{"id":632,"date":"2022-03-24T09:56:28","date_gmt":"2022-03-24T09:56:28","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=632"},"modified":"2022-03-24T04:20:07","modified_gmt":"2022-03-24T04:20:07","slug":"relation-between-pressure-and-volume","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/relation-between-pressure-and-volume\/","title":{"rendered":"\u09b0\u09c1\u09a6\u09cd\u09a7\u09a4\u09be\u09aa\u09c0\u09df \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u09c7 \u099a\u09be\u09aa, \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be \u0993 \u0986\u09df\u09a4\u09a8\u09c7\u09b0 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995"},"content":{"rendered":"

\u09b0\u09c1\u09a6\u09cd\u09a7\u09a4\u09be\u09aa\u09c0\u09df \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u09c7 \u099a\u09be\u09aa \u0993 \u0986\u09df\u09a4\u09a8\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 <\/b>(Relation Between Pressure And Volume During An Adiabatic Change)<\/b><\/h2>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf \u098f\u0995\u099f\u09bf \u09aa\u09be\u09a4\u09cd\u09b0\u09c7 \u098f\u0995 \u09ae\u09cb\u09b2 \u0986\u09a6\u09b0\u09cd\u09b6 \u0997\u09cd\u09af\u09be\u09b8 \u0986\u099b\u09c7\u0964 \u098f\u0987 \u0997\u09cd\u09af\u09be\u09b8\u09c7 dQ<\/span><\/span>\u00a0\u09aa\u09b0\u09bf\u09ae\u09be\u09a3 \u09a4\u09be\u09aa \u09aa\u09cd\u09b0\u09df\u09cb\u0997 \u0995\u09b0\u09bf\u0964 \u098f\u09a4\u09c7 \u0997\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be \u09ac\u09c3\u09a6\u09cd\u09a7\u09bf \u09aa\u09be\u09ac\u09c7 \u098f\u09ac\u0982 \u09b8\u09c7\u0987 \u09b8\u0999\u09cd\u0997\u09c7 \u0997\u09cd\u09af\u09be\u09b8 \u0995\u09bf\u099b\u09c1 \u0995\u09be\u099c \u0995\u09b0\u09ac\u09c7 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09a4\u09be\u09aa \u09a6\u09c1\u0987\u09ad\u09be\u09ac\u09c7 \u09ac\u09cd\u09af\u09df\u09bf\u09a4 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

\u09a7\u09b0\u09bf \u0986\u09df\u09a4\u09a8\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 dV<\/span><\/span> \u098f\u09ac\u0982 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 dT<\/span><\/span>.<\/span><\/p>\n

\u2234<\/span> \u09a4\u09be\u09aa\u0997\u09a4\u09bf\u09ac\u09bf\u09a6\u09cd\u09af\u09be\u09b0 \u09aa\u09cd\u09b0\u09a5\u09ae \u09b8\u09c2\u09a4\u09cd\u09b0 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n

dQ=C_vdT+PdV<\/span><\/span>\u00a0———————- (1.5)<\/span><\/p>\n

\u098f\u0996\u09be\u09a8\u09c7, C_v<\/span><\/span>=<\/span> \u09b8\u09cd\u09a5\u09bf\u09b0 \u0986\u09df\u09a4\u09a8\u09c7 \u0997\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u0986\u09aa\u09c7\u0995\u09cd\u09b7\u09bf\u0995 \u09a4\u09be\u09aa \u098f\u09ac\u0982 PdV<\/span><\/span>=<\/span> \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u099a\u09be\u09aa\u09c7 \u0997\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u09aa\u09cd\u09b0\u09b8\u09be\u09b0\u09a3\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0995\u09c3\u09a4 \u0995\u09be\u099c\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ae\u09be\u09a3\u0964 <\/span>\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \u09b0\u09c1\u09a6\u09cd\u09a7\u09a4\u09be\u09aa \u09aa\u09cd\u09b0\u0995\u09cd\u09b0\u09bf\u09df\u09be\u09df \u09ac\u09be\u0987\u09b0\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u0997\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u09a4\u09be\u09aa\u09c7\u09b0 \u0995\u09cb\u09a8\u09cb \u0986\u09a6\u09be\u09a8-\u09aa\u09cd\u09b0\u09a6\u09be\u09a8 \u0998\u099f\u09c7 \u09a8\u09be\u0964 <\/span>\u0985\u09a4\u098f\u09ac, dQ=0<\/span> <\/span><\/p>\n

\u2234<\/span> \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/span>1.5<\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, C_vdT+PdV=0<\/span><\/span><\/p>\n

\u0986\u09ac\u09be\u09b0, \u0986\u09a6\u09b0\u09cd\u09b6 \u0997\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7, PV=RT<\/span><\/span>,<\/span> \u098f\u0996\u09be\u09a8\u09c7 <\/span>R<\/span> \u09ae\u09cb\u09b2\u09c7\u09b0 \u0997\u09cd\u09af\u09be\u09b8 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u0964<\/span><\/p>\n

\u0989\u0995\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0995\u09c7 \u09ac\u09cd\u09af\u09ac\u0995\u09b2\u09a8 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\nPdV+VdP=RdT<\/span>\n

