Молекулярная физика и термодинамика. Учебное пособие
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ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɦ ɩɨɥɢɬɪɨɩɢɱɟɫɤɨɦɭ ɩɪɨɰɟɫɫɭ ɉɪɢ ɚɞɢɚɛɚɬɢɱɟɫɤɨɦ ɩɪɨɰɟɫɫɟ
ɭɱɚɫɬɨɤ ± ɬɟɩɥɨɨɛɦɟɧ ɧɟ ɩɪɨɢɫɯɨɞɢɬ
ɉɨɥɭɱɟɧɧɨɟ ɬɟɩɥɨ Q ɜɵɱɢɫɥɹɟɬɫɹ ɨɱɟɧɶ ɩɪɨɫɬɨ ɩɨɫɦɨɬɪɢɬɟ ɪɟɲɟɧɢɟ
ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɢ
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ɚ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɬɟɩɥɚ Q ɨɬɞɚɧɧɨɝɨ ɝɚɡɨɦ ɩɨɫɬɭɩɢɦ ɧɢɠɟɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ ɉɨɫɤɨɥɶɤɭ ɩɪɨɰɟɫɫ ɩɨɥɢɬɪɨɩɢɱɟɫɤɢɣ ɬɨ
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ɝɞɟ Ȟ – ɤɨɥɢɱɟɫɬɜɨ ɦɨɥɟɣ ɝɚɡɚ с ± ɟɝɨ ɦɨɥɹɪɧɚɹ
ɬɟɩɥɨɟɦɤɨɫɬɶ ɜ ɞɚɧɧɨɦ ɩɪɨɰɟɫɫɟ Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɢɯ
Т
Т
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Т
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Ɋɢɫ
ɧɚɣɬɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ ɤɨɬɨɪɚɹ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɭɫɥɨɜɢɹɦ
ɞɚɧɧɨɣ ɡɚɞɚɱɢ ɡɚɩɢɲɟɬɫɹ ɬɚɤ
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ɇɚɦ ɨɫɬɚɥɨɫɶ ɧɚɣɬɢ ɄɉȾ ɰɢɤɥɚ
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6. РЕАЛЬНЫЕ ГАЗЫ
ɉɨ ɦɟɪɟ ɭɜɟɥɢɱɟɧɢɹ ɩɥɨɬɧɨɫɬɢ ɝɚɡ ɫɬɚɧɨɜɢɬɫɹ ɜɫɟ ɦɟɧɟɟ ɩɨɯɨɠɢɦ ɧɚ ɢɞɟɚɥɶɧɵɣ ȼɫɟ ɛɨɥɶɲɟɟ ɡɧɚɱɟɧɢɟ ɩɪɢɨɛɪɟɬɚɸɬ ɫɢɥɵ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɨɥɟɤɭɥ ɝɚɡɚ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɉɨɥɭɱɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɫɨɫɬɨɹɧɢɹ ɝɚɡɚ ɜ ɤɨɬɨɪɨɦ ɛɵɥɨɛɵ ɬɨɱɧɨ ɭɱɬɟɧɨ ɬɚɤɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɹɜɥɹɟɬɫɹ ɨɱɟɧɶ ɫɥɨɠɧɨɣ ɡɚɞɚɱɟɣ ɉɪɢɛɥɢɠɟɧɧɵɣ ɢ ɞɨɜɨɥɶɧɨ ɝɪɭɛɵɣ ɭɱɟɬ ɨɫɧɨɜɧɵɯ ɨɫɨɛɟɧɧɨɫɬɟɣ ɷɬɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɩɪɢɜɨɞɢɬ ɤ ɭɪɚɜɧɟɧɢɸ ȼɚɧ ɞɟɪ ȼɚɚɥɶɫɚ
