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In figure, a resistance coil, wired to an external battery is placed inside a thermally insulated cylinder find with a frictionless piston and containing an ideal diatomic gas A current = 200 ma exists in the coil which has a resistance R=350? If the pressure of the gas remains constant what should be the speed v (in cm/s of the piston of mass m=10kg)?
![question image](https://images.topperlearning.com/topper/new-ate/top_mob1683352033773694969IMG_20230506_111542.jpg)
Asked by samarthghogare | 06 May, 2023, 11:17: AM
Let us assume the dissipated heat in the resistance coil is absorbed by the enclosed gas in the cylinder.
Rate of heat dissipation is
![begin mathsize 14px style fraction numerator d Q over denominator d t end fraction space equals space I to the power of 2 to the power of space end exponent R subscript c space equals space 0.2 space cross times space 0.2 space cross times 350 space equals space 14 space W end style](https://images.topperlearning.com/topper/tinymce/cache/5d6a471f87b2e5f2bcda69ce6968c184.png)
where i is current through coil and Rc is resistance of coil .
For diatomic gas that is hetaed at constant pressure , heat dissipation is
![begin mathsize 14px style fraction numerator d Q over denominator d t end fraction space equals space n space C subscript p subscript space end subscript fraction numerator d T over denominator d t end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/e450d5c3f76ac1e31c41b1ec8590308b.png)
where n is number of moles , Cp =(7/2) R is constant volume specific heat, R is universal gas constant and
(dT/dt) is rate of change of increase of temperature.
From eqn.(1) and eqn.(2), we get
![begin mathsize 14px style fraction numerator d T over denominator d t end fraction space equals space fraction numerator 14 over denominator n space C subscript p end fraction space equals space fraction numerator 14 over denominator n space cross times space 3.5 space R end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/e95707aa8dc73116fef11589c4c4f95b.png)
For ideal gas , we have
![begin mathsize 14px style V space equals space open parentheses fraction numerator n space R over denominator P end fraction close parentheses space T end style](https://images.topperlearning.com/topper/tinymce/cache/dac22ba9beee66b5ad1aeb6fd20d9680.png)
when pressure is kept constant , rate of change of volume is
![begin mathsize 14px style fraction numerator d V over denominator d T end fraction equals space open parentheses fraction numerator n R over denominator P end fraction close parentheses fraction numerator d T over denominator d t end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/af0184a7b87e3b55e47e649f702e63b0.png)
Above expression can be written as
![begin mathsize 14px style A space fraction numerator d h over denominator d t end fraction space equals space open parentheses fraction numerator n space R over denominator P end fraction close parentheses space fraction numerator d T over denominator d t end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/69e44661c28bd2028543b77fac3a9668.png)
where A is area of cross section of cylinder and dh/dt is rate of vertical displacement of piston with respect to time , i.e, (dh/dt) is speed of piston .
From eqn.(3) and (4) , we get speed of piston as
![begin mathsize 14px style fraction numerator d h over denominator d t end fraction space equals space open parentheses fraction numerator n cross times space R over denominator P cross times space A end fraction close parentheses space fraction numerator d T over denominator d t end fraction space equals space space open parentheses fraction numerator n cross times space R over denominator P cross times space A end fraction close parentheses space cross times fraction numerator 14 over denominator n space cross times space 3.5 space cross times R end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/69e760f64f057467f93a17c7bb7f16e7.png)
![begin mathsize 14px style fraction numerator d h over denominator d t end fraction space equals space fraction numerator 4 over denominator P space cross times A end fraction space equals space fraction numerator 4 over denominator 10 to the power of 5 cross times 10 cross times 10 to the power of negative 4 end exponent end fraction space equals space 0.04 space m divided by s space equals space 4 space c m divided by s end style](https://images.topperlearning.com/topper/tinymce/cache/922d0b0669bfe40819c6ee81b4364301.png)
Speed of piston = 4 cm/s
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