30 Te whakamahinga o te ture tuatahi o te thermodynamics i roto i ētahi tukanga thermodynamic (Isobaric Isothermal Isochoric)
1. E whakaatu ana te kauwhata i raro nei i te huringa thermodynamic e pā ana ki tētahi hau. Ko te mahi i mahia e te hau i runga i te tukanga ABCD ko …
Mōhiotia:
Pēhanga 1 (P1 ) = 2 x 105 Pa
Pēhanga 2 (P 2 ) = 4 x 10 5 Pa
Rōrahi 1 (V 1 ) = 1 m 3
Rōrahi 2 (V 2 ) = 3 m 3
E hiahiatia ana: Mahi i mahia e te hau i runga i te tukanga ABCD (W).
Rongoā:
Ko te mahi e mahia ana e te hau he rite ki te horahanga o ABCD.
W = (P 2 – P 1 )(V 2 – V 1 )
W = (4 x 10 5 – 2 x 10 5 )(3 – 1)
W = (2 x 10 5 )(2)
W = 4 x 10 5 Joule
2. Ko te mahi i mahia e te hau i runga i te tukanga ABC ko…
Mōhiotia:
Pēhanga 1 (P 1 ) = 3 x 10 5 Pa
Pēhanga 2 (P 2 ) = 6 x 10 5 Pa
Rōrahi 1 (V 1 ) = 20 cm 3 = 20 x 10 -6 m 3
Rōrahi 2 (V 2 ) = 60 cm 3 = 60 x 10 -6 m 3
E hiahiatia ana : Te mahi i mahia e te hau i runga i te tukanga ABC
Rongoā:
Te mahi i mahia e te hau = te horahanga o ABC.
W = ½ (P 2 – P 1 )(V 2 – V 1 )
W = 1/2 (6 x 10 5 – 3 x 10 5 )(60 x 10 -6 – 20 x 10 -6 )
W = 1/2 (3 x 10 5 )(40 x 10 -6 )
W = 1/2 (120 x 10 -1 Joules)
W = 1/2 (12 Joules)
W = 6 Joule
3. He hoahoa PV mō te hau i roto i tētahi ipu kati e whakaaturia ana i te pikitia i raro nei. Ko te mahi i mahia e te hau e whakaaturia ana i roto i tēhea tukanga.
otinga
He tukanga isobaric te Tukanga AB me te Tukanga DC (ka mau tonu te pēhanga) . He tukanga isobaric te Tukanga AD me te Tukanga BC ( kāore e rerekē te rōrahi ).
I oti te mahi mā te hau i te wā e whakawhanui ana te hau (tukanga DC).
Anei ngā raruraru me ngā otinga e iwa e pā ana ki te Tukanga Isobaric (Pēhanga Pūmau):
Te raru 1:
Ka pēhia he hau 3.0 m³ i te pēhanga pumau ki te 2.0 m³. Mēnā ko te pāmahana tīmatanga he 300 K, he aha te pāmahana whakamutunga?
Rongoā:
Mā te whakamahi i te whārite mō tētahi tukanga isobaric: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\), ka taea e tātou te whakaoti mō te pāmahana whakamutunga:
\[
\frac{3.0}{300} = \frac{2.0}{T_2} \e kī ana ko T_2 = \frac{2.0 \times 300}{3.0} = 200\, \text{K}
\]
Te raru 2:
Tātaihia te mahi i mahia e te 2 mole o te hau e whakawhanui ana i te āhua isobaric mai i te 1 rita ki te 3 rita i te 2 atm.
Rongoā:
Mā te whakamahi i te whārite mō te mahi i mahia i roto i te tukanga isobaric: \( W = P \Delta V \):
\[
W = 2\, \text{atm} \times (3 – 1)\, \text{rita} = 4\, \text{atm}\cdot\text{rita}
\]
Te raru 3:
He aha te panonitanga o te pūngao ā-roto ina whakamahanatia ngā mole e 4 o te hau diatomic tino pai i te pēhanga pumau mai i te 200 K ki te 300 K?
Rongoā:
Mā te whakamahi i te whārite mō te huringa o te pūngao ā-roto i te pēhanga pumau mō te hau rua-atomic: \( \Delta U = nC_pdT \), ko \( C_p = \frac{7}{2}R \):
\[
\Delta U = 4 \times \frac{7}{2} \times 8.314 \times (300 – 200) \approx 6990\, \text{J}
\]
Te raru 4:
Tātaihia te wera i tāpirihia i te wā o tētahi tukanga isobaric ina whakawhanuihia ngā mole e 2 o te hau monoatomic mai i te 5 rita ki te 10 rita i te 400 K.
