# CHAPTER 27 SPECIFIC HEAT CAPACITIES OF GASES

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1 HATE 7 SEIFI HEAT AAITIES OF GASES. N mole, W g/mol, m/s K.E. of the vessel Internal energy of the gas (/) mv (/) J n r(t) 8. T T k. 8.. m g, t.7 cal/g- J. J/al. dq du + dw Now, (for a rigid body) So, dw. So dq du. Q msdt cal Joule..., w or piston kg., A of piston cm o kpa, g m/s, x cm. mg dw pdv oadx A ndt dt n. J. n dq npdt np n ( ) n.. J.. H. al/g, H. al/g M g/ Mol, 8. 7 erg/mol- We know, al/g So, difference of molar specific heats M M al/g Now, J J 8. 7 erg/mol- J. 7 erg/cal.. 7.6, n mole, T K (a) Keeping the pressure constant, dq du + dw, T K, 7/6, m mole, dq du + dw n dt du + dt du npdt dt 7 dt dt 6 dt dt 7 6 DT dt 7dT dt 6 dt J. (b) Kipping olume constant, dv n dt dt J (c) Adiabetically dq, du dw n 8. T T T T J

2 6. m.8 g, cm L T k, a nt or n N T 8. atm. T Q Now, v.6 9. n dt p + v Specific Heat apacities of Gases Q n p dt.87.8 al cm, cm a, Q J (a) Q du + dw du + ( 6 ) du + du J (b) n 8. [ U nt for monoatomic] n (c) du n v dt v dndtu p v (d) v. (roved above) 8. Q Amt of heat given.8. Work done Q, Q W + U for monoatomic gas U Q Q Q n T Q nt nt Again Q n pdt Where > Molar heat capacity at const. pressure. nt ndt nt 9. K K T K T T K U dv K. dq du + dw mcdt dt + pdv msdt dt+ ms + K + K,, b dv dt b dt + dt b + r, b + b 7. df K. onsidering two gases, in Gas() we have,, p (Sp. Heat at const. ), v (Sp. Heat at const. ), n (No. of moles) p & p v v

3 v v v ( ) v & p Specific Heat apacities of Gases In Gas() we have,, p (Sp. Heat at const. ), v (Sp. Heat at const. ), n (No. of moles) p & p v v v v ( ) v v Given n : n : du nv dt & du nv dt nvdt v v r &p v () p So, [from () & ()] v ( ) () & p. p. p. v. v. n n mol (n + n ) dt n v dt + n v dt nv nv.. n n + + p.. n mole, J/mol-k, (a) Temp at A T a, a a nt a 6 T a a a k. n Similarly temperatures at point b k at it is 8 k and at D it is k. (b) For ab process, dq npdt [since ab is isobaric] T b T a ( ) 9 J For bc, dq du + dw [dq, Isochorie process] n dq du n v dt T c T a () J (c) Heat liberated in cd n p dt n T d T c Heat liberated in da n v dt T a T d J ( ) 7 J Ka Ka a d Ta Td cm Tc c b Tb cm 7.

4 . (a) For a, b is constant So, T T For b,c is constant So, T T 6 T T T T 6 6 k 9 k Specific Heat apacities of Gases (b) Work done Area enclosed under the graph cc kpa 6 J J (c) Q Supplied n v dt Now, n considering at pt. b T v Q bc dt a, b. 6 dt T 6.67 Q supplied to be n p dt [ p ] 6.9 (.67) dt.9 T (d) Q U + w Now, U Q w Heat supplied Work done (.9 +.9) 9.8. In Joly s differential steam calorimeter v ml m ( ) m Mass of steam condensed.9 g, L al/g. J/g m Mass of gas present g,,.9. v.89.9 J/g-K ( ) 6.. Since it is an adiabatic process, So const. (a) (b) T onst. T T Given L, L,..96. T T 7.. a, cc, T k (a).....? (b) T T (). T (). T. k k 7.7 cm c cm a b Ka Ka 7.

5 (c) Work done by the gas in the process m T T T( ) W T T 6.. [ ].7 J (, ). 8.., T 9 k, atm, p atm We know for adiabatic process, T T or (). (9). ().. T (). (9). T..78 T. T (.78) /.. K 9. Ka a, cm 6 m, T k,. (a) Suddenly compressed to cm (). (). (). 8 Ka Specific Heat apacities of Gases T T (). T ().- T 6 K (b) Even if the container is slowly compressed the walls are adiabatic so heat transferred is. Thus the values remain, 8 Ka, T 6 K.. Given (Initial ressure), (Initial olume) (a) (i) Isothermal compression, or, (ii) Adiabatic ompression or o + (b) (i) Adiabatic compression or (ii) Isothermal compression or. Initial pressure Initial olume (a) Isothermally to pressure + Adiabetically to pressure ( ) ( ) + (+)/ Final olume (+)/ 7.

