Solution to Set 2: Difference between revisions
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<math>(P+{{aN^2}\over v^2})(v-Nb)=NkT</math> | <math>(P+{{aN^2}\over v^2})(v-Nb)=NkT</math> | ||
say <math>V={v\over N}</math> | say <math>V={v\over N}</math> | ||
<math>Pv+{aN^2v\over v^2}-PNb-{aN^2Nb\over v^2}-NkT=0</math> | <math>Pv+{aN^2v\over v^2}-PNb-{aN^2Nb\over v^2}-NkT=0</math> | ||
by multiplying both sides by <math>v^2</math> we get | by multiplying both sides by <math>v^2</math> we get | ||
<math>{Pv^3}+aN^2V-PNbv^2-aN^3b-NkTv^2=0</math> | <math>{Pv^3}+aN^2V-PNbv^2-aN^3b-NkTv^2=0</math> | ||
by dividing both sides by <math>PN^2</math> we get | by dividing both sides by <math>PN^2</math> we get | ||
<math>{v^3\over N^3}+{av\over PN}-{bv^2\over N^2}-{ab\over P}-{kTv^2\over PN^2}=0</math> | <math>{v^3\over N^3}+{av\over PN}-{bv^2\over N^2}-{ab\over P}-{kTv^2\over PN^2}=0</math> | ||
so | so | ||
<math>V^3+V{a\over P}-V^2b-{ab\over P}-V^2{kT\over P}=0</math> | <math>V^3+V{a\over P}-V^2b-{ab\over P}-V^2{kT\over P}=0</math> | ||
and combining terms we get | and combining terms we get | ||
<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> | ||
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<math>P={nRT\over (v-nb)^2}-{an^2\over v^2}</math> | <math>P={nRT\over (v-nb)^2}-{an^2\over v^2}</math> | ||
Taking the derivative we get | Taking the derivative we get | ||
<math>{dP\over dv}={-nRT\over v-nb}+{2an^2\over v^3}=0</math> | <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 | 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> | <math>{2nRT\over (v-nb)^3}-{4an^2\over v^3(v-nb)}=0</math> | ||
Taking the derivative again we get | Taking the derivative again we get | ||
<math>{d^2P\over d^2v}={2nRT\over (v-nb)^3}-{6an^2\over v^4}=0</math> | <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 | 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> | <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 | Which reduces to | ||
<math>6(v-nb)=4v</math> | <math>6(v-nb)=4v</math> | ||
<math>6v-6nb=4v</math> | <math>6v-6nb=4v</math> | ||
<math>6nb=2v</math> | <math>6nb=2v</math> | ||
<math>3nb=v_c</math> | <math>3nb=v_c</math> | ||
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Now we can say | Now we can say | ||
<math>{-nRT\over v_c-nb}+{2an^2\over v_c^3}=0</math> | <math>{-nRT\over v_c-nb}+{2an^2\over v_c^3}=0</math> | ||
Plugging <math>v_c</math> in we get | Plugging <math>v_c</math> in we get | ||
<math>{-nRT\over (3nb-bn)^2}+{2an^2\over (3nb)^3}=0</math> | <math>{-nRT\over (3nb-bn)^2}+{2an^2\over (3nb)^3}=0</math> | ||
Now we solve for T | Now we solve for T | ||
<math>{-nRT\over (2bn)^2}+{2an^2\over 27n^3b^3}=0</math> | <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>{-RT\over 4b^2n}+{2a\over 27nb^3}=0</math> | ||
<math>{2a\over 27nb^3}={RT\over 4b^2n}</math> | <math>{2a\over 27nb^3}={RT\over 4b^2n}</math> | ||
<math>{8a\over 27b}=RT</math> | <math>{8a\over 27b}=RT</math> | ||
<math>{8a\over 27bR}=T_c</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> | Now <math>T_c</math> and <math>v_c</math> can be plugged in to find <math>P_c</math> | ||
say | say | ||
<math>P={nR({8a\over 27bR})\over 3nb-nb}-{an^2\over (3nb)^2}</math> | <math>P={nR({8a\over 27bR})\over 3nb-nb}-{an^2\over (3nb)^2}</math> | ||
<math>P={8na\over 27b(2nb}-{a\over 9b^2}</math> | <math>P={8na\over 27b(2nb}-{a\over 9b^2}</math> | ||
<math>P={4a\over 27b^2}-{a\over 9b^2}</math> | <math>P={4a\over 27b^2}-{a\over 9b^2}</math> | ||
<math>P={4a\over 27b^2}-{3a\over 27b^2}</math> | <math>P={4a\over 27b^2}-{3a\over 27b^2}</math> | ||
<math>P={a\over 27b^2}</math> | <math>P={a\over 27b^2}</math> | ||
<math>P_c={a\over 27b^2}</math> | <math>P_c={a\over 27b^2}</math> | ||
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Use <math>T_c={8a\over 27bR}, v_c=3nb, P_c={a\over 27b^2}, T_r={T\over T_c}, n=1</math> | Use <math>T_c={8a\over 27bR}, v_c=3nb, P_c={a\over 27b^2}, T_r={T\over T_c}, n=1</math> | ||
and | and | ||
<math>P={RT\over v-b}-{a\over v^2}</math> | <math>P={RT\over v-b}-{a\over v^2}</math> | ||