Rongoā:
Te whakamahi i te whārite mō te whakawhiti wera i roto i te tukanga isobaric: \( Q = nC_pdT \), ko \( C_p = \frac{5}{2}R \), me \( \frac{V_2}{T_2} = \frac{V_1}{T_1} \):
\[
Q = 2 \times \frac{5}{2} \times 8.314 \times \left(400 \times \frac{10}{5} – 400\right) = 3316\, \text{J}
\]
Te raru 5:
Tātaihia te huringa o te enthalpy o ngā mole e 3 o te hau triatomic kua whakawhanuitia mai i te 1 rita ki te 2 rita i te pāmahana o te 200 K.
Rongoā:
Mā te whakamahi i te tātai mō te huringa o te enthalpy mō tētahi tukanga isobaric: \( \Delta H = nC_pdT \), ko \( C_p = \frac{f}{2}R \) me \( f = 6 \):
\[
\Delta H = 3 \times \frac{6}{2} \times 8.314 \times \left(200 \times \frac{2}{1} – 200\right) = 4989\, \text{J}
\]
Te raru 6:
Tātaihia te rōrahi whakamutunga o ngā mole hau e 2 i te 300 K me te 5 rita i te tīmatanga ina whakamahanatia ki te 500 K i te pēhanga pumau.
Rongoā:
Mā te whakamahi i te whārite mō tētahi tukanga isobaric:
\[
\frac{V_1}{T_1} = \frac{V_2}{T_2} \e kī ana ko V_2 = \frac{V_1 \times T_2}{T_1} = \frac{5 \times 500}{300} \approx 8.33\, \text{rita}
\]
Te raru 7:
Kimihia te mahi i mahia e te hau monoatomic ina pēhia mai i te 6 rita ki te 3 rita i te 2 atm.
Rongoā:
Mā te whakamahi i te whārite mō te mahi i mahia i roto i te tukanga isobaric:
\[
W = P \Delta V = 2 \times (3 – 6) = -6\, \text{atm}\cdot\text{rita}
\]
Te raru 8:
Ka pāngia tētahi hau rua-atomic e te whakawhanuitanga isobaric mai i te 3 rita ki te 6 rita i te 300 K. Kimihia te huringa o te entropy.
Rongoā:
Mā te whakamahi i te whārite mō te huringa o te entropy i roto i tētahi tukanga isobaric mō tētahi hau diatomic, ko \( C_p = \frac{7}{2}R \):
\[
\Delta S = nC_p\ln\frac{T_2}{T_1} + nR\ln\frac{V_2}{V_1} = 2 \times \frac{7}{2} \times 8.314 \times \ln\frac{300 \times \frac{6}{3}}{300} + 2 \times 8.314 \times \ln\frac{6}{3} \approx 34.76\, \text{J/K}
\]
Te raru 9:
Tātaihia te whakawhiti wera i te wā e pēhia ana te 5 mole o te hau triatomic mai i te 10 rita ki te 5 rita i te 500 K.
Rongoā:
Mā te whakamahi i te whārite mō te whakawhiti wera i roto i tētahi tukanga isobaric, ko \( C_p = \frac{f}{2}R \) me \( f = 6 \):
\[
Q = 5 \times \frac{6}{2} \times 8.314 \times \left(500 \times \frac{5}{10} – 500\right) = -12473\, \text{J}
\]
Me whakarato ēnei raruraru i tētahi tirohanga whānui mō ngā ariā e pā ana ki te tukanga isobaric.
Anei ngā raruraru me ngā otinga e iwa e pā ana ki te Tukanga Isothermal (Pāmahana Pūmau):
Te raru 1:
Raruraru: Tātaihia te mahi e mahia ana e ngā mole e 2 o te hau pai e whakawhanui ana i te wera mai i te 1 rita ki te 2 rita i te 300 K.
Rongoā:
Mā te whakamahi i te whārite mō te mahi i mahia i roto i te tukanga isothermal mō te hau pai:
\[ W = nRT\ln\frac{V_2}{V_1} \]
\[
W = 2 \times 8.314 \times 300 \times \ln\frac{2}{1} \approx 3454\, \text{J}
\]
Te raru 2:
Raru: Kimihia te wera i whakawhitia ina pēhia ngā mole hau e 3 mā te whakamahi i te wera motuhake mai i te 4 rita ki te 2 rita i te 400 K.