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ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɦɵ ɡɚɩɢɫɚɥɢ ɞɥɹ 1 ɦɨɥɹ ɝɚɡɚ ɢ ɧɚɪɹɞɭ ɫ ɢɡɜɟɫɬɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ ɡɞɟɫɶ ɩɨɹɜɢɥɢɫɶ ɞɜɟ ɧɨɜɵɯ ɤɨɧɫɬɚɧɬɵ ±аɢbɤɨɬɨɪɵɟ ɧɚɡɵɜɚɸɬɫɹ постоянными Ван-дер-Ваальсаɉɟɪɜɚɹ ɢɡ ɧɢɯ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɢɥɵ ɩɪɢɬɹɠɟɧɢɹ ɦɟɠɞɭ ɦɨɥɟɤɭɥɚɦɢ ɚ ɜɬɨɪɚɹ ± ɫɢɥɵ ɨɬɬɚɥɤɢɜɚɧɢɹ Ɉɧɢ ɡɚɜɢɫɹɬ ɨɬ ɫɨɪɬɚ ɝɚɡɚ ɢ ɢɯɦɨɠɧɨɧɚɣɬɢɜɫɩɪɚɜɨɱɧɢɤɚɯɁɚɦɟɬɢɦɱɬɨɟɫɥɢ а ɢb ɩɪɢɧɹɬɶɪɚɜɧɵɦɢ ɧɭɥɸ ɦɵ ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɫɨɫɬɨɹɧɢɹ ɢɞɟɚɥɶɧɨɝɨ ɝɚɡɚ ± ɭɪɚɜɧɟɧɢɟ Ɇɟɧɞɟɥɟɟɜɚ ± Ʉɥɚɩɟɣɪɨɧɚȼɧɭɬɪɟɧɧɹɹ ɷɧɟɪɝɢɹ ɪɟɚɥɶɧɨɝɨ ɝɚɡɚ ɩɨɞɱɢɧɹɸɳɟɝɨɫɹ ɭɪɚɜɧɟɧɢɸ ȼɚɧ
ɞɟɪ ȼɚɚɥɶɫɚ ɡɚɜɢɫɢɬ ɧɟ ɬɨɥɶɤɨ ɨɬ ɟɝɨ ɬɟɦɩɟɪɚɬɭɪɵ ɧɨ ɢ ɨɬ ɨɛɴɟɦɚ ɡɚɧɢɦɚɟɦɨɝɨ ɝɚɡɨɦ Ⱦɥɹνɦɨɥɟɣ ɝɚɡɚ
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Задача 6.1. Какому давлению необходимо подвергнуть углекислый газ при температуре Т = 300 К, чтобы его плотность оказалась равной ȡ = 500 г/л? Расчет провести как для идеального газа, так и для газа Ван-дер-Ваальса.
ɋɱɢɬɚɹ ɝɚɡ ɢɞɟɚɥɶɧɵɦ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɭɪɚɜɧɟɧɢɟɦ Ɇɟɧɞɟɥɟɟɜɚ ±
Ʉɥɚɩɟɣɪɨɧɚ pV |
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ɉɪɢ ɪɚɫɱɟɬɚɯ ɦɵ ɭɱɥɢ ɱɬɨ ɦɨɥɹɪɧɚɹ ɦɚɫɫɚ ɭɝɥɟɤɢɫɥɨɝɨ ɝɚɡɚМ = ɝ ɦɨɥɶ
ɉɪɨɞɟɥɚɟɦ ɬɟ ɠɟ ɜɵɱɢɫɥɟɧɢɹ ɞɥɹ ɪɟɚɥɶɧɨɝɨ ɝɚɡɚ ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ
ɭɪɚɜɧɟɧɢɟɦ ɤɨɬɨɪɨɟ ɡɚɩɢɲɟɦ ɞɥɹνɦɨɥɟɣ ɝɚɡɚ
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Ɋɚɡɥɢɱɢɟ ɜɟɫɶɦɚ ɡɧɚɱɢɬɟɥɶɧɨɟ ± ɜ ɬɪɢ ɫ ɩɨɥɨɜɢɧɨɣ ɪɚɡɚ
Задача 6.2. Найти работу, совершаемую одним молем вандерваальсова газа при его изотермическом расширении от объема V1 до объема V2 при температуре Т.
Ɋɚɛɨɬɚ ɝɚɡɚ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɢɡɜɟɫɬɧɨɣ ɮɨɪɦɭɥɟ
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ɜ ɤɨɬɨɪɨɣ ɞɚɜɥɟɧɢɟ ɨɩɪɟɞɟɥɢɦ ɢɡ ɭɪɚɜɧɟɧɢɹ |
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Ɍɨɝɞɚ |
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Задача 6.3. Какое количество тепла надо сообщить Ȟ= 3 молям
углекислого газа, чтобы при расширении в вакуум от объема V1 = 5 л до V2 = 10 л его температура не изменилась? Газ считать вандерваальсовым.
