1
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我們來多做點題吧

2
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關於理想氣體物態方程的題目

3
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假設我有一罐氣體

4
00:00:06,680 --> 00:00:15,323
當前壓力是3個大氣壓

5
00:00:15,323 --> 00:00:19,756
假設罐子的體積是

6
00:00:19,756 --> 00:00:27,413
比如 9升

7
00:00:27,413 --> 00:00:30,136
現在 氣壓會如何改變

8
00:00:30,136 --> 00:00:39,280
如果體積從9升變爲3升

9
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從第一個理想氣體物態方程的影片中

10
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你可以大概感覺到

11
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如果你有一堆——保持——

12
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這很重要

13
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我們保持溫度不變

14
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這很重要 必須明白

15
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所以在我們的直覺

16
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在理想氣體物態方程背後 我們說

17
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看 如果我們有一定量的粒子

18
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以及一定的動能

19
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他們施加一定的壓力

20
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在容器壁上

21
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如果我們要讓容器變小

22
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我們有同樣數量的粒子

23
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n不變

24
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平均動能不變

25
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所以它們會更頻繁地撞上容器壁

26
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所以當我們使容器縮小時

27
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當體積變大

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當體積變小時

29
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氣壓應該升高

30
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所以我們看看能不能算出準確數字

31
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我們可以用理想氣體方程了

32
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壓力乘體積等於nRT

33
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現在 粒子數改變了嗎

34
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在我改變情況縮小了體積時？

35
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不！

36
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我們有一樣的粒子數

37
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我只是把容器縮小了

38
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所以n還是n R不變 那是個常數

39
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溫度不變

40
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所以我原先的壓力乘體積

41
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等於nRT

42
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而我的新壓力乘體積——

43
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讓我叫這個P1和V1

44
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然後P2是這個

45
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不好意思 那是V2

46
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所以V2是這個 我們要算出P2

47
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P2是什麽？

48
00:02:23,134 --> 00:02:31,354
我們知道P1乘V1等於nRT

49
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我們還知道既然氣體的溫度和

50
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莫耳數都不變

51
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P2乘V2等於nRT

52
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既然它們都等於同一個東西

53
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我們可以說壓力乘體積

54
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只要溫度不變

55
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就也不變

56
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所以P1乘V1也等於P2乘V2

57
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那P1是什麽？

58
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P1 我們原先的壓力 是3個大氣壓

59
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所以3大氣壓乘9升等於

60
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我們的新氣壓乘3升

61
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如果我們將等式兩邊同除以3

62
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3升就消掉了

63
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我們還有9大氣壓

64
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這有道理

65
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當你把體積縮小2/3

66
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或者當你使體積

67
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變爲原來的1/3

68
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你的壓力就會增加3倍

69
00:03:46,199 --> 00:03:51,571
所以這個乘3 這個乘1/3

70
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一般這是很有用的知識

71
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如果溫度不變

72
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那麽壓力乘體積

73
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也是一個常數

74
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現在 你可以更進一步

75
00:04:00,958 --> 00:04:06,878
如果我們看看PV＝nRT

76
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我們知道兩個量不變

77
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在我們做的大部分練習中

78
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就是分子數

79
00:04:13,535 --> 00:04:15,529
以及顯然R不會變

80
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所以如果我們將兩邊同除以T

81
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我們就有了PV/T＝nR

82
00:04:23,165 --> 00:04:24,918
或者可以說它等於一個常數

83
