Reason (R): On sudden expansion, no heat is supplied to system and hence gas does work at the expense of its internal energy.
Concept: Adiabatic expansion and First Law of Thermodynamics. Sudden expansion is a rapid process, approximated as adiabatic (( Q = 0 )). The gas does work ( W > 0 ). By \( \Delta U = Q - W ), ( \Delta U \) becomes negative, leading to a decrease in internal energy and thus cooling. Both (A) and (R) are true, and (R) explains (A).
Assertion (A): Total entropy change in one cycle of carnot engine is zero.
Reason (R): Entropy is a state function.
For any reversible cyclic process like a Carnot cycle, the net change in entropy of the working substance is zero. This is because entropy is a state function, meaning its value depends only on the state of the system, not the path taken. Hence, both A and R are true, and R correctly explains A.
Assertion (A): The efficiency of a carnot cycle depends on the nature of the gas used.
Reason (R): Adiabatic process is a part of carnot cycle and work done in adiabatic process does not depend on nature of gas.
The efficiency of a Carnot engine \(\eta = 1 - \frac{T_c}{T_h}\) depends only on the temperatures of the hot and cold reservoirs, not the nature of the working gas. Work done in an adiabatic process \(W = \frac{nR(T_1 - T_2)}{1 - \gamma}\) depends on \(gamma\) (ratio of specific heats), which is specific to the nature of the gas. Therefore, both Assertion (A) and Reason (R) are false.
Assertion (A): It is not possible for a system, unaided by an external agency to transfer heat from a body at lower temperature to another body a higher temperature.
Reason (R): According to Clausius statement “No process is possible whose sole result is the transfer of heat from a cooled object to a hotter object”.
Assertion (A) is the practical implication of the Clausius statement of the second law of thermodynamics: heat does not spontaneously flow from cold to hot. Reason (R) provides the exact wording of the Clausius statement. Thus, both A and R are true, and R is the correct explanation for A.
Assertion (A): In adiabatic compression, the temperature of system gets decreased.
Reason (R): Adiabatic compression is a slow process.
In adiabatic compression, work is done on the gas \(W =0\), leading to an increase in temperature. Thus (A) is false. Adiabatic processes are typically rapid to prevent heat exchange. Thus (R) is false. Both (A) and (R) are false.
Assertion (A): All processes in which P and V are proportional, take place at constant temperature.
Reason (R): Work done in a thermodynamical process is path independent.
If \(P propto V\) (i.e., \(P=kV\)), then for an ideal gas \(PV=nRT\) implies \(kV^2=nRT\), so \(T \propto V^2\). Thus, temperature is not constant. (A) is false. Work done \(W = \int PdV\) depends on the path taken on a \(P-V\) diagram, so it is path dependent. (R) is false. Both (A) and (R) are false.
Assertion (A): During adiabatic expansion of an ideal gas, temperature falls but entropy remains constant.
Reason (R): During adiabatic expansion, work is done by the gas using a part of internal energy and no heat exchange takes place the system and the surrounding.
During adiabatic expansion, \(Q=0\). The gas does work by using its internal energy, causing temperature to fall. For a reversible adiabatic process, entropy \(\Delta S=0\). (A) is true. (R) correctly explains the energy changes (no heat exchange, internal energy conversion to work) that lead to temperature drop and, for reversible processes, constant entropy. Both are true and (R) is the correct explanation of (A).
Assertion (A): Air quickly leaking out of a balloon becomes cooler.
Reason (R): The leaking air undergoes adiabatic expansion.
Air quickly leaking out of a balloon undergoes rapid expansion. This is an adiabatic process where the gas does work, leading to a decrease in internal energy and thus temperature.
Both (A) and (R) are true and (R) is the correct explanation of (A).
Assertion (A): If heat is supplied to an ideal gas in an isothermal process, the internal energy of the gas increases.
Reason (R): When an ideal gas expands adiabatically, it does positive work and its internal energy increases.
For an ideal gas in an isothermal process, temperature is constant, so internal energy \(Delta U = 0\). Thus (A) is false. In adiabatic expansion, work \(W > 0\) is done by the gas and heat \(Q = 0\), so \(Delta U = -W < 0\), meaning internal energy decreases. Thus (R) is false. Both (A) and (R) are false.
Assertion (A): In adiabatic expansion of monoatomic ideal gas, if volume increases by 12%, then pressure decreases by 20%.
Reason (R): In adiabatic process \(PV^{5/3} = \text{constant}\).
For a monoatomic ideal gas, \(\gamma = 5/3\), so \(PV^{5/3} = \text{constant}\). (R) is true. If volume increases by 12%, \(V_2 = V_1(1+0.12)\). Using \(P_1V_1^{\gamma} = P_2V_2^{\gamma}\), we get \(P_2 = P_1(1+0.12)^{-5/3}\). Using approximation \((1+x)^n approx 1+nx\) for small \(x\), \(P_2 \approx P_1(1 - (5/3)(0.12)) = P_1(1-0.20) = 0.8P_1\). Thus, pressure decreases by 20%. (A) is true. (R) correctly explains (A).