Solved Problems

Question 1:

A gas occupies one litre under atmospheric pressure. What will be the volume of the same amount of gas under 750 mm of Hg at the same temperature?

Solution.

Given V₁ = 1 litre

P₁ = 1 atm

V₂ = ? 

P₂ = 750 / 760 atm

Using

P₁V₁ = P₂V₂

We get

1 × 1 = 750 / 760 × V₂

V₂ = 1.0133 litre = 1013.3 ml

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Question 2:

How large a balloon could you fill with 4g of He gas at 22°C and 720 mm of Hg?

Solution.               

Given, P = 720 / 760 atm, T = 295K, w = 4g  

and m = 4 for He  

PV = w / M RT  

 = 720 / 760 × V = 4/4 × 0.0821 × 295  

∴ V = 25.565 litre    

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Question 3:

Calculate the temperature at which 28g N₂ occupies a volume of 10 litre at 

2.46 atm        

Solution.     

w = 28g, P = 2.46 atm, V = 10 litre, m = 28  

Now, PV = w / M RT (R = 0.0821 litre atm K^–1 mole^–1)  

T = 299.6 K    

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Question 4:

A gas occupies 300 ml at 27°C and 730 mm pressure what would be its volume at STP               

Solution            

V₂ = 300 / 1000 litre, P₂ = 730 / 760 atm, T₂ = 300K  

At STP, V₁= ? P₁= 1 atm, T₁= 273K  

P₂V₂ / T₂ = P₁V₁ / T₁ or V₁ = 0.2622 litre  

Volume at STP = 262.2 ml    

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Question 5:

In Victor Meyer’s experiment, 0.23g of a volatile solute displaced air which measures 112 ml at NTP. Calculate the vapour density and molecular weight of the substance.              

Solution 

Volume occupied by solute at NTP = Volume of air displaced at NTP   = 112 ml  

For volatile solute       PV = w / M RT  

at NTP, P = 1 atm, T = 273 K  

1 × 112/ 1000 = 0.23 / m × 0.0821 × 273   

 m = 46.02 and V.D. = 23.01

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Question 6:

A cylindrical balloon of 21 cm diameter is to be filled up with H₂ at NTP from a cylinder containing the gas at 20 atm at 27°C. The cylinder can hold 2.82 litre of water at NTP. Calculate the number of balloons that can be filled up. 

Solution

Volume of 1 balloon which has to be filled = 4/3 π (21/2)³ = 4851 ml   = 4.851 litre  

Let n balloons be filled, then volume of H₂ occupied by balloons = 4.851 × n  

Also, cylinder will not be empty and it will occupy volume of H₂ = 2.82 litre.  

∴ Total volume occupied by H₂ at NTP = 4.851 × n + 2.82 litre  

∴ At STP  

P₂ = 1 atm   Available H₂  

V₁= 4.851 × n + 2.82            P₂ = 20 atm  

T₁ = 273 K   T₂ = 300K  

P₁V₁ / T₂ = P₂V₂ / T₂  V₂ =  2.82 litre  

or 1 × (4.85 1n + 2.82 / 273) = 20 × 2.82 / 300 ∴ n = 10 

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Question 7:

A 20g chunk of dry ice is placed in an empty 0.75 litre wire bottle tightly closed what would be the final pressure in the bottle after all CO_­2 has been evaporated and temperature reaches to 25°C?    

Solution         

w = 20g dry CO₂ which will evaporate to develop pressure  

m = 44, V = 0.75 litre, P = ? T = 298K  

PV = W / m RT  

P × 0.75 = 20 / 44 5  0.0821 × 298  

P = 14.228 atm  

Pressure inside the bottle = P + atm pressure  = 14.828 + 1 = 15.828 atm

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Question 8:

The pressure of the atmosphere is 2 × 10^–6 mm at about 100 mile from the earth and temperature is – 180°C. How many moles are there in 1 ml gas at this attitude?    