\u09ac\u09be, dT=\\frac{PdV+VdP}{R}<\/span><\/span><\/p>\n

\u2234<\/span> \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/span>1.5<\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\nC_v \\big(\\frac{PdV+VdP}{R})+PdV=0<\/span>\n

\u09ac\u09be, C_vPdV+C_vVdP+RPdV=0<\/span><\/span><\/p>\n

\u09ac\u09be, C_vPdV+C_vVdP+(C_p-C_v)PdV=0<\/span><\/span>\u00a0 \u00a0 \u00a0[\\because R=C_p-C_v]<\/span><\/p>\n

\u09ac\u09be, C_vPdV+C_vVdP+C_pPdV-C_vPdV=0<\/span><\/span><\/p>\n

\u09ac\u09be, C_vVdP+C_pPdV=0<\/span><\/span><\/p>\n

\u09ac\u09be, VdP+\\frac{C_p}{C_v}PdV=0<\/span><\/span>\u00a0[C_v<\/span><\/span>\u00a0<\/span>\u09a6\u09cd\u09ac\u09be\u09b0\u09be<\/span>\u00a0<\/span>\u09ad\u09be\u0997<\/span>\u00a0<\/span>\u0995\u09b0\u09c7<\/span>]<\/span><\/p>\n

\u09ac\u09be, <\/span>VdP+\\gamma PdV=0<\/span><\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\\Big[\\because \\frac{C_p}{C_v}=\\gamma \\Big]<\/span><\/p>\n

\u09ac\u09be, \\frac{dP}{P}+\\gamma \\frac{dV}{V}=0<\/span><\/span>\u00a0[PV\u00a0<\/span>\u09a6\u09cd\u09ac\u09be\u09b0\u09be<\/span>\u00a0<\/span>\u09ad\u09be\u0997<\/span>\u00a0<\/span>\u0995\u09b0\u09c7<\/span>]<\/span><\/p>\n

\u098f\u0996\u09a8 \u09b8\u09ae\u09be\u0995\u09b2\u09a8 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n

\\log_{e}P+\\gamma \\log_{e}V<\/span><\/span>=<\/span>\u09a7\u09cd\u09b0\u09c1\u09ac\u0995<\/span>=\\log_{e}K<\/span><\/span>\u00a0,<\/span>\u098f\u0996\u09be\u09a8\u09c7<\/span>\u00a0K=<\/span>\u09a7\u09cd\u09b0\u09c1\u09ac\u0995<\/span><\/p>\n

\u09ac\u09be, \\log_{e}P+\\log_{e}V^{\\gamma}=\\log_{e}K<\/span><\/span><\/p>\n

\u09ac\u09be, \\log_{e}PV^{\\gamma}=\\log_{e}K<\/span><\/span><\/p>\n

\\therefore PV^{\\gamma} = K<\/span><\/span>=<\/span>\u09a7\u09cd\u09b0\u09c1\u09ac\u0995<\/span><\/p>\n

\u098f\u099f\u09bf\u0987 \u09b9\u09b2\u09cb \u099a\u09be\u09aa \u0993 \u0986\u09df\u09a4\u09a8\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u0964<\/span><\/p>\n

\u09af\u09a6\u09bf \u0986\u09a6\u09bf \u099a\u09be\u09aa \u0993 \u0986\u09df\u09a4\u09a8 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 P_1<\/span><\/span> \u0993 V_1<\/span><\/span> \u098f\u09ac\u0982 \u099a\u09c2\u09dc\u09be\u09a8\u09cd\u09a4 \u099a\u09be\u09aa \u0993 \u0986\u09df\u09a4\u09a8 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 P_2<\/span><\/span> \u0993 V_2<\/span><\/span> \u09b9\u09df, \u09a4\u09be\u09b9\u09b2\u09c7 P_1V_1^{\\gamma}=P_2V_2^{\\gamma}<\/span><\/span>=<\/span> \u09a7\u09cd\u09b0\u09c1\u09ac\u0995<\/span><\/p>\n

\u09b0\u09c1\u09a6\u09cd\u09a7\u09a4\u09be\u09aa\u09c0\u09df \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u09c7 \u0986\u09df\u09a4\u09a8 \u0993 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 (<\/b>Relation between volume and temperature in Adiabatic Change)<\/b><\/h2>\n

\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \u0986\u09a6\u09b0\u09cd\u09b6 \u0997\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7, PV=RT<\/span><\/span><\/p>\n\\therefore P=\\frac{RT}{V}<\/span>\n

\u09aa\u09c1\u09a8\u0983, \u0986\u09ae\u09b0\u09be \u09aa\u09be\u0987, PV^{\\gamma}<\/span><\/span>=<\/span> \u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u0964<\/span><\/p>\n

\u0989\u0995\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 <\/span>P-<\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09ac\u09b8\u09bf\u09df\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n

\\frac{RT}{V} \\times V^{\\gamma}<\/span>= <\/span>\u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09ac\u09be, RTV^{\\gamma-1}<\/span><\/span>=<\/span> \u09a7\u09cd\u09b0\u09c1\u09ac\u0995<\/span><\/p>\n