Ⱦɥɹ ɧɚɱɚɥɚ ɪɚɫɫɦɨɬɪɢɦ ɱɬɨ ɩɪɨɢɡɨɣɞɟɬ ɟɫɥɢ ɝɚɡ ɪɚɫɲɢɪɹɟɬɫɹ ɜ ɩɭɫɬɨɬɭ
ɛɟɡɬɟɩɥɨɨɛɦɟɧɚɫɨɤɪɭɠɚɸɳɢɦɢɬɟɥɚɦɢɊɚɛɨɬɵɨɧɧɟɫɨɜɟɪɲɚɟɬ ± ɬɨɟɫɬɶ
А Ɍɟɩɥɨɨɛɦɟɧɚ ɬɨɠɟ ɧɟɬ ± ɡɧɚɱɢɬ ɢ Q |
Ɍɨɝɞɚ ɢɡ ɩɟɪɜɨɝɨ ɧɚɱɚɥɚ |
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ɬɟɪɦɨɞɢɧɚɦɢɤɢ ɫɥɟɞɭɟɬ ɱɬɨ ɢ ǻU |
ɂ ɟɫɥɢ ɦɵ ɩɨɫɦɨɬɪɢɦ ɧɚ ɮɨɪɦɭɥɭ |
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ɬɨ ɭɜɢɞɢɦ ɱɬɨ |
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ɬɚɤ ɤɚɤV > V .Ⱥ ɷɬɨ ɡɧɚɱɢɬ ɱɬɨ ɩɪɢ ɪɚɫɲɢɪɟɧɢɢ ɪɟɚɥɶɧɨɝɨ ɝɚɡɚ ɜ ɩɭɫɬɨɬɭ ɟɝɨ
ɬɟɦɩɟɪɚɬɭɪɚ ɭɦɟɧɶɲɚɟɬɫɹ ± ɜ ɨɬɥɢɱɢɟ ɨɬ ɝɚɡɚ ɢɞɟɚɥɶɧɨɝɨ Ⱦɥɹ ɩɨɞɞɟɪɠɚɧɢɹ ɟɝɨ
ɬɟɦɩɟɪɚɬɭɪɵ ɧɟɢɡɦɟɧɧɨɣ ɝɚɡɭ ɧɟɨɛɯɨɞɢɦɨ ɫɨɨɛɳɢɬɶ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ
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Задача 6.4. Найти для вандерваальсова газа уравнение адиабаты в переменных T, V, если его теплоемкость при постоянном объеме равна cV .
Ⱥɞɢɚɛɚɬɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ ± ɷɬɨ ɩɪɨɰɟɫɫ ɩɪɨɬɟɤɚɸɳɢɣ ɛɟɡ ɬɟɩɥɨɨɛɦɟɧɚ ɫ
ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ ɩɨɷɬɨɦɭ ɩɟɪɜɨɟ ɧɚɱɚɥɨ ɬɟɪɦɨɞɢɧɚɦɢɤɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ
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ɇɚɦ ɨɫɬɚɥɨɫɶ ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɬɶ ɷɬɨ ɜɵɪɚɠɟɧɢɟ
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ɢ ɩɨɥɭɱɢɬɶ ɨɬɜɟɬ
R
T V b cV FRQVW
Задача 6.5. Один моль вандерваальсова газа, имеющий объем V1 и температуру Т1 , переведен в состояние с объемом V2 и температурой Т2 . Найти изменение энтропии газа, если известны его молярная теплоемкость сV и постоянная b.
ɉɪɢɪɚɳɟɧɢɟ ɷɧɬɪɨɩɢɢ ɜ ɯɨɞɟ ɩɪɨɰɟɫɫɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
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7. ЯВЛЕНИЯ ПЕРЕНОСА
