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11.2: Wiani

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    Malengo ya kujifunza

    Mwishoni mwa sehemu hii, utaweza:

    • Eleza wiani.
    • Tumia wingi wa hifadhi kutoka wiani wake.
    • Linganisha na kulinganisha densities ya vitu mbalimbali.

    Ambayo ina uzito zaidi, tani ya manyoya au tani ya matofali? Kitendawili hiki cha zamani kinacheza na tofauti kati ya wingi na wiani. Tani ni tani, bila shaka; lakini matofali yana wiani mkubwa zaidi kuliko manyoya, na hivyo tunajaribiwa kufikiria kuwa nzito (Kielelezo\(\PageIndex{1}\)).

    Rundo la manyoya linalopima tani na tani ya matofali huwekwa upande wowote wa ubao ambao una usawa kwenye msaada mdogo.
    Kielelezo\(\PageIndex{1}\): Tani ya manyoya na tani ya matofali yana wingi sawa, lakini manyoya hufanya rundo kubwa zaidi kwa sababu wana wiani mdogo sana.

    Uzito wiani, kama utakavyoona, ni tabia muhimu ya vitu. Ni muhimu, kwa mfano, katika kuamua kama kitu kinazama au kinaelea katika maji.

    Ufafanuzi: Uzito

    Uzito wiani ni wingi kwa kiasi cha kitengo.

    \[\rho = \dfrac{m}{V},\label{density}\]

    ambapo barua ya Kigiriki\(\rho\) (rho) ni ishara ya wiani,\(m\) ni wingi, na\(V\) ni kiasi kinachotumiwa na dutu hii.

    Katika kitendawili kuhusu manyoya na matofali, raia ni sawa, lakini kiasi kinachotumiwa na manyoya ni kubwa zaidi, kwani wiani wao ni mdogo sana. Kitengo cha SI cha wiani ni\(kg/m^3\), maadili ya mwakilishi hutolewa katika Jedwali\(\PageIndex{1}\). Mfumo wa metri awali ulipangwa ili maji yatakuwa na wiani wa\(1 \, g/cm^3\), sawa na\(10^3 \, kg/m^3\). Hivyo kitengo cha msingi cha molekuli, kilo, kilipangwa kwanza kuwa wingi wa 1000 ml ya maji, ambayo ina kiasi cha\(1000 \, cm^3\).

    Jedwali\(\PageIndex{1}\): Densities ya vitu mbalimbali
    Dutu \(\rho(10^3\frac{kg}{m^3} \, or \, \frac{g}{mL})\) Dutu \(\rho(10^3\frac{kg}{m^3} \, or \, \frac{g}{mL})\) Dutu \(\rho(10^3\frac{kg}{m^3} \, or \, \frac{g}{mL})\)
    \ (\ rho (10^3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” rowspan="1" style="Nakala-align:center; "> Yabisi \ (\ rho (10^3\ frac {kg} {m^3}\, au\,\ frac {g} {ml})\)” rowspan="1" style="text-align:center; "> Vinywaji \ (\ rho (10^3\ frac {kg} {m^3}\, au\,\ frac {g} {ml})\)” rowspan="1" style="text-align:center; "> Gesi
    Aluminium \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 2.7 Maji (4ºC) \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 1.000 Air \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">\(1.29 \times 10^{-3}\)
    Shaba \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 8.44 Damu \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 1.05 Dioksidi kaboni \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">\(1.98 \times 10^{-3}\)
    Copper (wastani) \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:katikati; "> 8.8 Maji ya bahari \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 1.025 Monoxide ya kaboni \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">\(1.25 \times 10^{-3}\)
    Dhahabu \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:katikati; "> 19.32 Mercury \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 13.6 Hidrojeni \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">\(0.090 \times 10^{-3}\)
    Chuma au chuma \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:katikati; "> 7.8 Pombe ya ethyl \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 0.79 Heliamu \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">\(0.18 \times 10^{-3}\)
    Kiongozi \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 11.3 Petroli \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 0.68 Methane \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">\(0.72 \times 10^{-3}\)
    Polystyrene \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 0.10 Glycer \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; ">1.26 Nitrojeni \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">\(1.25 \times 10^{-3}\)
    Tungsten \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 19.30 Mafuta \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 0.92 Oxide ya nitrous \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">\(1.98 \times 10^{-3}\)
    Uranium \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 18.70   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; "> Oksijeni \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">\(1.43 \times 10^{-3}\)
    Zege \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 2.30—3.0   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; "> Steam\(100^o\) \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">\(0.60 \times 10^{-3}\)
    Cork \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 0.24   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">
    Kioo, kawaida (wastani) \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 2.6   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">
    Granite \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 2.7   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">
    Ukonde wa dunia \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 3.3   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">
    Wood \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 0.3—0.9   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">
    Barafu (0°C) \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 0.917   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">
    Mfupa \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="Nakala-align:katikati; "> 1.7—2.0   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">   \ (\ rho (10 ^ 3\ frac {kg} {m ^ 3}\, au\,\ frac {g} {ml})\)” style="text-align:center; ">