00:04:24,918 --> 00:04:27,203
它對於任何係統都是一個常數

84
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如果我們不改變

85
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容器中的分子數的話

86
00:04:31,524 --> 00:04:33,373
所以 如果我們改變壓力——

87
00:04:33,373 --> 00:04:35,653
如果一開始有

88
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壓力1 體積1 以及某個溫度1

89
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那就等於這個常數

90
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如果我們改變其中一個

91
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我們回到

92
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壓力2 體積2 溫度2

93
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他們應該還等於這個常數

94
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所以他們相等

95
00:04:51,467 --> 00:04:55,350
舉例來說 假設我開始有

96
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1個大氣壓的壓力

97
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而體積是——

98
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我換一下單位來試一種不同的做法

99
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——2立方米

100
00:05:10,639 --> 00:05:20,209
假設溫度是27攝氏度

101
00:05:20,209 --> 00:05:21,742
我剛寫了攝氏度

102
00:05:21,742 --> 00:05:22,697
以爲我想讓你永遠記住

103
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你要轉換成開氏度

104
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所以27攝氏度加上273是

105
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正好300開氏度

106
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我們的新溫度是

107
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實際上我們來算一下新溫度

108
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會是什麽

109
00:05:41,418 --> 00:05:46,270
假設新壓力是2大氣壓

110
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壓力增加了

111
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我們讓容器縮小

112
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比如1立方米

113
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所以容器縮小了一半

114
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而壓力增加了一半

115
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所以你可以猜一下

116
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你知道我們讓壓力升高了

117
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我來讓容器更小一點

118
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實際上 算了

119
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我來讓壓力更大

120
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讓壓力變成5個大氣壓

121
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現在我們想知道第二個溫度是什麽

122
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我們建立起等式

123
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所以就得到

124
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2/300大氣壓乘立方米每開氏度

125
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等於2/T2 我們的新溫度

126
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然後我們有1500等於2T2

127
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兩邊同除以2

128
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得到了T2等於750開氏度

129
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這有道理 對吧

130
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我們讓壓力升高了這麽多

131
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而同時也減小了體積

132
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所以溫度必須擧升

133
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或者如果這麽想

134
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或許我們升高了溫度

135
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因此也使壓力

136
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變得高多了

137
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尤其是當我們減小了體積時

138
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我想最好的理解方法是

139
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壓力升高了這麽多

140
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它增加了五倍

141
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他從1大氣壓變爲5大氣壓

142
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因爲一面

143
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我們將體積縮小了1/2

144
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所以壓力應該加倍了

145
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我們就到了2大氣壓

146
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然後我們讓溫度高多了

147
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所以我們也在撞上容器壁

148
00:07:25,407 --> 00:07:27,901
我們使溫度變爲750開氏度

149
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所以溫度升高了兩倍還多

150
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於是我們得到了5大氣壓

151
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現在 另一件你可能會聽到的概念

152
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是發生什麽

153
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在標準溫度和氣壓下

154
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我把這些都刪了

155
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標準溫度和氣壓

156
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我把這些不需要的都刪掉

157
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標準溫度和氣壓

158
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我提到這個

159
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是因爲雖然它被稱作

160
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標準溫度和氣壓

161
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有時也稱作STP

162
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世界的一大不幸是

163
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人們還沒有完全標準化

164