ɉɪɢ ɧɚɪɭɲɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ ɫɢɫɬɟɦɚ ɫɬɪɟɦɢɬɫɹ ɜɟɪɧɭɬɶɫɹ ɜ ɪɚɜɧɨɜɟɫɧɨɟ ɫɨɫɬɨɹɧɢɟ ɗɬɨɬ ɩɪɨɰɟɫɫ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɜɨɡɪɚɫɬɚɧɢɟɦ ɷɧɬɪɨɩɢɢ ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɧɟɨɛɪɚɬɢɦ ɇɚɪɭɲɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɜɨɡɧɢɤɧɨɜɟɧɢɟɦ ɩɨɬɨɤɨɜ ɥɢɛɨ ɦɨɥɟɤɭɥ ɥɢɛɨ ɬɟɩɥɚ ɥɢɛɨ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɡɚɪɹɞɚ ɢ ɬ ɩ ȼ ɫɜɹɡɢ ɫ ɷɬɢɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɩɪɨɰɟɫɫɵ ɧɨɫɹɬ ɧɚɡɜɚɧɢɟ явлений переноса ɂɡɫɤɚɡɚɧɧɨɝɨɜɵɲɟɜɵɬɟɤɚɟɬɱɬɨɹɜɥɟɧɢɹɩɟɪɟɧɨɫɚɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɧɟɨɛɪɚɬɢɦɵɟ ɩɪɨɰɟɫɫɵ Ɇɵ ɪɚɫɫɦɨɬɪɢɦ ɬɪɢ ɹɜɥɟɧɢɹ ɩɟɪɟɧɨɫɚ ɞɢɮɮɭɡɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɢ ɜɧɭɬɪɟɧɧɟɟ ɬɪɟɧɢɟ ɢɥɢ ɜɹɡɤɨɫɬɶ ɩɪɢɱɟɦ ɨɝɪɚɧɢɱɢɦɫɹ ɫɥɭɱɚɟɦ ɤɨɝɞɚ ɨɬɤɥɨɧɟɧɢɹ ɨɬ ɪɚɜɧɨɜɟɫɢɹ ɧɟɜɟɥɢɤɢ
Эффективный диаметр молекулы.Ɇɨɥɟɤɭɥɵ ɝɚɡɚ ɧɚɯɨɞɹɫɶ ɜ ɬɟɩɥɨɜɨɦ ɞɜɢɠɟɧɢɢɧɟɩɪɟɪɵɜɧɨ ɫɬɚɥɤɢɜɚɸɬɫɹɞɪɭɝ ɫ ɞɪɭɝɨɦɌɟɪɦɢɧ ©ɫɬɨɥɤɧɨɜɟɧɢɟª ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɦɨɥɟɤɭɥɚɦ ɧɟ ɫɥɟɞɭɟɬ ɩɨɧɢɦɚɬɶ ɛɭɤɜɚɥɶɧɨ ɢ ɩɪɟɞɫɬɚɜɥɹɬɶ ɫɟɛɟ ɷɬɨɬ ɩɪɨɰɟɫɫ ɩɨɞɨɛɧɵɦ ɫɨɭɞɚɪɟɧɢɸ ɬɜɟɪɞɵɯ ɲɚɪɨɜ ɉɨɞ ɫɬɨɥɤɧɨɜɟɧɢɟɦ ɦɨɥɟɤɭɥɩɨɧɢɦɚɸɬɩɪɨɰɟɫɫɜɡɚɢɦɨɞɟɣɫɬɜɢɹɦɟɠɞɭɦɨɥɟɤɭɥɚɦɢɜɪɟɡɭɥɶɬɚɬɟ ɤɨɬɨɪɨɝɨ ɦɨɥɟɤɭɥɵ ɢɡɦɟɧɹɸɬ ɧɚɩɪɚɜɥɟɧɢɟ ɫɜɨɟɝɨ ɞɜɢɠɟɧɢɹ
Ɇɢɧɢɦɚɥɶɧɨɟ ɪɚɫɫɬɨɹɧɢɟ ɧɚ ɤɨɬɨɪɨɟ ɫɛɥɢɠɚɸɬɫɹ ɩɪɢ ɫɬɨɥɤɧɨɜɟɧɢɢ ɰɟɧɬɪɵ ɞɜɭɯ ɦɨɥɟɤɭɥ ɧɚɡɵɜɚɟɬɫɹ эффективным диаметром ɦɨɥɟɤɭɥɵ d ɪɢɫ ȼɟɥɢɱɢɧɚ
ς Σd 2 |
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ɧɚɡɵɜɚɟɬɫɹэффективным сечениемɦɨɥɟɤɭɥɵ |
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ɗɮɮɟɤɬɢɜɧɵɣ ɞɢɚɦɟɬɪ ɦɨɥɟɤɭɥ ɡɚɜɢɫɢɬ ɨɬ ɷɧɟɪɝɢɢ ɚ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɢ ɨɬ |
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ɬɟɦɩɟɪɚɬɭɪɵ ɋ ɩɨɜɵɲɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɷɮɮɟɤɬɢɜɧɵɣ ɞɢɚɦɟɬɪ ɦɨɥɟɤɭɥ
ɭɦɟɧɶɲɚɟɬɫɹ
Средняя длина свободного пробега Ɂɚ ɫɟɤɭɧɞɭ ɦɨɥɟɤɭɥɚ ɩɪɨɯɨɞɢɬ ɜ ɫɪɟɞɧɟɦ ɩɭɬɶ ɪɚɜɧɵɣ ɫɪɟɞɧɟɣ ɫɤɨɪɨɫɬɢ¢X² ɉɨɞ ɫɪɟɞɧɟɣ ɞɥɢɧɨɣ ɫɜɨɛɨɞɧɨɝɨ
