Unit of Stefan’s constant is: (2002)
From Stefan's law, $E = \sigma T^4$. The unit of emissive power $E$ is $\text{J}/(\text{s m}^2)$ or $\text{W}/\text{m}^2$. Therefore, the unit of $\sigma$ is $\text{W}/(\text{m}^2 \text{K}^4)$.
Two cylinders A and B of equal capacity are connected to each other via a stop cock. A contains an ideal gas at standard temperature and pressure. B is completely evacuated. The entire system is thermally insulated. The stop cock is suddenly opened. The process is: (2020)
As the entire system is thermally insulated, no heat exchange occurs with the surroundings ($Q = 0$). Therefore, the free expansion process is adiabatic.
In which of the following processes, heat is neither absorbed nor released by a system? (2019)
In an adiabatic process, the system is insulated from its surroundings, meaning the net heat exchange ($Q$) is zero.
A sample of $0.1 \text{ g}$ of water at $100^{\circ}\text{C}$ and normal pressure ($1.013 \times 10^{5} \text{ Nm}^{-2}$) requires $54 \text{ cal}$ of heat energy to convert to steam at $100^{\circ}\text{C}$. If the volume of the steam produced is $167.1 \text{ cc}$, the change in internal energy of the sample, is: (2018)
Heat supplied $Q = 54 \text{ cal} = 54 \times 4.18 \text{ J} = 225.72 \text{ J}$. Work done $W = P\Delta V = 1.013 \times 10^{5} \times (167.1 - 0.1) \times 10^{-6} \approx 16.92 \text{ J}$. By first law of thermodynamics, $\Delta U = Q - W = 225.72 - 16.92 = 208.8 \text{ J}$, which is closest to $208.7 \text{ J}$.
11. An ideal gas is compressed to half its initial volume by means of several processes. Which of the process results in the maximum work done on the gas? (2015 Re)
Work done on the gas is maximum in the adiabatic process as the area under the $P-V$ curve is maximum.
13. A monoatomic gas at a pressure $P$, having a volume $V$ expands isothermally to a volume $2V$ and then adiabatically to a volume $16V$. The final pressure of the gas is (take $\gamma = 5/3$): (2014)
For isothermal process $P_1V_1 = P_2V_2 \Rightarrow P \times V = P_2 \times 2V \Rightarrow P_2 = P/2$. For adiabatic process $P_2V_2^{\gamma} = P_3V_3^{\gamma} \Rightarrow (P/2)(2V)^{5/3} = P_3(16V)^{5/3} \Rightarrow P_3 = P/64$.
14. During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its temperature. The ratio of $\frac{C_p}{C_v}$ for the gas is: (2013)
$P \propto T^3$ or $P T^{-3} = \text{constant}$. We know that for an adiabatic process $P^{1-\gamma} T^{\gamma} = \text{constant}$ or $P T^{\frac{\gamma}{1-\gamma}} = \text{constant}$. Comparing the powers of $T$, $\frac{\gamma}{1-\gamma} = -3 \Rightarrow \gamma = -3 + 3\gamma \Rightarrow 2\gamma = 3 \Rightarrow \gamma = 3/2$.
When volume changes from $V$ to $2V$ at constant pressure $P$, then the change in internal energy will be: (1998)
Work done $\Delta W = P(2V - V) = PV$. Heat supplied $\Delta Q = n C_p \Delta T = \frac{\gamma PV}{\gamma-1}$. Thus, change in internal energy $\Delta U = \Delta Q - \Delta W = \frac{PV}{\gamma-1}$.
A gas of volume changes $2\text{ litre}$ to $10\text{ litre}$ at constant temperature $300\text{ K}$, then the change in internal energy will be: (1998)
Since the process takes place at a constant temperature (isothermal), the internal energy of an ideal gas depends only on temperature, so the change in internal energy is zero.
A sample of gas expands from volume $V_1$ to $V_2$. The amount of work done by the gas is greatest, when the expansion is: (1997)
On a $P-V$ diagram, the work done is represented by the area under the curve. For the same expansion volume, isobaric expansion maintains the highest pressure throughout, resulting in the maximum area and work done.