    Kama unaweza kuona kwa kuchunguza Jedwali\(\PageIndex{1}\), wiani wa kitu inaweza kusaidia kutambua muundo wake. Uzito wa dhahabu, kwa mfano, ni karibu mara 2.5 wiani wa chuma, ambayo ni karibu mara 2.5 wiani wa alumini. Uzito wiani pia unaonyesha kitu kuhusu awamu ya suala hilo na substructure yake. Angalia kwamba densities ya vinywaji na yabisi ni takribani kulinganishwa, sambamba na ukweli kwamba atomi zao ni katika kuwasiliana karibu. Uzito wa gesi ni mdogo sana kuliko zile za majimaji na yabisi, kwa sababu atomi katika gesi zinatenganishwa na kiasi kikubwa cha nafasi tupu.

    CHUKUA-NYUMBANI MAJARIBIO SUKARI NA

    Rundo la sukari na rundo la chumvi linaonekana sawa, lakini ni ipi lenye uzito zaidi? Ikiwa kiasi cha piles zote mbili ni sawa, tofauti yoyote katika wingi ni kutokana na msongamano wao tofauti (ikiwa ni pamoja na nafasi ya hewa kati ya fuwele). Ambayo unafikiri ina wiani mkubwa? Ulipata maadili gani? Ni njia gani uliyotumia kuamua maadili haya?

    Mfano\(\PageIndex{1}\): Calculating the Mass of a Reservoir From Its Volume

    Hifadhi ina eneo la uso\(50 \, km^2\) na kina cha wastani cha 40.0 m Ni umati gani wa maji unaofanyika nyuma ya bwawa? (Angalia Kielelezo\(\PageIndex{2}\) kwa mtazamo wa hifadhi kubwa-Tatu Gorges Bwawa tovuti kwenye Mto Yangtze katikati ya China.)

    Picha ya Tatu Gorges Mkwawa katikati ya China.
    Kielelezo\(\PageIndex{2}\): Tatu Gorges Mkwawa katikati China. Baada ya kukamilika mwaka 2008, hii ikawa mmea mkubwa wa umeme wa maji duniani, inayozalisha nguvu sawa na ile inayozalishwa na mitambo 22 ya nguvu za nyuklia yenye ukubwa wa wastani. Bwawa la saruji lina urefu wa 181 m na 2.3 km kote. Hifadhi iliyofanywa na bwawa hili ni urefu wa kilomita 660. Zaidi ya watu milioni 1 walihamishwa na kuundwa kwa hifadhi. (mikopo: Le Grand Portage)

    Mkakati

    Tunaweza kuhesabu kiasi\(V\) cha hifadhi kutoka kwa vipimo vyake, na kupata wiani wa maji\(\rho\) katika Jedwali\(\PageIndex{1}\). Kisha masi\(m\) inaweza kupatikana kutokana na ufafanuzi wa wiani (Equation\ ref {wiani}).

    Suluhisho

    Kutatua Equation\ ref {wiani} kwa\(m\) anatoa

    \[m = \rho V. \nonumber\]

    Kiasi\(V\) cha hifadhi ni eneo lake la uso\(A\) mara kina cha wastani\(h\):

    \[\begin{align*} V &= Ah \\[5pt] &= (50.0 \, km^2)(40.0 \, m) \\[5pt] &= \left [ (50.0 \, km^2)\left(\frac{10^3 \, m}{1 \, km}\right )\right ](40.0 \, m) \\[5pt] &= 2.00 \times 10^9 \, m^3 \end{align*}\]

    Uzito wa maji\(\rho\) kutoka Jedwali\(\PageIndex{1}\) ni\(1.000 \times 10^3 \, kg/m^3 \). Kubadilisha\(V\) na\(\rho\) katika kujieleza kwa wingi hutoa

    \[\begin{align*} m &= (1.00 \times 10^3 \, kg/m^3)(2.00 \times 10^9 \, m^3) \\[5pt] &= 2.00 \times 10^{12} \, kg.\end{align*}\]

    Majadiliano

    Hifadhi kubwa ina molekuli kubwa sana ya maji. Katika mfano huu, uzito wa maji katika hifadhi\(g\) ni\(mg = 1.96 \times 10^{13} \, N\) wapi kasi kutokana na mvuto wa Dunia (kuhusu\(9.80 \, m/s^2\)). Ni busara kuuliza kama bwawa lazima ugavi nguvu sawa na uzito huu mkubwa. Jibu ni hapana. Kama tutakavyoona katika sehemu zifuatazo, nguvu bwawa lazima ugavi inaweza kuwa ndogo sana kuliko uzito wa maji ambayo inashikilia nyuma.

    Muhtasari

    • Uzito wiani ni wingi kwa kiasi cha kitengo cha dutu au kitu. Katika fomu ya equation, wiani hufafanuliwa kama\[\rho = \dfrac{m}{V}. n\nonumber\]
    • Kitengo cha SI cha wiani ni\(kg/m^3\).

    faharasa

    wiani
    wingi kwa kiasi cha kitengo cha dutu au kitu