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❈❖◆❙❚❆◆❚❊❙
❈♦♥st❛♥t❡ ❞❡ ❆✈♦❣❛❞r♦ ❂ 6, 02 · 1023 ♠♦❧✕✶
❱♦❧✉♠❡ ♠♦❧❛r ❞♦ ❣ás ✐❞❡❛❧ ❂ ✷✷✱✹ ▲ ✭❈◆❚P✮
❈♦♥st❛♥t❡ ❞♦s ●❛s❡s ✐❞❡❛✐s ❂ 8, 21 · 10−2 ❛t♠ ▲ ❑✕✶
❂ ✽✱✸✶ ❏ ✭❑♠♦❧✮✕✶
▼❆❙❙❆❙ ▼❖▲❆❘❊❙ ✭❣✴♠♦❧✮
❍❂✶ ❙❂✸✷ ◆❛❂✷✸ ■❂✶✷✻✱✾
❈❂✶✷ ❈❧❂✸✺✱✺ ❋❂✶✾ ❋❡❂✺✺✱✽✺
◆❂✶✹ ❈❛❂✹✵ ❖❂✶✻ ❈✉❂✻✸✱✻
▼❆❚❊▼➪❚■❈❆
✶✳ ❘❡s♦❧✈❡r✿
x
4
−
2x− 1
3
=
x+ 1
6
✷✳ ❙❡ xy = 7✱ ❞❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡
2(x+y)2
2(x−y)2
✳
✸✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡ x ❡♠
x+
√
x2 − 1
x−
√
x2 − 1
−
x−
√
x2 − 1
x+
√
x2 − 1
= 8
√
3
✹✳ ❘❡❞✉③❛ ❛♦ ♥✉♠❡r❛❧ ♠❛✐s s✐♠♣❧❡s
√
2 ·
√
2 +
√
2 ·
√
2 +
√
2 +
√
2 ·
√
2−
√
2 +
√
2
✺✳ ❙❡ 4x = 3 ❡ 4y = 9✱ ❡♥tã♦ (0, 125)−4x+2y ✈❛❧❡✿
✻✳ ❙❡ log 2 = 0, 3✱ ❞❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦ q✉♦❝✐❡♥t❡
log5 32
log4 5
✳
✼✳ ❙❡ log3 7 = a ❡ log5 3 = b✱ ❡♥tã♦ log5 7 é ✐❣✉❛❧ ❛✿
✽✳ ❈❛❧❝✉❧❡ ❛ ❡①♣r❡ssã♦✿
(
2
5
)0
· (0, 01)2 · (0, 25)
1
2
✾✳ ❈❛❧❝✉❧❡ ♦ ✈❛❧♦r ❞❡






1
5
−
2
3



3
−
(
212
210
)
1
2



−






(0, 333...)
−
5
2
√
3
−
5


5
3


2
3
√
5






✶✵✳ ❈❛❧❝✉❧❡ ♦ ✈❛❧♦r ♥✉♠ér✐❝♦ ❞❛ ❡①♣r❡ssã♦
−x3 − 3
√
y
−a−1
♣❛r❛ x = −4✱ y = −8 ❡ a =
1
3
✳
✶✶✳ ❉❡t❡r♠✐♥❡ x ❡ y











xy
x+ y
=
6
5
1
y
−
1
x
=
1
6
✶✷✳ ❈❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❡ a ❡♠✿
√
a3 ·
√
a
48
√
