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An oxygen cylinder of volume 30 litres has an initial gauge pressure of 15 atm and a temperature of 27 °C. After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to 17 °C. Estimate the mass of oxygen taken out of the cylinder (R = 8.31 J mol^{–1} K^{–1}, molecular mass of O_{2} = 32 u)
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Solution 1
Volume of oxygen, V_{1} = 30 litres = 30 × 10^{–3} m^{3}
Gauge pressure, P_{1} = 15 atm = 15 × 1.013 × 10^{5} Pa
Temperature, T_{1} = 27°C = 300 K
Universal gas constant, R = 8.314 J mole^{–1} K^{–1}
Let the initial number of moles of oxygen gas in the cylinder be n_{1}.
The gas equation is given as:
`P_1V_1 = eta_1RT_1`
`:.n_1 = P_1V_1/RT_1`
`= (15.195 xx10^5 xx 30 xx 10^(3))/((8.314) xx 300) = 18.276`
Where,
m_{1} = Initial mass of oxygen
M = Molecular mass of oxygen = 32 g
∴m_{1} = n_{1}M = 18.276 × 32 = 584.84 g
After some oxygen is withdrawn from the cylinder, the pressure and temperature reduces.
Volume, V_{2} = 30 litres = 30 × 10^{–3} m^{3}
Gauge pressure, P_{2} = 11 atm = 11 × 1.013 × 10^{5} Pa
Temperature, T_{2} = 17°C = 290 K
Let n_{2} be the number of moles of oxygen left in the cylinder.
The gas equation is given as:
P_{2}V_{2} = n_{2}RT_{2}
`:. n_2= (P_2V_2)/(RT_2)`
`= (11.143 xx 10^5 xx 30 xx 10^(3))/8.314 xx 290 = 13.86`
But `n_2 = m_2/M`
Where,
m_{2} is the mass of oxygen remaining in the cylinder
∴m_{2} = n_{2}M = 13.86 × 32 = 453.1 g
The mass of oxygen taken out of the cylinder is given by the relation:
Initial mass of oxygen in the cylinder – Final mass of oxygen in the cylinder
= m_{1} – m_{2}
= 584.84 g – 453.1 g
= 131.74 g
= 0.131 kg
Therefore, 0.131 kg of oxygen is taken out of the cylinder.
Solution 2
Initial Volume, `V_1 = 30 litre = 30 xx 10^3 cm^3`
`= 30 xx 10^3 xx 10^(6) m^3 = 30 xx 10^(3) m^3`
Initla Pressure, P_1 = 15 atm
`= 15 xx1.013 xx 10^5 N m^(2)`
Inital temperature, `T_1 = (27 + 273) K = 300 K`
Initial number of moles
`mu_1 = (P_1V_1)/RT_1 = (15xx1.013 xx10^5 xx 30xx10^(3))/(8.31 xx 300) = 18.3`
Final Pressure, `P_2 = 11 atm`
`= 11 xx 1.013 xx 10^5 N m^(2)`
Final Volume, V`_2 = 30 litre = 30 xx 10^(3) cm^3`
Final temperature, `T_2 = 17 + 273 = 290 K`
Final number of moless, `mu_2 = (P_2V_2)/(RT_2) = (11xx 1.013xx 10^5xx 30xx10^(3))/(8.31 xx 290) = 13.9`
Number of moles takesout of cyclinder = 18.3  13.9 = 4.4
Mass of gas taken out of cylinder = 4.4 xx 32 g = 140.8 g = 0.141 kg
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