题目内容
如图所示,粗细均匀一端封闭、一端开口的细玻璃管总长为100cm,中央折成直角,使它一半水平,一半竖直,外界大气压为
=76cmHg不变,管内有30cm长的水银柱将一部分空气封闭于封闭端,当管内空气为-17℃时,空气柱长40cm.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241201449061207.gif)
(1)当管内气体温度上升到30℃时,管内空气柱有多长?
(2)要将管内水银全部排空,管内气温至少应大于多少摄氏度?(假设温度上升是缓慢的)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120144813952.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241201449061207.gif)
(1)当管内气体温度上升到30℃时,管内空气柱有多长?
(2)要将管内水银全部排空,管内气温至少应大于多少摄氏度?(假设温度上升是缓慢的)
(1) 45cm (2)T≥516.4L可全部排空水银.
本题的最高温度是否恰好在水银全部排空的时刻,极有可能成为意外的盲点,因为在排空水银的过程中压强在减小,排空的速度在加快,气体对外做功的速度也在加快,当然会引起内能减少的加快,如果做功的速度大于吸热的速度,温度就会降低,也就是说,温度的最高点应出现在做功的速度等于吸热的速度的那一刻.
(1)设水银恰好全部进入竖直管内,有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120144969948.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145171936.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145171947.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145187944.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145218931.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145234941.gif)
据
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241201452491180.gif)
有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241201452651196.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145374940.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145390951.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145405939.gif)
据
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241201454371178.gif)
有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241201455301385.gif)
解得:x=5cm
所以:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145405939.gif)
(2)设管口还留有水银柱h cm时,温度为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145577945.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145593948.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145608938.gif)
据
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241201457021171.gif)
有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241201457951389.gif)
整理:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145811951.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145577945.gif)
方程有解有其判别式:Δ=242-4(15
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145577945.gif)
解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824120145577945.gif)
即当 T≥516.4L可全部排空水银.
温度上升时气体体积膨胀,水平管内的水银逐渐减少,竖直管内的水银逐渐增加,对30℃时水平管内有无水银需做出判断;要将管内水银全部排空,应计算管内温度的极大值,而这一极大值不一定是出现在气体恰好全部排空的时刻.
[思路一]设水银恰好全部进入竖直管内时,求此时的温度并与30℃时的温度相比,看水平管内此时是否留有水银,进一步求解管内气体温度上升到30℃时,管内空气柱有多长;排空水银的过程中,设温度的极大值出现在排空过程中的某一点,列方程即可分析求解温度的极大值.
[思路二]设当管内气体温度上升到30℃时,水平管内留有长x的水银柱,求出结果看是否合理,进而确定当管内气体温度上升到30℃时,管内空气柱有多长;排空水银的过程中,设温度的极大值出现在排空过程中的某一点,列方程即可分析求解温度的极大值.
[思路三]设被封气体体积保持不变,用p/T=Δp/ΔT可以计算出压强的增量值,也可确定当管内气体温度上升到30℃时,水平管内有无水银;排空水银的过程中,设温度的极大值出现在排空过程中的某一点,列方程即可分析求解温度的极大值.
![](http://thumb2018.1010pic.com/images/loading.gif)
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