a10 · 24
√
a7
= 243
✶✸✳ ❯♠ ♠❡♥s❛❣❡✐r♦ ✈❛✐ ❞❡ A ❛té B ❞❡ ❜✐❝✐❝❧❡t❛✱ ❝♦♠
❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ✶✵ ❦♠✴❤ ❡ ✈♦❧t❛ ❞❡ B ❛ A✱ ❛ ♣é✱
❢❛③❡♥❞♦ ✹ ❦♠✴❤✳ ❈❛❧❝✉❧❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ AB✱
s❛❜❡♥❞♦✲s❡ q✉❡ ♦ t❡♠♣♦ t♦t❛❧ ❞❡ ✐❞❛ ❡ ✈♦❧t❛ ❢♦✐
❞❡ ✼ ❤♦r❛s✳
✶✹✳ ❙♦♠❛♥❞♦✲s❡ ✽ ❛
3
5
❞❡ ✉♠ ♥ú♠❡r♦✱ ♦❜té♠✲s❡ ✉♠❛
s♦♠❛ ✐❣✉❛❧ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s
9
10
❞♦ ♠❡s♠♦
♥ú♠❡r♦ ❡ ✶✸✳ ❉❡t❡r♠✐♥❛r ♦ ♥ú♠❡r♦✳
✶✺✳ ❆ ❧❡✐ ❞❡ ❞❡❝♦♠♣♦s✐çã♦ ❞♦ r❛❞✐✉♠✱ ♥♦ t❡♠♣♦ t ≥
0✱ é ❞❛❞❛ ♣♦r M(t) = C · e−kt✱ ♦♥❞❡ M(t) é ❛
q✉❛♥t✐❞❛❞❡ ❞❡ r❛❞✐✉♠ ♥♦ t❡♠♣♦ t❀ C ❡ k sã♦
❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s ❡ e é ♦ ♥ú♠❡r♦ ♥❡♣❡r✐❛♥♦
✭e = 2, 71828...✮✳ ❙❡ ❛ ♠❡t❛❞❡ ❞❛ q✉❛♥t✐❞❛❞❡
♣r✐♠✐t✐✈❛ M(0) ❞❡s❛♣❛r❡❝❡ ❡♠ ✶✻✵✵ ❛♥♦s✱ q✉❛❧
❛ q✉❛♥t✐❞❛❞❡ ♣❡r❞✐❞❛ ❡♠ ✶✵✵ ❛♥♦s❄
◗❯❮▼■❈❆
✶✳ P❛r❛ ♣r♦❞✉③✐r ✹✱✹✽ ▲ ❞❡ ❈❖✷ ♥❛s ❈◆❚P✱ ❝♦♥✲
❢♦r♠❡ ❛ r❡❛çã♦✿
CaCO3 −−→ CaO + CO2
é ♥❡❝❡ssár✐♦ q✉❡ q✉❛♥t✐❞❛❞❡✱ ❡♠ ❣r❛♠❛s✱ ❞❡
❈❛❈❖✸❄
❉❛❞♦✿ ❈❛❈❖✸❂✶✵✵ ❣✴♠♦❧
✷✳ ◗✉❛❧ ❛ ♠❛ss❛ ❛tô♠✐❝❛ ❞❡ X✱ s❡ ✹✻✱✺ ❣ ❞♦ ❝♦♠✲
♣♦st♦ ❳✸✭P❖✹✮✷ ❝♦♥té♠ ✶✽ ❣ ❞❡ X❄
✸✳ ❋❛③❡♥❞♦ r❡❛❣✐r á❝✐❞♦ ❝❧♦rí❞r✐❝♦ ✭❍❈❧✮ ❡♠ ❡①✲
❝❡ss♦ ❝♦♠ ❝❛r❜♦♥❛t♦ ❞❡ ❝á❧❝✐♦ ✭❈❛❈❖✸✮✱ ❢♦r❛♠
♦❜t✐❞♦s ✸✱✶ ▲ ❞❡ ❣ás✱ ♠❡❞✐❞♦s ❛ ✸✼♦❈ ❡ à ♣r❡s✲
sã♦ ❞❡ ✵✱✽✷ ❛t♠✳ ◗✉❛❧ ❛ ♠❛ss❛ ❞❡ ❝❛r❜♦♥❛t♦ ❞❡
❝á❧❝✐♦ q✉❡ r❡❛❣✐✉❄
✹✳ ❘❡❢❡r✐♥❞♦✲s❡ ❛ ✶✽✵ ♠▲ ❞❡ ❍✷❖ ✭❞❡♥s✐❞❛❞❡ ❂ ✶
