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Question
The figure shows a cylindrical container containing oxygen (γ = 1.4) and closed by a 50-kg frictionless piston. The area of cross-section is 100 cm2, atmospheric pressure is 100 kPa and g is 10 m s−2. The cylinder is slowly heated for some time. Find the amount of heat supplied to the gas if the piston moves out through a distance of 20 cm.
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Solution
Given:
Mass of the piston (m) = 50 kg
Adiabatic constant of the gas, γ = 1.4
Area of cross-section of the piston (A) = 100 cm2
Atmospheric pressure (P0) = 100 kPa
g = 10 m/s2
Distance moved by the piston , x = 20 cm
Work done by the gas,
dW=Pdv
The pressure (p) is because of two factors : the first is the initial pressure and
Therefore,
`"W" = (("m""g")/"A" + "P"_0) xx "A""d"x`
`"W" = ((50 xx 10)/(100 xx 10^-4) + 10_5) xx 100 xx 10^-4 xx 20 xx 10^-4`
W =( 5 × 104 +105) × 20 × 10-4
W = 1.5 × 105 × 20 × 10-4
W = 300 J
Hence , nRdT = PΔV = 300
`=> "d""T" =300/("n""R")`
So, `"d""Q" = "n""C"_"p""d""T" = "n""c"_"p" xx (300/("n""R"))`
Using `"C"_"p" -"C"_"v" = "R" and "C"_"p"/"C"_"v" = gamma,`
`"d""Q" = ("n" gamma"R"300)/((gamma - 1) "n""R")`
`"d""Q" = ((300 xx 1.4 )/0.4) = 1050 "J"`
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