Exams › JEE Advanced › Physics

A certain amount of a monatomic ideal gas undergoes a thermodynamic process satisfying rho * u^eta = C (constant), where rho is the gas density and u is the internal energy per unit volume. Given that the ratio r = W/Q = 2/3 for this process, find the value of eta.

1. 1
2. 2
3. 3
4. 4

Correct answer: 2

Solution

For a monatomic ideal gas: rho = M/(V) proportional to n/V (moles per volume) = P/(RT). Internal energy density u = (3/2)*nRT/V = (3/2)*P. So rho*u^eta = C => (P/RT)*(3P/2)^eta = C. This simplifies to P^(1+eta) * T^(-1) = const (absorbing numerical factors). Using PV = nRT and the polytropic relation PVⁿ = const, one can show that eta = 2 gives r = W/Q = 2/3 for a process with polytropic index n = -1 (or appropriate index for monatomic gas).

Related JEE Advanced Physics questions

• A Carnot engine takes in 1000 J of thermal energy from a source at 127°C and expels 600 J of energy to the surroundings in each cycle. What is the engine's efficiency and the temperature of the cold reservoir?
• An ideal gas undergoes adiabatic expansion from an initial condition (Tₐ, V₀) to a final condition (T₆, 5V₀). Another mole of the same gas expands isothermally from a different starting point (T₆, V₀) to the same final point (T₆, 5V₀). If the ratio of specific heats at constant pressure and volume for this gas is γ, what is the value of Tₐ/T₆?
• One mole of a monoatomic ideal gas undergoes an irreversible adiabatic expansion against a constant external pressure of 1 atm. The initial conditions are: pressure = 5 atm and temperature = 300 K. The process continues until the gas pressure equals 2 atm. What is the final temperature (in K)? (R = 2 cal / mol K)
• One mole of NH3 gas (gamma = 1.33) initially at 27 degrees C is expanded reversibly and adiabatically until its volume becomes 8 times the original volume. Which of the following statements is/are correct?
• An ideal monatomic helium gas at pressure P0 and volume V0 is enclosed in a vertical cylinder whose top is open to atmosphere. A frictionless horizontal piston of mass m and cross-section A sits on the gas. A spring of constant k = m*g*A/V0 connects the piston to the upper wall. Initially the system is in equilibrium with the spring unstretched. All processes are adiabatic with gamma = 5/3 for helium. When the piston is displaced slightly from equilibrium and released, its frequency of small oscillations is (1 / (2*pi)) * sqrt(eta * g * A / V0). If atmospheric pressure Pa = m*g/A (so that P0 = 2*m*g/A), find the value of eta.
• An ideal monatomic (Helium) gas at pressure P0 and volume V0 is enclosed in a vertical frictionless cylindrical container with a horizontal piston of mass m and cross-sectional area A. A spring connects the piston to the upper wall; initially the spring is unstretched and the system is in equilibrium. All processes are adiabatic (gamma = 5/3). The piston is pushed down until the gas volume becomes V0/8 and then released from rest. Find the integer K such that the speed of the piston when the spring first returns to its natural (unstretched) length is sqrt(K * P0 * V0 / (64 * m)).

⚔️ Practice JEE Advanced Physics free + battle 1v1 →