❣✴♠▲✮✱ ❛ss✐♥❛❧❡ ❱❊❘❉❆❉❊■❘❖ ✭❱✮ ♦✉ ❋❆▲❙❖
✭❋✮✿
✶
■✳ P♦ss✉✐ ✶✵ ♠♦❧s ❞❡ ♠♦❧é❝✉❧❛s✳
■■✳ ❙❡ ❞❡❝♦♠♣õ❡ ❡♠ ✶✻✵ ❣ ❞❡ ♦①✐❣ê♥✐♦✳
■■■✳ ❈♦♥té♠ 6, 02 · 1023 ♠♦❧é❝✉❧❛s✳
■❱✳ ❆♣r❡s❡♥t❛ ✶✵ át♦♠♦s ❞❡ ♦①✐❣ê♥✐♦✳
❱✳ ❈♦♥té♠ ✶✵ ♠♦❧s ❞❡ át♦♠♦s ❞❡ ❤✐❞r♦❣ê♥✐♦✳
✺✳ ❉❡t❡r♠✐♥❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ át♦♠♦s ❞❡ ♠❡r❝ú✲
r✐♦ ♣r❡s❡♥t❡s ♥✉♠ t❡r♠ô♠❡tr♦ q✉❡ ❝♦♥té♠ ✷✱✵
❣ ❞❡ss❡ ♠❡t❛❧ ✭❍❣ ❂ ✷✵✵ ❣✴♠♦❧✮
✻✳ ▲❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ❛ ❡①✐stê♥❝✐❛ ❞♦s ✸ ✐sót♦♣♦s
❞♦ ❤✐❞r♦❣ê♥✐♦ ❍✿ ❩❂✶❀ ❆❂✶
❍✿ ❩❂✶❀ ❆❂✷
❍✿ ❩❂✶❀ ❆❂✸
❡ ❞❡ ❛♣❡♥❛s ✉♠ ✐sót♦♣♦ ❞♦ ♦①✐❣ê♥✐♦ ✭❖✿ ❩❂✽❀
❆❂✶✻✮✱ ♦ ♥ú♠❡r♦ ❞❡ ♥ê✉tr♦♥s ✐♠♣♦ssí✈❡❧ ❞❡ s❡
❡♥❝♦♥tr❛r ♥✉♠❛ ♠♦❧é❝✉❧❛ ❞❡ á❣✉❛ é✿
❛✮ ✾ ❜✮ ✶✵ ❝✮ ✶✶ ❞✮ ✶✷ ❡✮ ✶✸
✼✳ ❆ ♦❜t❡♥çã♦ ❞❡ ❡t❛♥♦❧✱ ❛ ♣❛rt✐r ❞❡ s❛❝❛r♦s❡ ✭❛ç✉✲
❝❛r✮ ♣♦r ❢❡r♠❡♥t❛çã♦✱ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛
♣❡❧❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦✿
C12H22O11 +H2O −−→ 4C2H5OH+ 4CO2
❆❞♠✐t✐♥❞♦✲s❡ q✉❡ ♦ ♣r♦❝❡ss♦ t❡♥❤❛ r❡♥❞✐♠❡♥t♦
❞❡ ✶✵✵✪✱ ❝❛❧❝✉❧❡ ❛ ♠❛ss❛ ✭❡♠ ❦❣✮ ❞❡ ❛çú❝❛r
♥❡❝❡ssár✐❛ ♣❛r❛ ♣r♦❞✉③✐r ✉♠ ✈♦❧✉♠❡ ❞❡ ✺✵ ❧✐tr♦s
❞❡ ❡t❛♥♦❧✳
❉❛❞♦s✿ ❉❡♥s✐❞❛❞❡ ❞♦ ❡t❛♥♦❧ ❂ ✵✱✽ ❣✴❝♠✸❀
❙❛❝❛r♦s❡ ❂ ✸✹✷ ❣✴♠♦❧❀
❊t❛♥♦❧ ❂ ✹✻ ❣✴♠♦❧✳
✽✳ ❙❡❥❛♠ ♦s ❡❧❡♠❡♥t♦s ❆ ✭Z = 63 ❀ A = 150✮✱ B
❡ C ❞❡ ♥ú♠❡r♦s ❛t♦♠✐❝♦s ❝♦♥s❡❝✉t✐✈♦s ❡ ❝r❡s✲
❝❡♥t❡s ♥❛ ♦r❞❡♠ ❞❛❞❛✳ ❙❛❜❡♥❞♦ q✉❡ A ❡ B sã♦
