- #1
johnconnor
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A vessel is divided into two parts of equal volume by a partition in which there is a very small hole. Initially, each part contains gas at 300K and a low pressure, p. One part of the vessel is now heated to 600K while the other is maintained at 300K. If a steady state is established when the rate at which molecules pass through the hole from each side is the same, find the resulting pressure difference between the two parts.
Attempt:
I'm assuming that the number and mass of molecules inside remain the same and that the temperature of the two parts during the steady state is the same.
So we have
[tex]N_1+N_2=2N[/tex], where N is the number of molecules inside each part before heating and N1 and N2 denote the number of molecules inside each part after heating.
Also pressure is proportional to <c>2, implying T is proportional to <c>2, and that p is proportional to T.
We also have [tex]N_1<c>_1=N_2<c>_2[/tex], where N_i<c>_i denotes the rate at which molecules pass through the hole from one side to another.
So now [itex]<c>^2 \propto T
\text{and }N_1<c>_1=N_2<c>_2
\Rightarrow \dfrac{N_1}{N_2}= \dfrac{<c>_2}{<c>_1}
\Rightarrow \dfrac{N_1^2}{N_2^2}= \dfrac{<c>_2^2}{<c>_1^2}
\Rightarrow \dfrac{N_1^2}{N_2^2}= \dfrac{T_2}{T_1}
\Rightarrow \dfrac{N_1}{N_2}= \(\dfrac{T_2}{T_1})^{1/2}[/itex]
And I'm stuck. I'm supposed to find the difference of pressure in terms of p but how do I do that when the terms which I have introduced are nowhere close to p? The closest one I could get are p1 and p2. Help?
Attempt:
I'm assuming that the number and mass of molecules inside remain the same and that the temperature of the two parts during the steady state is the same.
So we have
[tex]N_1+N_2=2N[/tex], where N is the number of molecules inside each part before heating and N1 and N2 denote the number of molecules inside each part after heating.
Also pressure is proportional to <c>2, implying T is proportional to <c>2, and that p is proportional to T.
We also have [tex]N_1<c>_1=N_2<c>_2[/tex], where N_i<c>_i denotes the rate at which molecules pass through the hole from one side to another.
So now [itex]<c>^2 \propto T
\text{and }N_1<c>_1=N_2<c>_2
\Rightarrow \dfrac{N_1}{N_2}= \dfrac{<c>_2}{<c>_1}
\Rightarrow \dfrac{N_1^2}{N_2^2}= \dfrac{<c>_2^2}{<c>_1^2}
\Rightarrow \dfrac{N_1^2}{N_2^2}= \dfrac{T_2}{T_1}
\Rightarrow \dfrac{N_1}{N_2}= \(\dfrac{T_2}{T_1})^{1/2}[/itex]
And I'm stuck. I'm supposed to find the difference of pressure in terms of p but how do I do that when the terms which I have introduced are nowhere close to p? The closest one I could get are p1 and p2. Help?