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Question: A vessel of the volume \[50{\text{ litres}}\] contains an ideal gas at \[{0^ \circ }c\]. A portion o...

A vessel of the volume 50 litres50{\text{ litres}} contains an ideal gas at 0c{0^ \circ }c. A portion of the gas is allowed to leak out from it under isothermal conditions so that pressure inside falls by 0.80.8 atmosphere. The number of moles of gas leaked out is nearly
(a)1.51 mole1.51{\text{ }}mole
(b) 1.63 mole1.63{\text{ }}mole
(c) 1.98 mole1.98{\text{ }}mole
(d) 1.78 mole1.78{\text{ }}mole

Explanation

Solution

When the temperature of an ideal gas remains constant, the isothermal process describes the relationship between volume and pressure. The volume of a gas is directly proportional to the number of moles of gas under constant temperature and pressure. By using the formula, the number of moles of gas can be found. The value of the universal gas constant has to be known to use the formula.

Complete step-by-step solution:
Given,
Volume, V=50 litresV = 50{\text{ litres}}
Temperature,T=0cT = {0^ \circ }c
For an ideal gas, we use the formula,
PV=nRTPV = nRT
Where,
PP= pressure of the gas
VV= volume of the gas
nn= number of moles of gas
RR= ideal gas constant and its value is
R=0.08206 L.atm.mol1K1R = 0.08206{\text{ }}L.atm.mo{l^{ - 1}}{K^{ - 1}} and 8.314 k.Pa.L.mol1K18.314{\text{ }}k.Pa.L.mo{l^{ - 1}}{K^{ - 1}}
and TT= temperature of the gas at kelvin scale
initially,
PV=nRTPV = nRT
After the gas is leaked out final becomes,
PV=nRT(PP)V=(nn)RTP'V = n'RT \to \left( {P - P'} \right)V = \left( {n - n'} \right)RT
So,
No. of moles leaked out = (PP)VRT=0.8×500.0821×273=1.78moles\dfrac{{\left( {P - P'} \right)V}}{{RT}} = \dfrac{{0.8 \times 50}}{{0.0821 \times 273}} = 1.78moles

Note: When using the gas constantR=8.31 J/K.molR = 8.31{\text{ }}J/K.mol , we must enter the pressure PP in pascalsPaPa , volume in m3{m^3} , and temperature TT in kelvinKK .
When using the gas constant R=0.08206 L.atm.mol1K1R = 0.08206{\text{ }}L.atm.mo{l^{ - 1}}{K^{ - 1}}, pressure should be measured in atmospheres atm, volume measured in litres LL , and temperature measured in kelvin KK.
When the particles of a gas are so far apart that they do not exert any attractive forces on one another, the gas is said to be ideal. There is no such thing as an ideal gas in real life, but at high temperatures and low pressures (conditions in which individual particles move very swiftly and are very far apart from one another, with essentially no interaction), gases behave very similarly so the Ideal Gas Law is a good approximation.