ɩɪɨɛɟɝɚ ɩɨɧɢɦɚɸɬ ɜɟɥɢɱɢɧɭ ɪɚɜɧɭɸ ɩɭɬɢ ɩɪɨɣɞɟɧɧɨɦɭ ɦɨɥɟɤɭɥɨɣ ɡɚ ɜɪɟɦɹt ɞɟɥɟɧɧɨɦɭ ɧɚ ɱɢɫɥɨ ɫɬɨɥɤɧɨɜɟɧɢɣ Z ɤɨɬɨɪɨɟ ɢɫɩɵɬɚɥɚ ɦɨɥɟɤɭɥɚ ɡɚ ɷɬɨ ɜɪɟɦɹ t
¢O² l Z ¢X² t Z
ɝɞɟ ɫɪɟɞɧɟɚɪɢɮɦɟɬɢɱɟɫɤɭɸ ɫɤɨɪɨɫɬɶ ɝɚɡɨɜɵɯ ɦɨɥɟɤɭɥ ɦɨɠɧɨ ɪɚɫɫɱɢɬɚɬɶ ɩɨ ɮɨɪɦɭɥɟ
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ȼɵɜɟɞɟɦ ɮɨɪɦɭɥɭ ɞɥɹ ɫɪɟɞɧɟɣ ɞɥɢɧɵ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ Ȼɭɞɟɦ ɫɱɢɬɚɬɶ ɱɬɨ ɜɫɟ ɦɨɥɟɤɭɥɵɧɟɩɨɞɜɢɠɧɵ ɤɪɨɦɟ ɨɞɧɨɣ ɦɨɥɟɤɭɥɵɁɚ ɜɪɟɦɹ t ɦɨɥɟɤɭɥɚ ɩɪɨɣɞɟɬ ɪɚɫɫɬɨɹɧɢɟ l ¢X² t ɝɞɟ ¢X² ± ɫɪɟɞɧɹɹ ɚɪɢɮɦɟɬɢɱɟɫɤɚɹ ɫɤɨɪɨɫɬɶ ɦɨɥɟɤɭɥɵ ɉɪɢɱɟɦ ɨɧɚ ɡɚ ɷɬɨ ɜɪɟɦɹ t ɢɫɩɵɬɚɟɬ ɫɬɨɥɤɧɨɜɟɧɢɹ ɫɨ ɜɫɟɦɢ ɦɨɥɟɤɭɥɚɦɢ ɰɟɧɬɪɵ ɤɨɬɨɪɵɯ ɩɨɩɚɞɚɸɬ ɜɧɭɬɪɶ ɤɨɥɟɧɱɚɬɨɝɨ ɰɢɥɢɧɞɪɚ ɟɝɨ ɪɚɞɢɭɫ ɪɚɜɟɧ ɞɢɚɦɟɬɪɭ ɨɞɧɨɣ ɦɨɥɟɤɭɥɵ ɚ ɞɥɢɧɚ ɨɛɪɚɡɭɸɳɟɣ ɰɢɥɢɧɞɪɚ ɪɚɜɧɚ ɩɪɨɣɞɟɧɧɨɦɭ ɦɨɥɟɤɭɥɨɣ ɪɚɫɫɬɨɹɧɢɸl ¢X² t ɪɢɫ
d
Ɋɢɫ
ɑɢɫɥɨ ɫɬɨɥɤɧɨɜɟɧɢɣ ɞɜɢɠɭɳɟɣɫɹ ɦɨɥɟɤɭɥɵ ɛɭɞɟɬ ɪɚɜɧɨ ɤɨɥɢɱɟɫɬɜɭ
ɦɨɥɟɤɭɥ ɰɟɧɬɪɵ ɤɨɬɨɪɵɯ ɩɨɩɚɥɢ ɜ ɷɬɨɬ ɰɢɥɢɧɞɪ
Z nV nSl Sd n¢X² t
ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɡɚɩɢɫɚɬɶ
¢O² |
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ɝɞɟ ɜ ɮɨɪɦɭɥɭ ɜɜɟɞɟɧ ɤɨɷɮɮɢɰɢɟɧɬ 
ɭɱɢɬɵɜɚɸɳɢɣ ɞɜɢɠɟɧɢɹ ɜɫɟɯ ɦɨɥɟɤɭɥ ± ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɧɭɠɧɨ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɱɢɫɥɚ ɫɬɨɥɤɧɨɜɟɧɢɣ Z ɢɫɩɨɥɶɡɨɜɚɬɶ ɨɬɧɨɫɢɬɟɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ɦɨɥɟɤɭɥ ɚ ɧɟ ɫɤɨɪɨɫɬɶ ɦɨɥɟɤɭɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɬɟɧɤɢ ɫɨɫɭɞɚ Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɞɜɭɯ ɩɪɨɢɡɜɨɥɶɧɨ ɜɡɹɬɵɯ
ɦɨɥɟɤɭɥ ɤɚɤ ɩɨɤɚɡɵɜɚɟɬ ɪɚɫɱɟɬ |
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ɨɬɧ |
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Ɂɚɦɟɧɢɜ ɫɨɝɥɚɫɧɨ ʌd ɱɟɪɟɡ ı ɮɨɪɦɭɥɟ ɦɨɠɧɨ ɩɪɢɞɚɬɶ ɜɢɞ |
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ɉɪɢ ɩɨɫɬɨɹɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ n ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ p ɋɥɟɞɨɜɚɬɟɥɶɧɨ
