Solution to Set 2: Difference between revisions
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=Problem 1= | |||
=Part a= | ==Part a== | ||
<math>(P+{{aN^2}\over v^2})(v-Nb)=NkT</math> | <math>(P+{{aN^2}\over v^2})(v-Nb)=NkT</math> | ||
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<math>V^3-V^2(b+{kT\over P})+V{a\over P}-{ab\over P}=0</math> | <math>V^3-V^2(b+{kT\over P})+V{a\over P}-{ab\over P}=0</math> | ||
==part b== | |||
<math>P={nRT\over (v-nb)^2}-{an^2\over v^2}</math> | |||
Taking the derivative we get | |||
<math>{dP\over dv}={-nRT\over v-nb}+{2an^2\over v^3}=0</math> | |||
Multiply the derivative by <math>{-2\over v-nb}</math> to get | |||
<math>{2nRT\over (v-nb)^3}-{4an^2\over v^3(v-nb)}=0</math> | |||
Taking the derivative again we get | |||
<math>{d^2P\over d^2v}={2nRT\over (v-nb)^3}-{6an^2\over v^4}=0</math> | |||
By setting the derivatives equal to each other we get | |||
<math>{2nRT\over (v-nb)^3}-{4an^2\over v^3(v-nb)}={2nRT\over (v-nb)^3}-{6an^2\over v^4}</math> | |||
Which reduces to | |||
<math>6(v-nb)=4v</math> | |||
<math>6v-6nb=4v</math> | |||
<math>6nb=2v</math> | |||
<math>3nb=v_c</math> | |||
Now we can say | |||
<math>{-nRT\over v_c-nb}+{2an^2\over v_c^3}=0</math> | |||
Plugging <math>v_c</math> in we get | |||
<math>{-nRT\over (3nb-bn)^2}+{2an^2\over (3nb)^3}=0</math> | |||
Now we solve for T | |||
<math>{-nRT\over (2bn)^2}+{2an^2\over 27n^3b^3}=0</math> | |||
<math>{-RT\over 4b^2n}+{2a\over 27nb^3}=0</math> | |||
<math>{2a\over 27nb^3}={RT\over 4b^2n}</math> | |||
<math>{8a\over 27b}=RT</math> | |||
<math>{8a\over 27bR}=T_c</math> | |||
Now <math>T_c</math> and <math>v_c</math> can be plugged in to find <math>P_c</math> | |||
say | |||
<math>P={nR{(8a\over 27bR)}\over 3nb-nb}-{an^2\over (3nb)^2}</math> | |||
Revision as of 13:10, 30 April 2011
Problem 1
Part a
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (P+{{aN^2}\over v^2})(v-Nb)=NkT}
say Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V={v\over N}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Pv+{aN^2v\over v^2}-PNb-{aN^2Nb\over v^2}-NkT=0}
by multiplying both sides by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v^2} we get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {Pv^3}+aN^2V-PNbv^2-aN^3b-NkTv^2=0}
by dividing both sides by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle PN^2} we get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {v^3\over N^3}+{av\over PN}-{bv^2\over N^2}-{ab\over P}-{kTv^2\over PN^2}=0}
so
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^3+V{a\over P}-V^2b-{ab\over P}-V^2{kT\over P}=0}
and combining terms we get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^3-V^2(b+{kT\over P})+V{a\over P}-{ab\over P}=0}
part b
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P={nRT\over (v-nb)^2}-{an^2\over v^2}}
Taking the derivative we get
Multiply the derivative by to get
Taking the derivative again we get
By setting the derivatives equal to each other we get
Which reduces to
Now we can say
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {-nRT\over v_c-nb}+{2an^2\over v_c^3}=0}
Plugging Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_c} in we get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {-nRT\over (3nb-bn)^2}+{2an^2\over (3nb)^3}=0}
Now we solve for T
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {-nRT\over (2bn)^2}+{2an^2\over 27n^3b^3}=0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {-RT\over 4b^2n}+{2a\over 27nb^3}=0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {2a\over 27nb^3}={RT\over 4b^2n}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {8a\over 27b}=RT}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {8a\over 27bR}=T_c}
Now Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_c} can be plugged in to find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_c}
say
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P={nR{(8a\over 27bR)}\over 3nb-nb}-{an^2\over (3nb)^2}}