\u09ac\u09be, T\\times V^{\\gamma-1}<\/span><\/span>=<\/span> \u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span> [\u2235R=<\/span> \u09a7\u09cd\u09b0\u09c1\u09ac\u0995<\/span>]<\/span><\/p>\n

\u098f\u099f\u09bf\u0987 \u09b9\u09b2\u09cb \u09b0\u09c1\u09a6\u09cd\u09a7\u09a4\u09be\u09aa\u09c0\u09df \u09aa\u09cd\u09b0\u0995\u09cd\u09b0\u09bf\u09df\u09be\u09df \u0986\u09df\u09a4\u09a8 \u0993 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u0964<\/span><\/p>\n

\u09b0\u09c1\u09a6\u09cd\u09a7\u09a4\u09be\u09aa\u09c0\u09df \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u09c7 \u0986\u09a6\u09b0\u09cd\u09b6 \u0997\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u099a\u09be\u09aa \u0993 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 <\/b>(Pressure-temperature relationship for an ideal gas in Adiabatic Change)<\/b><\/h2>\n

\u0986\u09a6\u09b0\u09cd\u09b6 \u0997\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u098f\u0995 \u09ae\u09cb\u09b2 \u0997\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, PV=RT<\/span><\/span><\/p>\n

\u09ac\u09be, V=\\frac{RT}{P}<\/span><\/span><\/p>\n

\u09b0\u09c1\u09a6\u09cd\u09a7\u09a4\u09be\u09aa\u09c0\u09df \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 PV^{\\gamma}<\/span><\/span>=<\/span> \u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u0964<\/span><\/p>\n

V-<\/span>\u098f\u09b0 \u09ae\u09be\u09a8 \u09ac\u09b8\u09bf\u09df\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n

P\\Big(\\frac{RT}{P}\\Big)^{\\gamma}<\/span>=<\/span>\u09a7\u09cd\u09b0\u09c1\u09ac\u0995<\/span><\/p>\n

\u09ac\u09be,P\\times R^{\\gamma} \\times T^{\\gamma} \\times P^{-\\gamma}<\/span><\/span>=<\/span>\u09a7\u09cd\u09b0\u09c1\u09ac\u0995<\/span><\/p>\n

\u09ac\u09be,P^{1-\\gamma} \\times T^{\\gamma}=\\frac{\u09a7\u09cd\u09b0\u09c1\u09ac\u0995}{R^{\\gamma}}<\/span><\/span><\/p>\n

\u09ac\u09be,T^{\\gamma}P^{1-\\gamma}<\/span><\/span>=<\/span>\u09a7\u09cd\u09b0\u09c1\u09ac\u0995<\/span><\/p>\n

\u098f\u0987 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0989\u09ad\u09df \u09aa\u09be\u09b6\u09c7 \\gamma<\/span>\u00a0<\/span> \u09ae\u09c2\u09b2 \u09a6\u09bf\u09df\u09c7 \u09ad\u09be\u0997 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n

\\therefore TP^{\\frac{1-\\gamma}{\\gamma}}<\/span>=<\/span>\u09a7\u09cd\u09b0\u09c1\u09ac\u0995<\/span><\/p>\n

\u0987\u09b9\u09be\u0987 \u09b0\u09c1\u09a6\u09cd\u09a7\u09a4\u09be\u09aa\u09c0\u09df \u09aa\u09cd\u09b0\u0995\u09cd\u09b0\u09bf\u09df\u09be\u09df \u0986\u09a6\u09b0\u09cd\u09b6 \u0997\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u0995\u09cd\u09b7\u09a4\u09cd\u09b0\u09c7 \u099a\u09be\u09aa \u0993 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u0964<\/span><\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

\u09b0\u09c1\u09a6\u09cd\u09a7\u09a4\u09be\u09aa\u09c0\u09df \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u09c7 \u099a\u09be\u09aa \u0993 \u0986\u09df\u09a4\u09a8\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 (Relation Between Pressure And Volume During An Adiabatic Change) \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf \u098f\u0995\u099f\u09bf \u09aa\u09be\u09a4\u09cd\u09b0\u09c7 \u098f\u0995 \u09ae\u09cb\u09b2 \u0986\u09a6\u09b0\u09cd\u09b6 \u0997\u09cd\u09af\u09be\u09b8 \u0986\u099b\u09c7\u0964 \u098f\u0987 \u0997\u09cd\u09af\u09be\u09b8\u09c7 \u00a0\u09aa\u09b0\u09bf\u09ae\u09be\u09a3 \u09a4\u09be\u09aa \u09aa\u09cd\u09b0\u09df\u09cb\u0997 \u0995\u09b0\u09bf\u0964 \u098f\u09a4\u09c7 \u0997\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be \u09ac\u09c3\u09a6\u09cd\u09a7\u09bf \u09aa\u09be\u09ac\u09c7 \u098f\u09ac\u0982 \u09b8\u09c7\u0987 \u09b8\u0999\u09cd\u0997\u09c7 \u0997\u09cd\u09af\u09be\u09b8<\/p>\n

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