✐só❜❛r♦s ❡ q✉❡ B ❡ C sã♦ ✐sót♦♥♦s✱ ♣♦❞❡♠♦s ❝♦♥✲
❝❧✉✐r q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ♠❛ss❛ ❞♦ ❡❧❡♠❡♥t♦ C é
✐❣✉❛❧ ❛✿
✾✳ ❯♠❛ ❛♠♦str❛ ❞❡ ✶✺✱✹ ❣ ❞❡ ✉♠❛ ♠✐st✉r❛ ❞❡ ❑■
❡ ◆❛■ ❝♦♥té♠ ✉♠ t♦t❛❧ ❞❡ ✵✱✶✵✵ ♠♦❧ ❞❡ ✐♦❞❡t♦✳
❉❡t❡r♠✐♥❡ ❛ ♠❛ss❛ ❞❡ ❑■ ♥❡st❛ ♠✐st✉r❛✳
✶✵✳ ❉❡t❡r♠✐♥❡ ♦ ✈♦❧✉♠❡ ❞❡ ❙❖✷ ❣❛s♦s♦✱ ♠❡❞✐❞♦ ♥❛s
❈◆❚P✱ ♥❡❝❡ssár✐♦ ♣❛r❛ tr❛♥s❢♦r♠❛r ❝♦♠♣❧❡t❛✲
♠❡♥t❡ ✷✺✵ ❝♠✸ ❞❡ s♦❧✉çã♦ ❛q✉♦s❛ ✵✱✶✵✵ ♠♦❧❛r
❞❡ ◆❛❖❍ ❡♠ s♦❧✉çã♦ ❞❡ ◆❛✷❙❖✸✳
✶✶✳ ❖ ❝♦❜r❡ é ✉♠ ♠❡t❛❧ ❡♥❝♦♥tr❛❞♦ ♥❛ ♥❛t✉r❡③❛ ❡♠
❞✐❢❡r❡♥t❡s ♠✐♥❡r❛✐s✳ ❙✉❛ ♦❜t❡♥çã♦ ♣♦❞❡ ♦❝♦rr❡r
♣❡❧❛ r❡❛çã♦ ❞❛ ❝❛❧❝♦s✐t❛ ✭❈✉✷❙✮ ❝♦♠ ❛ ❝✉♣r✐t❛
✭❈✉✷❖✮✿
Cu2S + 2Cu2O −−→ 6Cu + SO2
❉❡t❡r♠✐♥❡ ❛ ♠❛ss❛ ❞♦ ❝♦❜r❡ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡
✷✵✵ ❣ ❝❛❧❝♦s✐t❛ ❝♦♠ ✷✵✱✺✪ ❞❡ ✐♠♣✉r❡③❛s ❡ ❝✉✲
♣r✐t❛ s✉✜❝✐❡♥t❡ ♥✉♠❛ r❡❛çã♦ ❝♦♠ ✻✵✪ ❞❡ r❡♥❞✐✲
♠❡♥t♦✳
✶✷✳ ❈♦♥s✐❞❡r❡ ❛ r❡❛çã♦ q✉í♠✐❝❛ r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛
❡q✉❛çã♦✿
2Fe2S3 + 6H2O+ 3O2 −−→ 4Fe(OH)3 + 6S
❈❛❧❝✉❧❡ ❛ q✉❛♥t✐❞❛❞❡ ✭❡♠ ♠♦❧s✮ ❞❡ ❋❡✭❖❍✮✸ q✉❡
♣♦❞❡ s❡r ♣r♦❞✉③✐❞❛ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ♠✐st✉r❛ q✉❡
❝♦♥t❡♥❤❛ ✶✱✵ ♠♦❧ ❞❡ ❋❡✷❙✸✱ ✷✱✵ ♠♦❧s ❞❡ ❍✷ ❡ ✸✱✵
♠♦❧s ❞❡ ❖✷✳
✶✸✳ ❆ t❡♦r✐❛ ❞❛s ❝♦❧✐sõ❡s é ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❞❛s
✈❡❧♦❝✐❞❛❞❡s ❞❡ r❡❛çõ❡s ❜✐♠♦❧❡❝✉❧❛r❡s ♥❛ ❢❛s❡ ❣❛✲
s♦s❛ ✉t✐❧✐③❛♥❞♦ ✉♠ ♠♦❞❡❧♦ ❡♠ q✉❡ ❞✉❛s ♠♦❧é❝✉✲