Rongoā:
Te whakamahi i te whārite mō te whakawhiti wera i roto i te tukanga isothermal: \( Q = W \):
\[
Q = 3 \times 8.314 \times 400 \times \ln\frac{2}{4} \approx -3462\, \text{J}
\]
Te raru 3:
Tātaihia te huringa o te entropy ina pāngia ngā mole hau e 4 e te whakawhanuitanga isothermal mai i te 1 rita ki te 5 rita i te 300 K.
Rongoā:
Mā te whakamahi i te whārite mō te huringa o te entropy i roto i tētahi tukanga isothermal:
\[
\Delta S = nR\ln\frac{V_2}{V_1} = 4 \times 8.314 \times \ln\frac{5}{1} \approx 46.15\, \text{J/K}
\]
Te raru 4:
Tātaihia te mahi ka oti ina pēhia te 1 mole o te hau pai mā te whakamahi i te wera-whakawera mai i te 6 rita ki te 3 rita i te 200 K.
Rongoā:
Mā te whakamahi i te whārite mō te mahi i mahia i roto i te tukanga isothermal:
\[
W = 1 \times 8.314 \times 200 \times \ln\frac{3}{6} \approx -575\, \text{J}
\]
Te raru 5:
Tātaihia te huringa o te pūngao ā-roto mō te 3 mole o te hau i te wā o te whakawhānui isothermal mai i te 2 rita ki te 6 rita i te pāmahana pumau.
Rongoā:
Mō tētahi tukanga isothermal o tētahi hau pai, ko te huringa o te pūngao ā-roto he kore:
\[
\Delta U = 0\, \text{J}
\]
Te raru 6:
Tātaihia te whakawhiti wera ina whakawhanuitia ā-werawera ngā mole hau e 5 mai i te 3 rita ki te 6 rita i te 250 K.
Rongoā:
Mā te whakamahi i te whārite mō te whakawhiti wera i roto i tētahi tukanga isothermal:
\[
Q = 5 \times 8.314 \times 250 \times \ln\frac{6}{3} \approx 2879\, \text{J}
\]
Te raru 7:
He aha te mahi e mahia ana e ngā mole hau e 2 i te wā o te kōpeketanga isothermal mai i te 4 rita ki te 2 rita i te 150 K?
Rongoā:
Mā te whakamahi i te whārite mō te mahi i mahia i roto i te tukanga isothermal:
\[
W = 2 \times 8.314 \times 150 \times \ln\frac{2}{4} \approx -1151\, \text{J}
\]
Te raru 8:
Kimihia te huringa o te entropy ina pēhia ngā mole hau e 3 i roto i te āhua wera mai i te 5 rita ki te 1 rita i te 500 K.
Rongoā:
Mā te whakamahi i te whārite mō te huringa o te entropy i roto i tētahi tukanga isothermal:
\[
\Delta S = 3 \times 8.314 \times \ln\frac{1}{5} \approx -34.77\, \text{J/K}
\]
Te raru 9:
Tātaihia te mahi i mahia e ngā mole e 4 o te hau pai e whakawhanui ana i te werawera mai i te 2 rita ki te 8 rita i te 100 K.
Rongoā:
Mā te whakamahi i te whārite mō te mahi i mahia i roto i te tukanga isothermal:
\[
W = 4 \times 8.314 \times 100 \times \ln\frac{8}{2} \approx 2304\, \text{J}
\]
Ka hipokina e ēnei raruraru ngā ariā e pā ana ki te tukanga isothermal, pērā i te mahi i mahia, te whakawhiti wera, me te huringa o te entropy.
Anei ngā raruraru e iwa me ngā otinga e pā ana ki te Tukanga Isochoric (Rōrahi Pūmau):
Te raru 1:
Tātaihia te huringa o te pūngao ā-roto mō ngā mole e 3 o tētahi hau pai monoatomic ina whakanuia te pāmahana mai i te 200 K ki te 400 K i te rōrahi pumau.
