 생성하며 이 반응의 평형 상수는 2.2 10⁻¹⁰입니다. 평형 농도를 계산하기 위해 값은 변화 및 평형 농도와 함께 ICE표에 게시됩니다.

식에서 평형 농도를 대체하면 K는 x 제곱을 0.66-x로 나눈 값과 같습니다. K가 매우 작기 때문에 이염화 카르보닐의 초기 농도 x의 변화는 무시할 수 있는 것으로 예상됩니다. 따라서 0.66-x는 0.66으로 근사할 수 있습니다.

식을 풀면 x는 1.2 10⁻⁵ 몰과 같습니다. 여기서 x는 초기 0.66 몰농도의 이염화 카르보닐의 0.0018 퍼센트에 불과하며 이는 허용되는 최대값인 5 퍼센트보다 훨씬 적습니다. 따라서 작은 x가정이 여기서 유효하게 됩니다.

ICE 표의 x값을 사용하면 이염화 카르보닐의 평형 농도는 유효 숫자로 0.66몰이며 일산화탄소와 염소의 농도는 모두 1.2 x 10⁻⁵ 몰로 됩니다.

<ol>
	<li>Lim, Kieran F. "화학에서 그래픽 계산기 및 스프레드시트 사용: 평형 문제 해결." 화학 교육 저널 85, no. 10 (2008): 1347. <a href="https://pubs.acs.org/doi/pdf/10.1021/ed085p1347">https://pubs.acs.org/doi/pdf/10.1021/ed085p1347</a></li>
</ol>

작은 x 가정

반응의 평형 상수가 작은 경우 평형 위치는 반응물에 유리합니다. 이러한 반응에서 반응물의 초기 농도가 높고 K_c 값이 작으면 농도의 변화는 무시할 수 있습니다. 이러한 상황에서 평형 농도는 초기 농도와 거의 같습니다. 이 추정은 일부 평형 농도가 초기 농도와 같다고 가정하여 평형 계산을 단순화하는 데 사용할 수 있습니다. 그러나 이러한 가정을 하려면 약산이나 약염기의 농도 변화, 즉 x가 초기 농도의 5% 미만이어야 합니다. x가 5%보다 크면 평형 방정식을 풀기 위해 2차 공식을 사용해야 합니다.

대수 단순화 가정을 사용한 평형 농도 계산

0.15 M HCN 용액의 평형 농도는 얼마일까요?



<img alt="Eq1" src="/files/ftp_upload/11477/11477_Equation_1.png" style="text-align: center; height: 50px;" />



x를 사용하여 평형 상태에서 각 생성물의 농도를 나타내면 이 ICE 표가 제공됩니다.

<div class="overflow">
<table style="margin-left: auto; margin-right: auto;">
	<tbody>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;"></td>
			<td style="color: black; border: 1px solid black; text-align: center;">HCN(아크)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">H+ (아크)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">CN- (아크)</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">초기 농도(M)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.15</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">변경(M)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">-x</td>
			<td style="color: black; border: 1px solid black; text-align: center;">+x</td>
			<td style="color: black; border: 1px solid black; text-align: center;">+x</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">평형 농도 (M)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.15 - x</td>
			<td style="color: black; border: 1px solid black; text-align: center;">Ⅹ</td>
			<td style="color: black; border: 1px solid black; text-align: center;">Ⅹ</td>
		</tr>
	</tbody>
</table>
</div>

평형 농도 항을 K_c 표현으로 대체합니다.

<img alt="Eq2" src="/files/ftp_upload/11477/11477_Equation_2.png" style="text-align: center; height: 95px;" />

이차 형식으로 다시 배열하고 x를 구합니다.

<img alt="Eq3" src="/files/ftp_upload/11477/11477_Equation_3.png" style="text-align: center; height: 95px;" />

따라서 [H^+] = [CN^–] = x = 8.6 × 10^–6 M 및 [HCN] = 0.15 – x = 0.15 M입니다.

이 경우 농도 변화는 초기 농도(작은 K의 결과)보다 훨씬 작으므로 초기 농도는 무시할 수 있는 변화를 경험합니다.

<img alt="Eq4" src="/files/ftp_upload/11477/11477_Equation_4.png" style="text-align: center; height: 50px;" />

이 근사법을 사용하면 2차 방정식의 근을 풀 필요가 없는 계산에 대한 보다 편리한 수학적 접근 방식이 가능합니다.

<img alt="Eq5" src="/files/ftp_upload/11477/11477_Equation_5.png" style="text-align: center; height: 135px;" />

계산된 x 값은 실제로 초기 농도보다 훨씬 적습니다.

<img alt="Eq6" src="/files/ftp_upload/11477/11477_Equation_6.png" style="text-align: center; height: 50px;" />

그래서 근사치가 정당화되었습니다. 이 단순화된 접근 방식으로 근사치를 정당화하지 못하는 x 값을 산출하려면 근사치를 만들지 않고 계산을 반복해야 합니다.

이 문서는 에서 발췌되었습니다 <a href="https://openstax.org/books/chemistry-2e/pages/13-4-equilibrium-calculations">Openstax, Chemistry 2e, Section 13.4 Equilibrium Calculations</a><a href="https://openstax.org/books/chemistry-2e/pages/13-4-equilibrium-calculations">.</a>

반응이 작은 평형 상수를 갖는 경우, 평형 위치는 반응물을 선호합니다. 이러한 반응에서 농도의 미미한 변화는 ...