ɫɪɟɞɧɹɹ ɞɥɢɧɚ ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɞɚɜɥɟɧɢɸ
O ~ p
ȼɵɲɟ ɨɬɦɟɱɚɥɨɫɶ ɱɬɨ ɷɮɮɟɤɬɢɜɧɵɣ ɞɢɚɦɟɬɪ ɦɨɥɟɤɭɥɵ ɭɛɵɜɚɟɬ ɫ ɪɨɫɬɨɦ
ɬɟɦɩɟɪɚɬɭɪɵ ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɷɬɢɦ ɩɪɢ ɩɨɜɵɲɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɞɥɢɧɚ
ɫɜɨɛɨɞɧɨɝɨ ɩɪɨɛɟɝɚ ɭɜɟɥɢɱɢɜɚɟɬɫɹ
Диффузия. Ⱦɢɮɮɭɡɢɟɣ ɧɚɡɵɜɚɟɬɫɹ ɨɛɭɫɥɨɜɥɟɧɧɨɟɬɟɩɥɨɜɵɦɞɜɢɠɟɧɢɟɦ
ɦɨɥɟɤɭɥ ɫɚɦɨɩɪɨɢɡɜɨɥɶɧɨɟ ɜɵɪɚɜɧɢɜɚɧɢɟ ɤɨɧɰɟɧɬɪɚɰɢɣ ɜ ɫɦɟɫɢ ɧɟɫɤɨɥɶɤɢɯ ɜ
ɩɪɨɫɬɟɣɲɟɦ ɫɥɭɱɚɟ ɞɜɭɯ ɪɚɡɥɢɱɧɵɯ ɜɟɳɟɫɬɜ ɗɬɨɬ ɩɪɨɰɟɫɫ ɧɚɛɥɸɞɚɟɬɫɹ ɜ
ɬɜɟɪɞɵɯ ɠɢɞɤɢɯ ɢ ɝɚɡɨɨɛɪɚɡɧɵɯ ɫɪɟɞɚɯ Ɇɵ ɨɝɪɚɧɢɱɢɦɫɹ ɪɚɫɫɦɨɬɪɟɧɢɟɦ
ɬɨɥɶɤɨ ɝɚɡɨɨɛɪɚɡɧɵɯ ɫɪɟɞ Ɂɚɤɨɧ Ɏɢɤɚ ɞɥɹ ɞɢɮɮɭɡɢɢ |
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'm D |
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ɝɞɟD± ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ ddxU ± ɩɪɨɟɤɰɢɹ ɧɚ ɨɫɶxɝɪɚɞɢɟɧɬɚ ɩɥɨɬɧɨɫɬɢ
ɝɚɡɚ S ± ɩɥɨɳɚɞɶ ɩɥɨɳɚɞɤɢ ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɤ ɨɫɢ x
't± ɜɪɟɦɹ ɞɢɮɮɭɡɢɢ
Ʉɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ ɛɭɞɟɬ ɨɩɪɟɞɟɥɹɬɶɫɹ ɜɵɪɚɠɟɧɢɟɦ
D |
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ɉɨɞɫɬɚɜɢɜ ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ¢Ξ² ɢ¢O² ɩɨɥɭɱɢɦ ɱɬɨ
D a |
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nV |
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ɂɡ ɜɵɬɟɤɚɟɬ ɱɬɨ ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ ɱɢɫɥɭ ɦɨɥɟɤɭɥ ɜ ɟɞɢɧɢɰɟ ɨɛɴɟɦɚ ɚ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɢ ɞɚɜɥɟɧɢɸp
D a p
ɉɪɢ ɩɨɜɵɲɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ D ɪɚɫɬɟɬ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɤɚɤ 
T ɧɚɩɨɦɧɢɦ ɱɬɨ ı ɫɥɟɝɤɚ ɡɚɜɢɫɢɬ ɨɬT