Solution       

Given, P = 2×10^–6 / 760 atm  

T = – 180 + 273 = 93 K  

V = 1 ml = 1 / 1000 litre  

PV = nRT  

2 × 10^–6 / 760 = n × 0.0821 × 93  

n = 3.45 × 10^–13 mole    

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Question 9:

50 litre of dry N₂ is passed through 36g of H₂O at 27°C. After passage of gas, there is a loss of 1.20g in water. Calculate vapour pressure of water.    

Solution 

The water vapours occupies the volume of N₂ gas i.e. 50 litre  

∴ For H₂O vapour V = 50 litre, w = 1.20g, T = 300K, m = 18  

PV = w / m RT or P × 50 = 1.2 / 18 × 0.0821 × 300  

∴ P = 0.03284 atm   = 24.95 mm    

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Question 10:

A mixture of CO and CO₂ is found to have a density of 1.50g/litre at 30°C and 730 mm. What is the composition of the mixture?    

For a mixture of CO and CO₂, d = 1.50 g/litre  

P = 730 / 760 atm, T = 303K  

PV = w / m RT; PV w / Vm  RT  

730 / 760 = 1.5 / m × 0.0821 × 303 =  ∴ 38.85  

i.e. molecular weight of mixture of CO and CO₂ = 38.85  

Let % of mole of CO be a in mixture then  

Average molecular weight = a × 28 + (100 – a) 44 / 100  

38.85 = 28a + 4400 – 44a / 100  

a = 32.19  

Mole % of CO = 32.19  

Mole % of CO₂ = 67.81    

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Question 11:

The average speed of an ideal gas molecule at 27°C is 0.3 m sec^–1. Calculate average speed at 927°C.    

Solution          

u_av= √8RT / πM  

Given u_av = 0.3 m sec^–1 at 300K  

u₁= 0.3 = √8R × 300 / πM  

at T = 273 + 927 = 1200K  

u₂ = √8R × 1200 / πM  

u₂ / 0.3 = √1200 / 300 u₂ = 0.6 m sec^–1  

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Question 12:

Pure O₂ diffuses through an aperture in 224 seconds, whereas mixture of O₂ and another gas containing 80% O₂ diffuses from the same in 234 sec. What is the molecular weight of gas?    

Solution       

For gaseous mixture 80% O₂, 20% gas  

Average molecular weight M_m = 32 × 80 + 20 × m / 100  

Now, for diffusion of gaseous mixture and pure O₂  

r_O2 / r_m = √M_m / M_O2 or V_O2 / E_O2 × E_m / V_m = √M_m / M_O2 

∴ 1 / 224 × 234 / 1 = √M_m / 32 ∴ M_m= 34.92    ∴ 32 × 80 + 20 × m / 100 = 34.92  

∴ m = 46.6    

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Question 13:

Calculate the pressure exerted by 10^–23 gas molecules, each of mass 10^–22 g in a container of volume one litre. The rms speed is 10⁵ cm sec^–1.    

Solution              

Given, n = 10²³ m = 10^–22g V = 1 litre = 10³cc  

u_rms = 10⁵cm sec^–1  

PV = 1/3 mnu²_rms  

P × 10³ = 1/3 × 10^–22 × 10²³ × (10⁵)²  

P = 3.3 × 10⁷dyne cm^–2    

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Question 14:

Calculate the temperature at which CO₂ has the same rms speed to that of O₂ at STP.    

Solution

r_rms of O₂ = √3RT / M  at STP, u_rms of O₂ = √3R × 273 / 32   

For CO₂ u_rms CO₂ = √3Rt / 44

Given both are same; 3R × 273 / 32 = 2RT / 44  

∴ T =375.38 K = 102.38°C    

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Question 15:

Calculate the compressibility factor for CO₂, if one mole of it occupies 0.4 litre at 300K and 40 atm. Comment on the result.    

Solution         

Compressibility factor (Z) = PV / nRT  

Z = 40 × 0.4 / 1 × 0.0821 × 300 = 0.65  

Z value is lesser than 1 and thus, nRT > PV. In order to have Z = 1, volume of CO₂must have been more at same P and T or CO₂ is more compressible than ideal gas.
 

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