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A veritcal cylinder closed at both ends is divided into two parts by a frictionless piston, each part containing one mole of air. At temperature 300 K, the volume of the upper part is four times than that of the lower part. At what temperature will the volume of the upper part be three times than that of the lower part ? |
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Answer» Solution :At 300 K, the position of the piston in the cylinder o volume, say V, is represented as : Let `p_(1)` and `p_(2)` be the pressure at the upper and lower part of the cylinder respectively. Let the pressure at the lower part due to the weight of the piston of the cylinder be `p_(0)`. `therefore p_(2) = p_(1) + p_(0)`. ....(1) In the two PARTS of the cylinder, the no. of moles of air are same. As the temperature is also same (300 K), We have `p_(1) xx (4V)/(5) = p_(2) xx (V)/(5)`, or `4p_(1) = p_(2) = p_(1) + p_(0), p_(1) = (p_(0))/(3)` ......(2) Now let the temperature be T at which the volume of the upper part will be three times than that of the lower part. Let the PRESSURES at the uppr and lower parts be `p._(1)` and `p._(2)` respectively. Thus, `p._(2) = p._(1) + p_(0)` ........(3) Again in both the parts, temperature and no. of moles are the same, we have, `p._(1) xx (3V)/(4) = p._(2) xx (V)/(4)`. or `3p._(1) = p._(2) = p._(1) + p_(0)` or `p._(1) = (p_(0))/(2)`. .........(4) From (2) and (4), we have , `p._(1) = (3)/(2)p_(1)` ......(5) Now, for the upper part of the cylinder at temperature T, We have, `p._(1)xx (3V)/(4) = RT ""(n = 1)` Substituting `p._(1)` from (5), we have, `(3)/(2)p_(1) xx (3V)/(4) = RT` `(9)/(8) p_(1) V = RT` .......(6) Further, for the upper part of the cylinder at 300 K, We have, `p_(1) xx (4V)/(5) = R(300)` ......(7) From (6) and (7), we get , T = 421.9 K. Second method At 300 K, `P_(1)xx (4V)/(5) = R xx 300` and `P_(2) xx (V)/(5) = R xx 300` At a temperature T (say), `P._(1) xx (3V)/(4) = RT` and `P._(2)xx (V)/(4) = RT` Substituting `P_(1), P_(2), P._(1)` and `P._(2)` from the above equations in `P_(2) - P_(1) = P._(2) - P._(1)` we get, T = 421.9 K. |
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