Теплопроводность. Ɉɩɵɬ ɩɨɤɚɡɵɜɚɟɬ ɱɬɨ ɟɫɥɢ ɜ ɧɟɤɨɬɨɪɨɣ ɫɪɟɞɟ ɬɜɟɪɞɨɣ ɠɢɞɤɨɣ ɢɥɢ ɝɚɡɨɨɛɪɚɡɧɨɣ ɫɨɡɞɚɬɶ ɜɞɨɥɶ ɨɫɢzɝɪɚɞɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɵ ɬɨ ɜɨɡɧɢɤɚɟɬ ɩɨɬɨɤ ɬɟɩɥɚ Ɋɚɫɫɦɨɬɪɢɦ ɩɨɬɨɤɚ ɬɟɩɥɚ ɜ ɝɚɡɟ ɨɫɧɨɜɵɜɚɹɫɶ ɧɚ ɦɨɥɟɤɭɥɹɪɧɨ ɤɢɧɟɬɢɱɟɫɤɢɯ ɩɪɟɞɫɬɚɜɥɟɧɢɹɯ ȿɫɥɢ ɬɟɦɩɟɪɚɬɭɪɚ ɝɚɡɚ ɜ ɪɚɡɧɵɯ ɦɟɫɬɚɯ ɪɚɡɥɢɱɧɚ ɬɨ ɢ ɫɪɟɞɧɹɹ ɷɧɟɪɝɢɹ ɦɨɥɟɤɭɥ ɜ ɷɬɢɯ ɦɟɫɬɚɯ ɛɭɞɟɬ ɪɚɡɥɢɱɧɚ ɉɟɪɟɦɟɳɚɹɫɶ ɜɫɥɟɞɫɬɜɢɟ ɬɟɩɥɨɜɨɝɨ ɞɜɢɠɟɧɢɹ ɢɡ ɨɞɧɢɯ ɦɟɫɬ ɜ ɞɪɭɝɢɟ ɦɨɥɟɤɭɥɵ ɩɟɪɟɧɨɫɹɬ ɡɚɩɚɫɟɧɧɭɸ ɢɦɢ ɷɧɟɪɝɢɸ ɗɬɨɬ ɩɟɪɟɧɨɫ ɷɧɟɪɝɢɢ ɢ ɨɛɭɫɥɨɜɥɢɜɚɟɬ ɩɪɨɰɟɫɫ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɝɚɡɚɯ
ɉɟɪɟɧɨɫ ɷɧɟɪɝɢɢ ɦɨɥɟɤɭɥ ɩɪɨɢɫɯɨɞɢɬ ɜ ɩɪɨɰɟɫɫɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ Ɂɚɤɨɧ Ɏɭɪɶɟ ɞɥɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
Q F |
dT |
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ɝɞɟF ± ɤɨɷɮɮɢɰɢɟɧɬɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ |
dT |
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±ɩɪɨɟɤɰɢɹɧɚɨɫɶ x ɝɪɚɞɢɟɧɬɚ |
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dx |
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ɬɟɦɩɟɪɚɬɭɪɵ
Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɝɚɡɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɭɪɚɜɧɟɧɢɸ
F¢O²¢X²U cV
ȼ ɮɨɪɦɭɥɭ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ K ɜɯɨɞɢɬ ɭɞɟɥɶɧɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ cV ɢɞɟɚɥɶɧɨɝɨ ɝɚɡɚ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɨɛɴɟɦɟ ɤɨɬɨɪɭɸ ɦɨɠɧɨ
ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ |
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c |
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ɍɱɬɟɦ ɱɬɨ |
X ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ |
T |
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ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ |
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ɚ Uc |
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m |
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nV |
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ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚinɉɨɥɭɱɢɦ |