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標準氣壓和溫度是什麽

165
00:08:13,742 --> 00:08:15,916
我在維基百科上查了一下

166
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你可能會看到的

167
00:08:16,882 --> 00:08:19,986
在大部分物理課以及標準化考試上的

168
00:08:19,986 --> 00:08:23,935
標準溫度是0攝氏度

169
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也就是 當然 273開氏度

170
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標準氣壓是1個大氣壓

171
00:08:30,302 --> 00:08:31,241
在維基百科上

172
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他們寫成101.325千帕斯卡

173
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或者101,000帕斯卡多一點

174
00:08:41,341 --> 00:08:44,246
當然 一帕斯卡是一牛頓每平方米

175
00:08:44,246 --> 00:08:45,968
在所有這些中 單位都是

176
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最難掌握的部分

177
00:08:47,665 --> 00:08:49,741
但假如我們設

178
00:08:49,741 --> 00:08:50,676
這些不同的

179
00:08:50,676 --> 00:08:52,195
標準溫度與壓力

180
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來自於不同的標準設定組織

181
00:08:54,878 --> 00:08:55,783
他們不能完全達成一致

182
00:08:55,783 --> 00:08:57,259
但如果我們把這個當成

183
00:08:57,259 --> 00:09:00,889
標準溫度和壓力的定義

184
00:09:00,889 --> 00:09:04,604
所以我們設溫度

185
00:09:04,604 --> 00:09:07,227
等於0攝氏度

186
00:09:07,227 --> 00:09:11,203
也就等於273開氏度

187
00:09:11,203 --> 00:09:15,255
壓力我們設爲1大氣壓

188
00:09:15,255 --> 00:09:16,075
也可以寫成

189
00:09:16,075 --> 00:09:22,440
101.325或3/8千帕

190
00:09:22,440 --> 00:09:26,349
我的問題是如果我有一種理想氣體

191
00:09:26,349 --> 00:09:30,021
在標準溫度與壓力的情況下

192
00:09:30,021 --> 00:09:36,453
1升中有多少莫耳？

193
00:09:36,453 --> 00:09:37,583
不 我換一種說法

194
00:09:37,583 --> 00:09:40,868
1莫耳會占多少升？

195
00:09:40,868 --> 00:09:43,785
我再多說一點

196
00:09:43,785 --> 00:09:46,384
n等於1莫耳

197
00:09:46,384 --> 00:09:48,940
我想算出體積是什麽

198
00:09:48,940 --> 00:09:50,657
如果我有1莫耳氣體

199
00:09:50,657 --> 00:09:55,556
我有6.02乘10的23次方個這種氣體氣體的分子

200
00:09:55,556 --> 00:09:58,456
在標準壓力1大氣壓

201
00:09:58,456 --> 00:10:01,002
以及標準溫度273度

202
00:10:01,002 --> 00:10:03,455
氣體的體積是多少？

203
00:10:03,455 --> 00:10:07,745
我們應用PV＝nRT

204
00:10:07,745 --> 00:10:10,096
壓力是1大氣壓

205
00:10:10,096 --> 00:10:11,748
但記住我們用的是大氣壓

206
00:10:11,748 --> 00:10:15,362
1大氣壓乘體積

207
00:10:15,362 --> 00:10:16,656
這是我們要解的

208
00:10:16,656 --> 00:10:18,043
我用紫色做

209
00:10:18,043 --> 00:10:22,007
等於1莫耳 我們有1莫耳這種氣體

210
00:10:22,007 --> 00:10:29,312
乘R 乘溫度 乘273

211
00:10:29,312 --> 00:10:31,786
現在這是開氏度 這是莫耳

212
00:10:31,786 --> 00:10:39,508
我們想讓體積是升

213
00:10:39,508 --> 00:10:41,562
所以用哪一個版本的R呢？

214
00:10:41,562 --> 00:10:44,414
我們用的大氣壓

215
00:10:44,414 --> 00:10:46,609
我們想要升的體積

216
00:10:46,609 --> 00:10:48,029
當然 我們有開氏度的莫耳

217
00:10:48,029 --> 00:10:50,531
所以用這個版本 0.082

218
00:10:50,531 --> 00:10:52,210
那麽這是1

219
00:10:52,210 --> 00:10:54,866
我們可以忽略這個1

220
00:10:54,866 --> 00:10:56,388
因此體積等於

221
00:10:56,388 --> 00:11:02,204
0.082乘273開氏度

222
00:11:02,204 --> 00:11:19,229
0.082乘273等於22.4升

223
00:11:19,229 --> 00:11:21,429
所以如果我有任何一種理想氣體

224
00:11:21,429 --> 00:11:24,079
所有的氣體都不是完全理想化的理想

225
00:11:24,079 --> 00:11:25,475
但如果我有一種理想氣體

226
00:11:25,475 --> 00:11:26,930
它在標準溫度

227
00:11:26,930 --> 00:11:29,099
也就是0攝氏度

228
00:11:29,099 --> 00:11:30,423
或者水的熔點

229
00:11:30,423 --> 00:11:32,423
也就是273開氏度

230
00:11:32,423 --> 00:11:33,713
我有一莫耳這種氣體

231
00:11:33,713 --> 00:11:37,559
它在標準壓力 1大氣壓

232
00:11:37,559 --> 00:11:42,479
那個氣體應該占22.4升

233
00:11:42,479 --> 00:11:44,796
如果你想知道多少立方米

234
00:11:44,796 --> 00:11:46,385
它會占

235
00:11:46,385 --> 00:11:50,987
你只用把22.4升乘

236
00:11:50,987 --> 00:11:53,236
現在 這是多少立方米

237
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對於每一立方米都有1000升

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我知道這看起來很多 但這是真的

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只要想想一立方米有多大

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所以這就是0.0224立方米

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如果你有1大氣壓的某種東西 一莫耳

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在0攝氏度情況下

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反正這其實是

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很有用的數字

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人們會經常說你有2莫耳

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在標準溫度與氣壓

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它會占多少升？

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好的 1莫耳占這麽多

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2莫耳在標準溫度與氣壓

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就會占兩倍體積

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因爲你只是用PV＝nRT

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然後加倍

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其他都不變

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壓力 其他一切都是固定的

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所以如果你把莫耳數加倍

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就會把體積加倍

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或者如果你把莫耳數減半

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就會把它的體積減半

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所以一個很有用的知識點是在升的單位中

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以及標準溫度與氣壓的情況下

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當標準溫度與氣壓

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被定義爲1大氣壓以及273開氏度

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一種理想氣體會占22.4升的體積