❧❛s ❝♦❧✐❞❡♠ ❞❡✈✐❞♦ ❛ ✉♠ ♠í♥✐♠♦ ❞❡ ❡♥❡r❣✐❛ ❝✐♥é✲
t✐❝❛✳ ❊ss❡ ♠í♥✐♠♦ ❞❡ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ é s✐♠✐❧❛r
à ❊♥❡r❣✐❛ ❞❡ ❆t✐✈❛çã♦ (Ea) ❞❛ r❡❛çã♦✳ ❆ ♣❛rt✐r
❞❡ss❡ ♠♦❞❡❧♦✱ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r q✉❡ ❛ ❝♦♥st❛♥t❡
❞❡ ✈❡❧♦❝✐❞❛❞❡ k2 ❞❡ ✉♠❛ r❡❛çã♦ ❜✐♠♦❧❡❝✉❧❛r é✿
k2 = Pσ
(
8kT
πµ
)1/2
NAe
−Ea/RT ✭■✮
◆❛ ❡q✉❛çã♦ ✭■✮✱ T é ❛ t❡♠♣❡r❛t✉r❛ ❞♦ s✐st❡♠❛
✭❡♠ ❦❡❧✈✐♥✮❀ P ✱ σ✱ ❡ µ sã♦ ♣❛râ♠❡tr♦s q✉❡ ❞❡✲
♣❡♥❞❡♠ ❞❛ ♦r✐❡♥t❛çã♦ ❞❛s ♠♦❧é❝✉❧❛s q✉❡ ❝♦❧✐✲
❞❡♠✱ ❞❛ ár❡❛ ❞❡ ❝♦❧✐sã♦ ❡ ❞❛ ♠❛ss❛ ❞❛s ♠♦❧é✲
❝✉❧❛s❀ k✱ π✱ NA ❡ R sã♦ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s❀
e ∼= 2, 72✳
❈♦♥s✐❞❡r❛♥❞♦ ❛ ❡q✉❛çã♦ ✭■✮✱ ❛ss✐♥❛❧❡ ❛ ❛❧t❡r♥❛✲
t✐✈❛ ❈❖❘❘❊❚❆✳
❛✮ ❛ ❡q✉❛çã♦ ✭■✮ t❛♠❜é♠ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦
k2 = Pσ
(
8kT
πµ
)1/2
NAe
RT/Ea .
❜✮ ➚ ♠❡❞✐❞❛ q✉❡ ❛ t❡♠♣❡r❛t✉r❛ T ❞✐♠✐♥✉✐✱ ❛
❝♦♥st❛♥t❡ ❞❡ ✈❡❧♦❝✐❞❛❞❡ k2 ❞❛ r❡❛çã♦ ❛✉♠❡♥t❛✳
❝✮ ❆ ❝♦♥st❛♥t❡ ❞❡ ✈❡❧♦❝✐❞❛❞❡ k2 ❞❛ r❡❛çã♦ é ❞✐✲
r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ à t❡♠♣❡r❛t✉r❛ T ✳
❞✮ ❆ ❡♥❡r❣✐❛ ❞❡ ❛t✐✈❛çã♦ ♣♦❞❡ s❡r ❞❛❞❛ ♣❡❧❛
❡q✉❛çã♦
Ea = −RT loge
[
k2
NAPσ
(
8kT
πµ
)1/2
]
.
❡✮ ➚ ♠❡❞✐❞❛ q✉❡ ❛ t❡♠♣❡r❛t✉r❛ T ❛✉♠❡♥t❛✱ ❛
❝♦♥st❛♥t❡ ❞❡ ✈❡❧♦❝✐❞❛❞❡ k2 t♦r♥❛✲s❡ ❝❛❞❛ ✈❡③
♠❛✐s ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❡♥❡r❣✐❛ ❞❡ ❛t✐✈❛çã♦ Ea✳
✷