Name: estimating equilibrium with small changes
Uploaded: 2020-10-19T09:47:37.000Z
Duration: 2 min 20 s
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The Small x Assumption

Założenie małego x jest przybliżeniem, które można zastosować w pewnych warunkach, aby uprościć rozwiązywanie wyrażenia równowagi i uniknąć używania wzoru kwadratowego. 
Jeśli stała równowagi dla reakcji jest mała, a początkowe stężenia reagentów są wystarczająco wysokie, możliwe jest, że tylko niewielka ilość reagentów zostanie przekształcona w produkty.
Gdy warunki te są spełnione, stężenia równowagowe można przybliżyć przy założeniu, że zmiana stężeń początkowych jest nieistotna. Należy jednak potwierdzić, że zmiana stężenia, x, jest mniejsza niż 5% stężeń początkowych, aby wykazać, że założenie było słuszne.
Rozkład 0,66 molowego dichlorku karbonylu wytwarza gazowy tlenek węgla i chlor gazowy, a stała równowagi dla tej reakcji wynosi 2,2 × 10−10.
Aby obliczyć stężenia równowagowe, wartości są zestawiane w tabeli ICE wraz ze zmianami i stężeniami równowagowymi.
Podstawiając stężenia równowagowe w wyrażeniu, K równa się x do kwadratu podzielone przez 0,66 minus x. Ponieważ K jest bardzo mały, oczekuje się, że zmiana początkowego stężenia x dichlorku karbonylu będzie nieistotna.
Dlatego 0,66 minus x można przybliżyć do 0,66. Po rozwiązaniu x równa się 1,2 × 10−5 zębów trzonowych.
W tym przypadku x wynosi tylko 0,0018 procent początkowego stężenia molowego 0,66 dichlorku karbonylu, czyli znacznie mniej niż maksymalne dozwolone 5%. Zatem założenie małego x jest tutaj poprawne.
Przyjmując wartość x w tabeli ICE, stężenie równowagowe dichlorku karbonylu wynosi nadal 0,66 mola ze znacznymi wartościami, podczas gdy stężenia tlenku węgla i chloru wynoszą zarówno 1,2, jak × 10−5 molowych.

<ol>
	<li>Lim, Kieran F. "Korzystanie z kalkulatorów graficznych i arkuszy kalkulacyjnych w chemii: rozwiązywanie problemów równowagi." Journal of Chemical Education 85, nr 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>

założenie małego x

Jeśli reakcja ma małą stałą równowagi, położenie równowagi faworyzuje reagenty. W reakcjach takich może dojść do znikomej zmiany stężenia, jeśli początkowe stężenia reagentów są wysokie, a wartość Kc mała. W takich okolicznościach stężenie równowagowe jest w przybliżeniu równe stężeniu początkowemu. Oszacowanie to można wykorzystać do uproszczenia obliczeń równowagi, zakładając, że niektóre stężenia równowagowe są równe stężeniom początkowym. Aby jednak przyjąć takie założenie, zmiana stężenia słabego kwasu lub zasady, tj. x, musi być mniejsza niż 5% jego początkowego stężenia. Jeśli x jest większe niż 5%, wówczas do rozwiązania równania równowagi należy zastosować wzór kwadratowy.

Obliczanie stężeń równowagowych przy użyciu założeń upraszczających algebrę

Jakie są stężenia w stanie równowagi 0,15 M roztworu HCN?

<img alt="Eq1" src="/files/ftp_upload/11477/11477_Equation_1.png" style="text-align: center; height: 50px;" />

Użycie x do przedstawienia stężenia każdego produktu w równowadze daje tę tabelę ICE.

<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 (aq)           </td>
			<td style="color: black; border: 1px solid black; text-align: center;">H+ (aq)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">CN− (aq)</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Stężenie początkowe (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;">Zmień (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;">Stężenie równowagowe (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>

Zastąp warunki stężenia równowagowego w wyrażeniu Kc

<img alt="Eq2" src="/files/ftp_upload/11477/11477_Equation_2.png" style="text-align: center; height: 95px;" />

uporządkuj do postaci kwadratowej i oblicz x

<img alt="Eq3" src="/files/ftp_upload/11477/11477_Equation_3.png" style="text-align: center; height: 95px;" />

Zatem [H+] = [CN–] = x = 8,6 × 10–6 M i [HCN] = 0,15 – x = 0,15 M.

Należy zauważyć, że w tym przypadku zmiana stężenia jest znacznie mniejsza niż stężenie początkowe (konsekwencja małego K ), a więc początkowe stężenie podlega pomijalnej zmianie:

<img alt="Eq4" src="/files/ftp_upload/11477/11477_Equation_4.png" style="text-align: center; height: 50px;" />

To przybliżenie pozwala na bardziej celowe matematyczne podejście do obliczeń, które pozwala uniknąć konieczności rozwiązywania pierwiastków równania kwadratowego:

<img alt="Eq5" src="/files/ftp_upload/11477/11477_Equation_5.png" style="text-align: center; height: 135px;" />

Obliczona wartość x jest w rzeczywistości znacznie mniejsza niż stężenie początkowe

<img alt="Eq6" src="/files/ftp_upload/11477/11477_Equation_6.png" style="text-align: center; height: 50px;" />

zatem przybliżenie było uzasadnione. Jeżeli to uproszczone podejście dałoby wartość x, która nie uzasadniałaby przybliżenia, obliczenia musiałyby zostać powtórzone bez dokonywania przybliżenia.

Założenie małego x

Jeśli reakcja ma małą stałą równowagi, położenie równowagi faworyzuje reagenty. W takich reakcjach znikoma zmiana stężenia może ...

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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

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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.

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Estimating Equilibrium with Small Changes

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