6 Specific Heat apacities of Gases (b) Adiabetically to pressure to ( + ) () Isothermal to pressure / (+)/ Final olume (+)/. nt Given Ka a, cm 6 m, T k (a) n T 8. (b) ( ) moles. 6.6 J/mole (c) Given Ka a,? cm 6 m,. cm 6 m, T k, T? Since the process is adiabatic Hence ( 6 ) ( 6 ) a 78 Ka (d) Q W + U or W U [U, in adiabatic] n dt ( ) J J (e) U n dt J. A B For A, the process is isothermal A A A A A For B, the process is adiabatic, A A A A A ( B ) A ( B ) B B B B B For,, the process is isobaric T T T T T Final pressures are equal. T. B. p A B. A : B : :. : : :. Initial ressure Initial olume Final ressure Final olume Given,, Isothermal workdone nt Ln 7.6

7 Specific Heat apacities of Gases Adiabatic workdone Given that workdone in both cases is same. Hence nt Ln ( )ln nt nt nt ( ) ln ( ) ln We know T const. in adiabatic rocess. T T, or T ( ) T () ( ) Or, T - T or T T (ii) From (i) & (ii) ( ) ln T T T ( ) ln T T T nt (i) [ ].., T k, Lv l (a) The process is adiabatic as it is sudden, ( ). (). / (b) Ka n a W [T T ] T (). T (.). T.. T nt n n w [T T ] K. T ( in m ) (. ). (c) Internal Energy, dq, du dw ( 8.8)J 8.8 J 8 J. (d) Final Temp.. k k. (e) The pressure is kept constant. The process is isobaric. 8.8 J 8 J. Work done ndt ( ) Final Temp K (.). J. Initial Temp (f) Initial volume Final volume L T T T Work done in isothermal ntln ln / T ln.9 (g) Net work done W A + W B + W J. L. 7.7

8 Specific Heat apacities of Gases 6. Given. We know fro adiabatic process T onst. So, T T (eq) As, it is an adiabatic process and all the other conditions are same. Hence the above equation can be applied. So, T. T.. T T. T. T So, T :T : 7. cm,. J/mol-k, T k, 7 cm (a) No. of moles of gas in each vessel, T 8. 7 (b) Heat is supplied to the gas but dv dq du n dt.8. dt dt for (A).8. For (B) dt.8. 7 A.8. A B A [For container A] T T A. 7. cm of Hg..8. B 7 [For ontainer B] B.8. B A cm of Hg. T T Mercury moves by a distance B A.. m. 8. mhe. g,.67, g/mol, mh? 8/mol. Since it is an adiabatic surrounding He dq n dt. dt. (.67 ) dt (i) m H n dt dt m. dt [Where m is the rqd. Mass of H ] Since equal amount of heat is given to both and T is same in both. Equating (i) & (ii) we get. m dt dt m g.67 A He / / T T / / T T : B H 9. Initial pressure, Initial Temperature T Initial olume (a) For the diathermic vessel the temperature inside remains constant A B, Temperature T o For adiabatic vessel the temperature does not remains constant. The process is adiabatic T T T T ( ) T T T T 7.8

9 p ( ) (b) When the values are opened, the temperature remains T through out nt +, nt ( n n )T [Total value after the expt + ] nt nt. For an adiabatic process, v onst. There will be a common pressure when the equilibrium is reached Hence For left For ight () ( ) () ( ) () Equating for both left & right or () ( ) / / Similarly / / / / / / / For right () / / / Specific Heat apacities of Gases For left.() (b) Since the whole process takes place in adiabatic surroundings. The separator is adiabatic. Hence heat given to the gas in the left part Zero. (c) From () Final pressure Again from () / / / ( ) or y / / /. A cm m, M. g. kg, atm pascal, L cm. m. L 8 cm.8 m,. atm The process is adiabatic / / () () (AL). (AL).. log log.9. / / T / / T.9 m / v m/s 7.9

10 Specific Heat apacities of Gases. 8 m/s, T, oh.89 kg/m, r 8. J/mol-k, At ST, a, We know sound Again, o J/mol-k.8 (8) (8).8 Again, or J/mol-k. g kg, cm 6 m cal/mol-ki. J/mol-k J/mol-k 8. ( ) (8.) 8. Since the condition is ST, atm pa m/s. Given o.7 g/cm.7 kg/m,. a, 8. J/mol-k,. KHz. Node separation in a Kundt tube 6 cm, cm m So, - 6 m/s We know, Speed of sound o (6)..7 (6) But J/mol-k Again So, J/mol-k. Hz, T Hz,. cm 6.6 m 6.6 (66 ) m/s m [v nt t M ot m T (66 ) m v J/mol-k, J/mol-k. o m T ] (66 ) 8. (66 )

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