so | so | ||
<math>P_rP_c={RT_rT_c\over v_rv_c-b}-{a\over (v_rv_c)^2}</math> | <math>P_rP_c={RT_rT_c\over v_rv_c-b}-{a\over (v_rv_c)^2}</math> | ||
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<math>(3({v\over v_c})-1)(({P\over P_c})+{3\over ({v_c\over v})^2})={8({T\over T_c})}</math> | <math>(3({v\over v_c})-1)(({P\over P_c})+{3\over ({v_c\over v})^2})={8({T\over T_c})}</math> | ||
also | also | ||
<math>{P_cv_c\over T_c}={({a\over 27b^2})(3nb)\over {8a\over 27b}}</math> | <math>{P_cv_c\over T_c}={({a\over 27b^2})(3nb)\over {8a\over 27b}}</math> | ||
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<math>{P_cv_c\over T_c}={3n\over 8}</math> | <math>{P_cv_c\over T_c}={3n\over 8}</math> | ||
==Part D== | |||
<math>K=({dv\over dP})_T</math> and <math>T=T_c</math> | |||
<math>K_T={1\over v}({dv\over dP})</math> | |||
<math>{dP\over dv}</math> at <math>v_c={-nkT\over 2n^2b^2}+{2an^2\over 27n^3b^3}</math> | |||
<math>={-kT\over 4nb^2}+{2a\over 27nb^3}</math> | |||
<math>={1\over 4nb^2}(-kT+{8a\over 27b})</math> | |||
<math>={1\over 4nb^2}({8a\over 27b}-kT)\approx T-T_c</math> | |||
Revision as of 14:04, 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
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 {dP\over dv}={-nRT\over v-nb}+{2an^2\over v^3}=0}
Multiply the derivative 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 {-2\over v-nb}}
to 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 {2nRT\over (v-nb)^3}-{4an^2\over v^3(v-nb)}=0}
Taking the derivative again 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 {d^2P\over d^2v}={2nRT\over (v-nb)^3}-{6an^2\over v^4}=0}
By setting the derivatives equal to each other 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 {2nRT\over (v-nb)^3}-{4an^2\over v^3(v-nb)}={2nRT\over (v-nb)^3}-{6an^2\over v^4}}
Which reduces to
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 6(v-nb)=4v}
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 6v-6nb=4v}
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 6nb=2v}
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 3nb=v_c}
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}}
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={8na\over 27b(2nb}-{a\over 9b^2}}
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={4a\over 27b^2}-{a\over 9b^2}}
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={4a\over 27b^2}-{3a\over 27b^2}}
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={a\over 27b^2}}
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={a\over 27b^2}}
Part C
Use 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={8a\over 27bR}, v_c=3nb, P_c={a\over 27b^2}, T_r={T\over T_c}, n=1}
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 P={RT\over v-b}-{a\over v^2}}
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 P_rP_c={RT_rT_c\over v_rv_c-b}-{a\over (v_rv_c)^2}}
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_r({a\over 27b^2})={RT_r({8a\over 27bR})\over v_r(3nb)-b}-{a\over v_r^2(3nb)^2}}
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_r({a\over 27b^2})={8aT_r\over 27b^2(3v_r-1)}-{a\over v_r^29b^2}}
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_r({a\over 27})={8aT_r\over 27(3v_r-1)}-{a\over v_r^29}}
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_r({a\over 27})+{a\over v_r^29}={8aT_r\over 27(3v_r-1)}}
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_r\over 27}+{1\over v_r^29}={8T_r\over 27(3v_r-1)}}
multiply 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 27(3v_r-1)}
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 (3v_r-1)({P_r}+{3\over v_r^2})={8T_r}}
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 (3({v\over v_c})-1)(({P\over P_c})+{3\over ({v_c\over v})^2})={8({T\over T_c})}}
also
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_cv_c\over T_c}={({a\over 27b^2})(3nb)\over {8a\over 27b}}}
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_cv_c\over T_c}={3n\over 8}}
Part D
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 K=({dv\over dP})_T} and
at