Education

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JoVE Core

화학

하나

Chemical Equilibrium

화학 평형

KEY TERMS AND CONCEPTS

주요 용어 및 개념

estimating equilibrium with small changes

A small equilibrium constant can make equilibrium calculations easier. In these reactions, the equilibrium position favors the reactants. When the initial reactant concentration is high and the Kc value is small, the change in concentration may be negligible. In that case, the equilibrium concentration is close to the initial concentration.

This idea helps simplify weak acid and weak base problems. Chemists can assume that some equilibrium concentrations are equal to the initial concentrations. The change in concentration is written as x. To use the shortcut, x must be less than 5% of the initial concentration. If x is greater than 5%, the quadratic formula is needed instead.

A solution of HCN shows how the approximation works. For a 0.15 M HCN solution, an ICE table gives HCN, H+, and CN− with initial concentrations of 0.15, 0, and 0 M. The change is −x for HCN and +x for each product, so the equilibrium concentrations are 0.15 − x, x, and x. Substituting these terms into the Kc expression and solving gives [H+] = [CN−] = 8.6 × 10−6 M and [HCN] = 0.15 M.

In this example, the calculated x is much smaller than the initial concentration. That means the initial HCN concentration changes very little. The shortcut is justified, and the calculation avoids solving a quadratic equation. If the result had not met the 5% rule, the calculation would need to be repeated without the approximation.

Estimating Equilibrium with Small Changes

estimating-equilibrium-with-small-changes

the equilibrium constant

Chemical Equilibrium and Kc Rules

Chemical equilibrium and the equilibrium constant Kc describe how reactants and products are balanced in a reaction. In the oxidation of sulfur dioxide, a mixture that starts with reactants only has no product at first. As the reaction moves forward toward equilibrium, reactant concentrations decrease and product concentration increases. When equilibrium is reached, the concentrations of both stay constant. A reaction can also begin with only products present. In that case, the reaction moves...

homogeneous equilibria for gaseous reactions

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Relating Kc and Kp in Gas Equilibria

Gas-phase equilibrium can be written using either concentration or pressure. The equilibrium constant may be shown as Kc, which uses molar concentration, or as Kp, which uses partial pressure of the reactants and products. The link between Kc and Kp comes from the ideal gas equation and the definition of molarity. Molarity is the number of moles divided by volume. In the ideal gas equation, P stands for partial pressure, V for volume, n for number of moles, R for the gas constant, and T for...

le chatelier's principle: changing concentration

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Shifting Equilibrium by Changing Concentration

Chemical equilibrium is a dynamic balance. At equilibrium, the forward and reverse reactions happen at equal rates. If the system is stressed in a way that changes one rate more than the other, it is no longer at equilibrium and will shift until balance is restored. This behavior is described by Le Châtelier’s principle. Changes in concentration are a major stress on an equilibrium system. Reaction rates depend mainly on concentration and temperature. For that reason, changing the amount of a...

calculating the equilibrium constant

Finding Kc from Equilibrium Data

The equilibrium constant, Kc, can be found from the equilibrium concentrations of reactants and products. When those concentrations are known, the calculation is straightforward. The values are placed into the Kc expression for the reaction. One example uses nitrogen dioxide and dinitrogen tetroxide. When 0.10 mol NO2 is added to a 1.0-L flask at 25 °C, the concentrations change until equilibrium is reached. At equilibrium, [NO2] is 0.016 M and [N2O4] is 0.042 M, and these values are used to...

reaction quotient

Using Q to Track Equilibrium Changes

Reaction quotient Q helps track how far a reversible reaction has moved toward equilibrium. For a balanced reaction such as m A + n B ⇌ x C + y D, Q comes directly from the stoichiometric coefficients in the equation. When concentrations are used, the expression is written as Qc. For gases, a similar form uses partial pressures. These expressions use relative concentration and pressure values, not absolute values. That means reaction quotients are dimensionless and have no units. As a reaction...

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Chemical Equilibrium in Reversible Reactions

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Equilibrium concentrations can be found from initial concentrations and an equilibrium constant. This skill matters in many areas of science and technology, including pharmaceutical product formulation and dosing. After a drug is ingested or injected, it may take part in several chemical equilibria that change its final concentration in the body system of interest. Scientists use these quantitative equilibrium relationships to help calculate a dosage that will produce the desired therapeutic...

le chatelier's principle: changing volume (pressure)

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Gas-phase equilibrium can shift when the volume of the system changes. In these reactions, a change in pressure changes concentration because the partial pressure of an ideal gas is proportional to its molar concentration. That means pressure changes affect equilibrium in the same way as concentration changes. When the volume of a gas mixture changes, the concentrations of reactants and products change together. Since every gas in the equilibrium occupies the same space, the same volume change...

le chatelier's principle: changing temperature

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Le Chatelier's Principle in Heating and Cooling

Le Chatelier's principle helps explain how chemical equilibrium responds to heating and cooling. A system at equilibrium will shift to restore balance after a temperature change, but the equilibrium constant, K, will not stay the same. Instead, the mixture reaches a new equilibrium with a different composition. This temperature effect is different from a concentration change, which shifts equilibrium without changing K. To understand why temperature matters, the reaction can be written with...

the small x assumption

A small equilibrium constant can make equilibrium calculations easier. In these reactions, the equilibrium position favors the reactants. When the initial reactant concentration is high and the Kc value is small, the change in concentration may be negligible. In that case, the equilibrium concentration is close to the initial concentration. This idea helps simplify weak acid and weak base problems. Chemists can assume that some equilibrium concentrations are equal to the initial concentrations.

key terms and concepts

Learn how a small equilibrium constant lets students use the 5% rule and an ICE table to estimate equilibrium concentrations without solving a quadratic.

Estimating Equilibrium with Small Changes | JoVE K12

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Estimating Equilibrium with Small Changes

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