States of Matter

NCERT Solution

Part 1

Question 1: What will be the minimum pressure required to compress 500 dm³ of air at 1 bar to 200 dm³ at 30° C?

Answer: Given, p₁ = 1 bar, V₁ = 500 dm³, V₂ = 200 dm³

As per Boyle’s Law: p₁V₁ = p₂V₂

Or, `1xx500=p_2xx200`

Or, `p_2=(500)/(200)=2.5` bar

Question 2: A vessel of 120 mL capacity contains a certain amount of gas at 35° C and 1.2 bar pressure. The gas is transferred to another vessel of volume 180 mL at 35°C. What would be its pressure?

Answer: Given, p₁ = 1.2 bar, V₁ = 120 mL, V₂ = 180 mL

As per Boyle’s Law: p₁V₁ = p₂V₂

Or, `1.2xx120=p_2xx180`

Or, `p_2=(1.2xx120)/(180)=0.8` bar

Question 3: Using the equation of state pV = nRT; show that at a given temperature density of a gas is proportional to gas pressure p.

Answer: `pV=nR\T`

Or, `p=(nR\T)/V`

We know that n = constant mass of gas ÷ molar mass of gas `=m/M`

So, `p=(mR\T)/(MV)`

Since `m/V=ρ`

So, `p=(ρR\T)/M`

So, `p∝ρ` proved

Question 4: At 0° C, the density of a certain dioxide of a gas at 2 bar is same as that of dinitrogen at 5 bar. What is the molecular mass of the oxide?

Answer: Molar mass of dinitrogen = 28 u

From previous question, we found

`p=(ρR\T)/M`

Or, `ρ=(pM)/(RT)`

As R is a constant and given temperature is same

So, `p_1M_1=p_2M_2`

Or, `2xx\M_1=5xx28`

Or, `M_1=(5xx28)/2=70` u

Question 5: Pressure of 1 g of an ideal gas A at 27° C is found to be 2 bar. When 2 g of another ideal gas B is introduced in the same flask at same temperature the pressure becomes 3 bar. Find the relationship between their molecular masses.

Answer: Let us assume that molar masses of A and B are M_A and M_B respectively. So, their number of moles can be as follows:

`n_A=1/(M_A)`

`n_B=2/(M_B)`

Given, p_A = 2 bar, p_A + p_B = 3 bar

So, p_B = 3 – 2 = 1 bar

Using the equation, pV = nRT, we get

`p_AV=n_A\RT`

And, `p_BV=n_B\RT`

So, `(p_A)/(p_B)=(n_A)/(n_B)`

`=(1/(M_A))/(2/(M_B))`

Or, `(p_A)/(p_B)=(M_B)/(2M_A)`

Or, `(M_B)/(M_A)=(2p_A)/(p_B)=2xx2/1=4`

Or, `M_B=4M_A`

Question 6: The drain cleaner, Drainex contains small bits of aluminum which react with caustic soda to produce dihydrogen. What volume of dihydrogen at 20° C and one bar will be released when 0.15 g of aluminum reacts?

Answer: The chemical reaction of this process is given by following equation:

2Al + 2NaOH + H₂ → 1NaAlO₂ + 3H₂

In this reaction, 2 M of aluminium produces 3 M of dihydrogen gas.

Molar mass of Al = 2 × 27 = 54 g

Volume of hydrogen produced at STP = 3 × 22720 mL

So, volume of hydrogen produce on reaction with 0.15 g

`=(3xx22720xx0.15)/(54)=189.33` mL

Now, we have: V₁ = 189.33 mL, p₁ 1 bar, p₂ = 1 bar, T₁ = 273 K and T₂ = 20 + 273 = 293 K

Volume V₂ can be calculated as follows:

According to ideal gas equation:

`(p_1V_1)/(T_1)=(p_2V_2)/(T_2)`

Or, `V_2=(p_1V_1T_2)/(p_2T_1)`

`=(1xx189.33xx293)/(1xx273)=203` mL

Question 7: What will be the pressure exerted by a mixture of 3.2 g of methane and 4.4 g of carbon dioxide contianed in a 9 dm³ flask at 27° C?

Answer: Molar mass of methane = 16 u and molar mass of CO₂ = 24 u

So, n of methane = `(3.2)/(16)=0.2`

And, n of carbon dioxide = `(4.4)/(44)=0.1`

R = 8.314 J K^-1 mol^-1

Given temperature = 27 + 273 = 300 K

Using pV = nRT

`p=(nR\T)/V`

So, p_methane `=(0.2xx0.08314xx300)/9=0.55` atm

And, p_{carbon dioxide} `=(0.1xx0.08314xx300)/9=0.27` atm

So, pressure exerted by both gases together = 0.55 + 0.27 = 0.82 atm

Question 8: What will the pressure of the gaseous mixture when 0.5 L of H₂ at 0.8 bar and 2.0 L of of dioxygen at 0.7 bar are introduced in a 1L vessel at 27° C?

Answer: Calculation of partial pressure of H₂: p₁ = 0.8 bar, p₂ = ?, V₁ = 0.5 L and V₂ = 1 L

`p_1V_1=p_2V_2`

Or, `0.8xx0.5=p_2xx1`

Or, `p_2=0.8xx0.5=0.4` bar

Calculation of partial pressure of O₂: p₁ = 0.7 bar, p₂ = ?, V₁ = 2 L and V₂ = 1 L

`p_1V_1=p_2V_2`

Or, `0.7xx2=p_2xx1`

Or, `p_2=1.4` bar

Total pressure = 0.4 + 1.4 = 1.8 bar

Question 9: Density of a gas is found to be 5.46 g/dm³ at 27° C at 2 bar pressure. What will be its density at STP?

Answer: We know: `d=p/(RT)`

So, `(d_1)/(d_2)=(p_1T_2)/(p_2T_1)`

Given, d₁ = 5.46, p₁ = 2 bar,T₁ = 300 K, p₂ = 1 bar,T₂ = 273 K

We have, `(5.46)/(d_2)=(2xx273)/(1xx300)`

Or, `d_2=(5.46xx300)/(2xx273)=3` g/dm³

Question 10: 34.05 mL of phosphorus vapor weighs 0.0625 g at 546° C and 0.1 bar pressure. What is the molar mass of phosphorus?

Answer: Given: V₁ = 34.05 mL, T₁ = 546 + 273 = 819 K, p₁ = 0.1 bar, m = 0.0625 g

Let us calculate the volume at STP (T₂ = 273 K, p₂ = 1 bar

`(p_1V_1)/(T_1)= (p_2V_2)/(T_2)`

Or, `(0.1xx34.05)/(819)=(1xx\V_2)/(273)`

Or, `V_2=(0.1xx34.05xx273)/(819)=1.135`

Since 11.35 mL weighs 0.0625 g at STP

So, 22720 mL weighs `(0.0625)/(11.35)xx22720=125` g

So, molar mass = 125 g mol^-1