Ideal Gas Equation - UNSOLVED PRACTICE SET

Q7. Write the ideal gas equation and explain the meaning of each symbol with proper units.

Q8. Derive the relation between the universal gas constant R and Boltzmann's constant k.

Q9. What is meant by STP? Write the values of pressure, temperature, and molar volume at STP.

Q10. Show that the density of an ideal gas is given by ρ = PM/RT, where M is the molar mass.

Q11. Two identical containers hold hydrogen and oxygen at the same temperature and pressure. Which container has more molecules? Explain using the ideal gas equation.

Q12. A balloon is filled with helium gas at 27°C and 1 atm. If it is taken to a height where the temperature is –23°C and pressure is 0.5 atm, what happens to its volume? Calculate the ratio of final volume to initial volume.

Q13. Starting from Boyle's law, Charles's law, and Avogadro's law, derive the ideal gas equation PV = nRT step by step. Explain how each law contributes to the final equation. Also derive the alternative forms: PV = NkT and PM = ρRT.

Q14. Discuss the limitations of the ideal gas equation. Under what conditions do real gases deviate significantly from ideal gas behaviour? Explain using the van der Waals equation and discuss the significance of the constants 'a' and 'b'.

Q15. A student has a container of volume 10 L containing an unknown gas at 2 atm and 300 K.

(i) Calculate the number of moles of gas in the container.

(ii) If the gas is found to have a mass of 56 g, calculate its molar mass and identify the gas.

(iii) If the temperature is raised to 400 K at constant volume, what is the new pressure?

(iv) If the gas is now allowed to expand isothermally to 20 L, what is the final pressure?

Q16. A cylinder of volume 0.05 m³ contains 0.5 moles of nitrogen gas at 300 K.

(i) Calculate the pressure of the gas.

(ii) Calculate the density of the gas. (Molar mass of N₂ = 28 g/mol)

(iii) If 0.5 more moles of nitrogen are added at the same temperature, what is the new pressure?

(iv) If the cylinder is now heated to 400 K at constant volume, what is the final pressure?

(v) Draw a P-T diagram showing the processes in parts (iii) and (iv).

(Given: R = 8.31 J mol⁻¹ K⁻¹)

Q17. An air bubble of volume 1.0 cm³ is at the bottom of a lake 40 m deep where the temperature is 4°C. The bubble rises to the surface where the temperature is 27°C. Atmospheric pressure is 1.0 × 10⁵ Pa and density of water is 1000 kg/m³.

(i) Calculate the pressure at the bottom of the lake.

(ii) Calculate the volume of the bubble when it reaches the surface.

(iii) Explain why the bubble expands as it rises, using the ideal gas equation.

(iv) What would happen to the bubble if the temperature at the surface were the same as at the bottom?

Q18. A hydrogen balloon is to be designed to lift a payload of 100 kg. The balloon is filled with hydrogen gas at 27°C and 1 atm. The molar mass of hydrogen is 2 g/mol and that of air is 29 g/mol.

(i) Calculate the minimum volume of hydrogen required to just lift the payload. (Ignore the mass of the balloon material)

(ii) Calculate the number of moles of hydrogen needed.

(iii) If the balloon is heated to 57°C at constant pressure, what happens to its lifting capacity? Explain.

(iv) Why is helium preferred over hydrogen for passenger balloons despite helium providing less lift?

(Given: R = 8.31 J mol⁻¹ K⁻¹, g = 9.8 m/s²)