P-T diagram of an ideal gas having three different densities ρ₁, ρ₂, ρ₃ (in three different cases) is shown in the figure. Which of the following is correct :

Solution

Physics sir · 4 steps

Write the gas law in terms of density, and the slope of each P-T line tells you its density.

1. Bring in density. Start from PV = nRT with n = m/M. Since ρ = m/V, this becomes P = (ρ R)/MT.

   P = ((ρ R)/M) T ⇒ slope = (ρ R)/M

2. Read the slopes. For a fixed density each case is a straight line through the origin, and R and M are the same in every case. A steeper line therefore means a larger density.
3. Compare the lines. The ρ₁ line is the steepest, then ρ₂, and the ρ₃ line is the flattest. So ρ₁ > ρ₂ > ρ₃.
4. So. ρ₁ > ρ₂ is the correct statement, which is option (a).

The idea to keep. For an ideal gas, P = ρ RT/M. On a P-T graph the slope is proportional to density, so the steeper line has the denser gas.

Answer from the NTA final answer key. Solution by PrepWiser.