JEE Advanced Chemistry: Kinetic Theory questions with solutions

Q1. For a non-linear N-atomic ideal gas molecule with all degrees of freedom (translational, rotational, and vibrational) fully excited, find the value of Cp/Cv - 1.

1. 1 / (3N - 3)
2. 1 / N
3. 1 / (3N - 3) only (without the leading 1+)
4. 1 / (N - 1)

Answer: 1 / (3N - 3)

With all modes excited, Cv = (3N-3)R and Cp = (3N-2)R for a non-linear N-atom molecule, giving Cp/Cv - 1 = R/Cv = 1/(3N-3).

Q2. On a Maxwell-Boltzmann speed distribution plot (distribution function f(v) vs speed v), three curves are labelled I, II, and III. The most probable speed (Vmp) is proportional to sqrt(2RT/M). Which of the following correctly identifies the three peaks I (leftmost), II (middle), and III (rightmost) corresponding to Vmp values of different gases at given temperatures?

1. Vmp of N2 (300K); Vmp of H2(300K); Vmp of O2(400K)
2. Vmp of H2 (300K); Vmp of N2(300K); Vmp of O2(400K)
3. Vmp of O2 (400K); Vmp of N2(300K); Vmp of H2(300K)
4. Vmp of N2 (300K); Vmp of O2(400K); Vmp of H2(300K)

Answer: Vmp of N2 (300K); Vmp of O2(400K); Vmp of H2(300K)

Vmp is proportional to sqrt(T/M). Computing the ratio: N2 at 300K gives sqrt(300/28)=3.27, O2 at 400K gives sqrt(400/32)=3.54, H2 at 300K gives sqrt(300/2)=12.25. Ranking I (lowest Vmp, leftmost peak) to III (highest Vmp, rightmost peak): I = N2(300K), II = O2(400K), III = H2(300K).

Q3. For a diatomic ideal gas, which of the following options is/are correct?

1. gamma = 1.40
2. C_P = 7R/2
3. C_V = 5R/2
4. gamma = 1.67

Answer: C_V = 5R/2

For a diatomic gas with 5 degrees of freedom: C_V = 5R/2, C_P = 7R/2, and gamma = C_P/C_V = 1.40. The value gamma = 1.67 corresponds to a monoatomic gas.

Q4. At what temperature does the most probable speed (U_mp) of SO2 equal the rms speed (U_rms) of O2 at 300 K?

1. 150 K
2. 600 K
3. 750 K
4. 900 K

Answer: 900 K

U_mp(SO2) = sqrt(2RT1/64) and U_rms(O2) = sqrt(3R*300/32). Equating and squaring: 2T1/64 = 3*300/32 → T1 = (3*300*64)/(2*32) = 900 K.