Rongoā:
Mā te whakamahi i te whārite mō te huringa o te pūngao ā-roto i te rōrahi pumau: \( \Delta U = nC_v\Delta T \), ko \( C_v = \frac{3}{2}R \):
\[
\Delta U = 3 \times \frac{3}{2} \times 8.314 \times (400 – 200) \approx 3741\, \text{J}
\]
Te raru 2:
Tātaihia te whakawhiti wera mō ngā mole e 2 o te hau takirua ina whakaitihia te pāmahana mai i te 300 K ki te 200 K i te rōrahi pumau.
Rongoā:
Mā te whakamahi i te whārite mō te whakawhiti wera i te rōrahi pumau: \( Q = \Delta U = nC_v\Delta T \), ko \( C_v = \frac{5}{2}R \):
\[
Q = 2 \times \frac{5}{2} \times 8.314 \times (200 – 300) \approx -4157\, \text{J}
\]
Te raru 3:
Tātaihia te huringa o te entropy mō ngā mole e 4 o te hau triatomic ina piki te pāmahana mai i te 100 K ki te 300 K i te rōrahi pumau.
Rongoā:
Mā te whakamahi i te whārite mō te huringa o te entropy i te rōrahi pumau, ko \( C_v = \frac{f}{2}R \) me \( f = 6 \):
\[
\Delta S = nC_v\ln\frac{T_2}{T_1} = 4 \times \frac{6}{2} \times 8.314 \times \ln\frac{300}{100} \approx 115.36\, \text{J/K}
\]
Te raru 4:
He aha te mahi e mahia ana e te hau i roto i te tukanga isochoric?
Rongoā:
I te mea he pumau te rōrahi i roto i te tukanga isochoric, kāore he mahi e mahia ana:
\[
W = 0\, \text{J}
\]
Te raru 5:
Tātaihia te whakawhiti wera mō te 5 mole o tētahi hau pai monoatomic ina whakamataohia mai i te 500 K ki te 300 K i te rōrahi pumau.
Rongoā:
Mā te whakamahi i te whārite mō te whakawhiti wera i te rōrahi pumau, ko \( C_v = \frac{3}{2}R \):
\[
Q = 5 \times \frac{3}{2} \times 8.314 \times (300 – 500) \approx -6232\, \text{J}
\]
Te raru 6:
Tātaihia te huringa o te pūngao ā-roto mō te 1 mole o te hau rua-atomic ina whakanuia te pāmahana mai i te 150 K ki te 250 K i te rōrahi pumau.
Rongoā:
Mā te whakamahi i te whārite mō te huringa o te pūngao ā-roto i te rōrahi pumau, ko \( C_v = \frac{5}{2}R \):
\[
\Delta U = 1 \times \frac{5}{2} \times 8.314 \times (250 – 150) \approx 2079\, \text{J}
\]
Te raru 7:
Tātaihia te huringa o te entropy mō ngā mole e 3 o te hau monoatomic ina heke te pāmahana mai i te 600 K ki te 300 K i te rōrahi pumau.
Rongoā:
Mā te whakamahi i te whārite mō te huringa o te entropy i te rōrahi pumau, ko \( C_v = \frac{3}{2}R \):
\[
\Delta S = 3 \times \frac{3}{2} \times 8.314 \times \ln\frac{300}{600} \approx -34.59\, \text{J/K}
\]
Te raru 8:
He aha te whakawhiti wera mō ngā mole e 2 o te hau triatomic ina whakanuia te pāmahana mai i te 200 K ki te 400 K i te rōrahi pumau?
Rongoā:
Mā te whakamahi i te whārite mō te whakawhiti wera i te rōrahi pumau, ko \( C_v = \frac{6}{2}R \):
\[
Q = 2 \times \frac{6}{2} \times 8.314 \times (400 – 200) \approx 4986\, \text{J}
\]
Te raru 9:
Tātaihia te huringa o te pūngao ā-roto mō ngā mole e 4 o tētahi hau pai monoatomic ina whakamahanatia mai i te 250 K ki te 350 K i te rōrahi pumau.
Rongoā:
Mā te whakamahi i te whārite mō te huringa o te pūngao ā-roto i te rōrahi pumau, ko \( C_v = \frac{3}{2}R \):
\[
\Delta U = 4 \times \frac{3}{2} \times 8.314 \times (350 – 250) \approx 4986\, \text{J}
\]
Ka kapi i ēnei raruraru ngā āhuatanga maha o te tukanga isochoric, pērā i te whakawhiti wera, ngā huringa o te pūngao ā-roto, me te entropy, mō ngā momo hau rerekē.