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F a |
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in |
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m |
nV |
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ɂɡ ɫɥɟɞɭɟɬ ɱɬɨ ɜ ɨɬɥɢɱɢɟ ɨɬ ɤɨɷɮɮɢɰɢɟɧɬɚ ɞɢɮɮɭɡɢɢ ɤɨɷɮɮɢɰɢɟɧɬɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢɝɚɡɚɧɟɡɚɜɢɫɢɬɨɬɱɢɫɥɚɦɨɥɟɤɭɥɜɟɞɢɧɢɰɟ ɨɛɴɟɦɚ ɚ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɢ ɨɬ ɞɚɜɥɟɧɢɹ p nkT ɉɪɢ ɩɨɜɵɲɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜɨɡɪɚɫɬɚɟɬ ɧɟɫɤɨɥɶɤɨ ɛɵɫɬɪɟɟ ɱɟɦ T ɗɬɨ ɨɛɭɫɥɨɜɥɟɧɨ ɬɟɦ ɱɬɨ ɷɮɮɟɤɬɢɜɧɨɟ ɫɟɱɟɧɢɟ ı ɫɥɟɝɤɚ ɡɚɜɢɫɢɬ ɨɬT
Явление внутреннего трения (явление вязкости).ȼ ɫɢɫɬɟɦɟ ɫɨɡɞɚɟɬɫɹ ɧɚɩɪɢɦɟɪ ɧɚɩɪɚɜɥɟɧɧɨɟ ɞɜɢɠɟɧɢɟ ɝɚɡɚ ɠɢɞɤɨɫɬɢ ɜɞɨɥɶ ɫɬɟɧɨɤ ɫɨɫɭɞɚ
Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɫɤɨɪɨɫɬɢ ɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɜɢɠɟɧɢɹ ɩɨ ɫɟɱɟɧɢɸ ɛɭɞɟɬ
ɧɟɪɚɜɧɨɦɟɪɧɵɦ ȿɫɥɢ ɪɚɡɛɢɬɶ ɨɛɴɟɦ ɝɚɡɚ ɠɢɞɤɨɫɬɢ ɧɚ ɨɬɞɟɥɶɧɵɟ ɫɥɨɢ ɬɨ ɢɯ
ɫɤɨɪɨɫɬɶ ɛɭɞɟɬ ɪɚɡɧɨɣ ± ɜɛɥɢɡɢ ɫɬɟɧɤɢ ɨɧɚ ɛɭɞɟɬ ɧɚɢɦɟɧɶɲɟɣ ɚ ɜɞɚɥɢ ɨɬ ɧɟɟ ± ɧɚɢɛɨɥɶɲɟɣɆɨɥɟɤɭɥɵɤɚɠɞɨɝɨɫɥɨɹɞɜɢɠɭɬɫɹ ɫɨɫɤɨɪɨɫɬɶɸ ɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɜɢɠɟɧɢɹ ɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɭɱɚɫɬɜɭɸɬ ɜ ɬɟɩɥɨɜɨɦ ɞɜɢɠɟɧɢɢ Ɂɚ ɫɱɟɬ ɬɟɩɥɨɜɨɝɨ ɞɜɢɠɟɧɢɹ ɦɨɥɟɤɭɥɵ ɛɭɞɭɬ ɩɟɪɟɯɨɞɢɬɶ ɢɡ ɨɞɧɨɝɨ ɫɥɨɹ ɜ ɞɪɭɝɨɣ ɩɪɢ ɷɬɨɦ ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɟɧɨɫ ɢɦɩɭɥɶɫɚ ɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɜɢɠɟɧɢɹ p 
ɜɫɥɟɞɫɬɜɢɟ ɷɬɨɝɨ ɧɚ ɝɪɚɧɢɰɟ ɫɥɨɟɜ ɜɨɡɧɢɤɚɸɬ ɫɢɥɵ ɬɪɟɧɢɹ ɤɨɬɨɪɵɟ ɬɨɪɦɨɡɹɬ
ɞɜɢɠɟɧɢɟ ɛɨɥɟɟ ɛɵɫɬɪɨɝɨ ɫɥɨɹ ɢ ɭɜɟɥɢɱɢɜɚɸɬ ɫɤɨɪɨɫɬɶ ɛɨɥɟɟ ɦɟɞɥɟɧɧɨɝɨ ɫɥɨɹ
ȼ ɹɜɥɟɧɢɢ ɜɹɡɤɨɫɬɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɧɟɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɹɟɬɫɹ
ɫɤɨɪɨɫɬɶ ɧɚɩɪɚɜɥɟɧɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫɥɨɟɜ ɝɚɡɚ ɜɫɥɟɞɫɬɜɢɟ ɱɟɝɨ ɩɪɨɢɫɯɨɞɢɬ